Upload File

spectral functions, the geometric power of eigenvalues,

  • Home
  • Education
  • Spectral Functions, The Geometric Power of Eigenvalues,
138
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications Spectral Functions, The Geometric Power of Eigenvalues, Pedro Fernando Morales Department of Mathematics Baylor University pedro [email protected] Athens, Ohio, 10/18/2012 Pedro Fernando Morales Math Department Spectral Functions

Upload: pedro-morales

Post on 04-Jul-2015

368 views

Category:

Education


4 download

Report

TRANSCRIPT

Page 1: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral FunctionsThe Geometric Power of Eigenvalues

Pedro Fernando Morales

Department of MathematicsBaylor University

pedro moralesbayloredu

Athens Ohio 10182012

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 2: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 3: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 4: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 5: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 6: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 7: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Outline

1 Introduction

2 Laplace-type Operators

3 Heat kernel

4 Zeta function

5 Regularization

6 Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 8: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 9: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 10: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 11: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 12: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 13: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 14: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 15: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 16: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 17: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 18: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 19: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ

(eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 20: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Eigenvalues (a little more formal)

bull Point spectrum of an operator

bull P minus λI not bounded below (not injective)

bull Pφ = λφ (eigenvalue equation)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 21: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 22: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 23: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 24: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 25: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Philosophical question

What is an eigenvalue

bull A way to linearize a problem

bull Measures the distortion of a system

bull Decomposes an object into simpler pieces

bull Determines the resolution of a method

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 26: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 27: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 28: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 29: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Other names and similar ideas

bull Fourier expansion

bull Characters of representations

bull Decomposition into irreducibles

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 30: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 31: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 32: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 33: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 34: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 35: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Differential Operators

bull M a compact manifold

bull E a vector bundle over M

bull P Γ(E )rarr Γ(E ) a differential operator over M

bull With boundary conditions if partM 6= empty

Eigenvalue Equation

Pφ = λφ

where φ isin Γ(E )

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 36: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

(Vibrating membrane)

Pedro Fernando Morales Math Department

Spectral Functions

membranewmv
Media File (videox-ms-wmv)

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 37: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 38: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 39: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 40: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifold

E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 41: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M

V isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 42: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )

nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 43: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E

g metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 44: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 45: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Generalized Laplace equation

M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M

Laplace-type

P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as

P = minusg ijnablaEi nablaE

j + V

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 46: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric

(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 47: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)

Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 48: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues

Bounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 49: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below

Tend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 50: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties Laplace-type Operators

Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 51: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 52: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 53: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 54: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 55: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 56: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Recall

Heat Equation

∆u = ut

for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution

we use the heat kernel to find the heat distribution at any time

bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0

bull limtrarr0

K (t x y) = δ(x minus y)

bull K (t x y) = 0 for x or y in partD

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 57: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 58: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Solving the heat equation

(Tf )(t x) =

intDK (t x y)f (y)dy

solves the heat equation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 59: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 60: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 61: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 62: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat kernel for a Laplace-type operator

Likewise for a Laplace-type operator

Heat Kernel

bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)

bull limtrarr0

intMK (t x y)f (y) = f (x)forallf isin L2(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 63: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 64: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 65: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 66: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Relation with eigenvalues

K (t x y) = etP

=sumλ

eminustλφλ(x)φλ(y)

where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 67: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 68: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 69: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 70: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Heat Kernel Asymptotic Expansion

For small values of t

Asymptotic Expansion

Kt(t x x) simsum

k=0121

bk(x)tkminusd2

Heat kernel coefficients

ak =

intMbk(x)dx

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 71: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 72: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 73: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 74: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 75: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Properties

bull Provide geometric information about the manifold

bull a0 is the volume of M

bull a12 is the volume of the boundary partM

bull ak has curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 76: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 77: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)

Recall K (t) =sum

λ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 78: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 79: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 80: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 81: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 82: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Spectral Functions

The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =

sumλ eminustλ

Zeta function

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 83: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 84: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 85: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta function

Problem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 86: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem

only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 87: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2

All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 88: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Zeta function

Zeta function

The zeta function associated with the operator P is defined by

ζP(s) =sumλ

λminuss

eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 89: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 90: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expression

Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 91: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence

Rather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 92: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Regularization

Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 93: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 94: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot =

minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 95: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 96: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn

=1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 97: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 98: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n

=1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 99: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 100: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1

We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 101: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Example

