Localized Perturbations of Localized Perturbations of Integrable BilliardsIntegrable Billiards

Saar RahavSaar Rahav

Technion, Haifa, May 2004Technion, Haifa, May 2004

OutlineOutline

• Motivation: spectral statistics and universality

• What is a perturbation at a point• Two ways for spectral statistics• Connection to star graphs• Periodic orbits and spectral statistics• Dependence on location of perturbation• Conclusions

Spectral statistics of dynamical systems

Dynamical systems exhibit spectral statistics of “random matrices”

• Time reversal symmetry statistics of random real symmetric matrices (GOE)

• No time reversal symmetry random Hermitian matrices (GUE)

•Integrable systems Poissonian statistics (uncorrelated levels)

Spectral statistics do not depend on details: Universality

How to explain this universality?

Some properties of the dynamical system itself must be universal (sum rules).

Examples for different dynamical systems:

Sinai billiard - chaotic

Ray splitting – pseudointegrable, non universal

Šeba billiard – singular billiard

Non universal

The statistical measures

Nearest neighbor spacing distribution ( )P s

Spectral form factor ( , ) ( ) ( ) i tosc oscK E t d d E d E e

Semiclassical analysis of spectral statistics

Trace formulae:

( ) exp ( ) /osc p pp

d E d A iS E Gutzwiller,

Berry and Tabor

Relates the density of states and periodic orbits

For large energies (semiclassical limit) pS

Contributions with random phases in the cosine

The sum is dominated by almost equal actions:p pS S

The semiclassical Form factor

*

,

( ) cos ( ) / 2p pp p p p

p p

S SK t A A t T T

Berry

The diagonal approximation:

2( ) j jj

K t A t T

Take only pairs with the same action i jS S

Berry

Evaluated using the Hannay & Ozorio de Almeida sum rule

GUE

( ) 2 GOE

1 Integrable

K

The correct short time asymptotics !

Validity of RMT Sum rules for periodic orbitsHigher order terms? corrections GOE were calculated by Sieber and Richter, see also Muller, Haake, et. al.

What are singular billiards?

“Physical” point of view:

Integrable Quantum systems, with local perturbation

a L

0r

The diffraction constant ( , ) ( )D E k T q

is proportional to the scattering amplitude and satisfy the optical theorem

2

2

0

1, ,

8D d D

The perturbation can be described be means of scattering theory:

0 00 0 0( , , ) ( , , ) ( , , ) ( , ) ( , , )G r r k G r r k G r r rk D G r k (without the boundary)

Geometrical theory of diffraction, Keller.

“Mathematical” point of view:

The self-adjoint extension of a Hamiltonian 0H r

One can define a family of extensions, with a simple Green function:

0 0( , ; ) , ; ( , ) ( , , ) ( , , )G r r z G r r z T z G r r z G r r z Zorbas

1

0 0 0 0( , ) 1 ( ) ( , , ) ( , , ) ( ) ( , , ) ( , , )i iT z e i z dr G r r z G r r i e i z dr G r r z G r r i

is related to the scattering strength

The new eigenvalues are the poles of ,T z

For closed systems: 1

( ) ( ), , n n

n n

r rG r r z

z E

A quantization condition for new eigenvalues

2 20 02 2

sin 1 1( ) ( ) 0

1 cos 1 1n

n nn nn n n

Er r

E z E E

Why singular billiards?

•Dynamics intermediate between integrable and chaotic

•Important diffraction effects

•Simple system

•What is the spectral statistics?

•New universality class for spectral statistics?

•A new ‘test’ for periodic orbit theory

Two approaches for spectral statistics of singular billiards

1. Periodic orbits

Simple scattering without a boundary

0 0 0 00( , , ) ( , , ) ( , , ) ( , , )G r r k G r r k G r rr k DG r k

The boundary can be added using an integral equation

1 1ˆ ˆ ˆ1 1 1 2 1

1

1 12 ( , ) ( , ) ( , )

n n

n

osc n n n n n n nn

dd ds ds G r r G r r G r r

n dk

In the semiclassical limit,

0

1( , , ) ik r rG r r k e

k r r

The integrals over the