1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1

Special case ofinfinsumn=0

rn =1

1minus r

infinsumn=0

2n =1

1minus 2= minus1

Convergent only for |r | lt 1We just made an analytic continuation

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 102: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 103: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 104: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Analytic continuation

ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer

Residues

Res ζP (s) =ad2minussΓ(s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 105: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 106: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 107: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)

ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 108: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)

Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 109: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero

No curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 110: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 111: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Riemann Zeta

bull Only one pole (simple) at s = 1

bull Res ζR(1) = 1

a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 112: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifold

d = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 113: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1

length=radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 114: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 115: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 116: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 117: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditions

eigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 118: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 119: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Manifoldd = 1length=

radicπ

Operator

minus d2

dx2φ = λφ

Dirichlet boundary conditionseigenvalues n2 n isin N

ζP(s) = ζR(2s)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 120: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behavior

The geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 121: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 122: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Meromorphic structure

Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues

Weylrsquos law

Let N(λ) be the number of eigenvalues less than λ then

N(λ) sim 1

(4π)d2Γ(d2)Vol(M)λd2

where d = dim(M)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 123: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized Trace

Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 124: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional Determinant

Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 125: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimension

PhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 126: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics

Casimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 127: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 128: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)

Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 129: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Applications

Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian

ECas =infinsumn=1

radicλn

One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients

ad2minusz = Γ(z) Res ζP(z)

for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 130: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Casimir Effect

Is a quantum field effect that arises when considering vacuumfluctuations

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 131: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 132: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 133: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Why is so important

bull Believed to explain the stability of an electron

bull Very sensitive to the geometry of the space (Quantum andComsmological implications)

bull Provides a better understanding of the zero-point energy

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 134: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 135: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 136: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 137: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Conclusions

bull Eigenvalues know a lot of the geometry of a system

bull Describe the dynamics in a geometric object

bull New information appear when regularizing expressions

bull Useful to describe high energy systems (quantum physics)

Pedro Fernando Morales Math Department

Spectral Functions

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications
Page 138: Spectral Functions, The Geometric Power of Eigenvalues,

Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications

Questions

Thank you

Pedro Fernando Morales Math Department

Spectral Functions

  • Introduction
  • Laplace-type Operators
  • Heat kernel
  • Zeta function
  • Regularization
  • Applications

Group Theory in Quantum Mechanics Lecture 5 Spectral … · 2017-01-30 · Group Theory in Quantum Mechanics Lecture 5 (1.31.17) Spectral Decomposition with Repeated Eigenvalues (Quantum

Salient spectral geometric features for shape matching and ...jinghua/publication/jh2_tvc_09.pdf · Salient spectral geometric features for shape matching and retrieval of geometry

Spectral Methods for Mesh Processing and Analysishaoz/pubs/zhang_eg07star... · 2007-06-26 · Spectral methods for mesh processing and analysis rely on the eigenvalues, eigenvectors,

The spectral radius of connected graphs with the ... · The spectral radius of A: The largest absolute value of the eigenvalues of A. For A(G);Q(G), the corresponding spectral radius

Eigenvalues and geometric representations of graphs L á szl ó Lov á sz Microsoft Research

Spectral theory of foliation geometric operators

ON THE NOTION OF SPECTRAL DECOMPOSITION IN LOCAL GEOMETRIC LANGLANDS · 2018. 9. 7. · ON THE NOTION OF SPECTRAL DECOMPOSITION IN LOCAL GEOMETRIC LANGLANDS 3 More5 intriguing is

Spectral Coarsening of Geometric Operatorshsuehtil/pdf/specCoarse_low.pdf · Spectral Coarsening of Geometric Operators HSUEH-TI DEREK LIU,University of Toronto, Canada ALEC JACOBSON,University

Eigenvalues and the Laplacian of a graph - UCSD …fan/research/cb/ch1.pdf · CHAPTER 1 Eigenvalues and the Laplacian of a graph 1.1. Introduction Spectral graph theory has a long

Spectra of Adjacency and Laplaciansco/Spectral/Lesson1Spectra.pdf · Spectra of Laplacian Matrices Properties: Possitive definiteness (all eigenvalues >= 0). The smallest eigenvalue

Some geometric bounds on eigenvalues of elliptic PDEs Evans Harrell Georgia Tech harrell Spectral Theory Network 25 July, 2004 Cardiff

A geometric approach to spectral subtraction · 2013-08-21 · A new geometric approach to spectral subtraction is proposed in the present paper that addresses these shortcomings

Geometric analysis and spectral synthesismilne.ruc.dk/~booss/BBB_pub_2010ff/2018_04_25_IMFUFAseminar.… · Geometric analysis Decomposition of space/manifolds/surfaces & recomposing

SPECTRAL CLUSTERING FROM A GEOMETRIC VIEWPOINTmath.gmu.edu/~tsauer/pre/Clustering.pdf · 2015-05-14 · SPECTRAL CLUSTERING FROM A GEOMETRIC VIEWPOINT TYRUS BERRY AND TIMOTHY SAUER

Geometric Approach to Spectral Substraction

Scientific Computing Seminar AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues * C. BekasY. Saad Comp. Science & Engineering Dept

Etudes in Spectral Theory [10pt] - Victor Ivriiweyl.math.toronto.edu/victor_ivrii_Publications/... · 2019-06-17 · Distribution of Eigenvalues Variational Methods Theproofof(1)byWeylwasbasedonthisformulaforboxes,integral

Bounds on sums of graph eigenvalues · smallest eigenvalues, and some other spectral properties, such as determinants, satisfy simple inequalities and provide information about the

Sharp geometric bounds for eigenvalues of Schrödinger operators Evans Harrell Georgia Tech harrell OTQP Praha 12 září 2006

Spectral Graph TheorySpectral Graph Theory Social and Technological Networks Rik Sarkar University of Edinburgh, 2018. Spectral methods • Understanding a graph using eigenvalues

SGT Toolbox A Playground for Spectral Graph Theory · 2019-08-18 · Spectral Graph Theory 1.1 Motivation Spectral graph theory (SGT) deals with the eigenvalues and eigenvectors of

Eigenvalues, Expanders and Gaps between Primes2017/02/05  · The importance of spectral graph theory is also demonstrated by the large number of books in which eigenvalues are studied

An Experimental Study into Spectral and Geometric ...reports-archive.adm.cs.cmu.edu/anon/2015/CMU-CS-15-149.pdfAn Experimental Study into Spectral and Geometric Approaches to Data

Graph Partitioning II: Spectral Methods - Carlos …Spectral graph theory The study of the eigenvalues and eigenvectors of a graph matrix – Adjacency matrix – Laplacian matrix

1 Geometric Representations of Graphs · 8 CHAPTER 1. EIGENVALUES OF GRAPHS The matrices PG and NG have the same eigenvalues, and so all eigenvalues of PG are real. Example 1.1.1

A geometric approach to spectral subtraction," Speech

Some geometric bounds on eigenvalues of elliptic PDEs

FUSE: Full Spectral Clustering - kdd.org · (a) SYN1 (b) eigenvalues (c) ZP (d) FUSE Figure 1: Clustering results of ZP and FUSE on our SYN1 data ((b) gives the top 10 eigenvalues

Geometric analysis and spectral synthesisthiele.ruc.dk/~booss/BBB_pub_2010ff/2018_04_25_IMFUFAseminar.p… · Geometric analysis Decomposition of space/manifolds/surfaces & recomposing

Spectral theory of foliation geometric operatorshurder/papers/25manuscript.pdfSpectral theory of foliation geometric operators Steven HURDER∗ Department of Mathematics (m/c 249)

GEOMETRIC ASPECTS OF MULTIPARAMETER SPECTRAL THEORYkosir/papers/GEOMST.pdf · GEOMETRIC ASPECTS OF MULTIPARAMETER SPECTRAL THEORY 3 we may always replace A by its completion A0⁄.Since

Lectures on Spectral Graph Theoryfan/cbms.pdfCHAPTER 1 Eigenvalues and the Laplacian of a graph 1.1. Introduction Spectral graph theory has a long history. In the early days, matrix

GEOMETRIC ASPECTS OF OPTIMIZATION AND APPLICATIONS TO SPECTRAL …helper.ipam.ucla.edu/publications/sal2016/sal2016_12604.pdf · 2016-02-16 · GEOMETRIC ASPECTS OF OPTIMIZATION AND

Constrained Clustering by Spectral Kernel Learningmmlab.ie.cuhk.edu.hk/2009/ICCV09_CCSKL.pdf · 2012-10-31 · clustering spectral embedding without using eigenvalues 2. Figure 1

Surgery of quantum graphs: eigenvalues and heat kernels · 2018-03-26 · Spectral inequalities Heat kernels Spectral estimates for quantum graphs Proposition (Nicaise 1987) L 1(G)≥ˇ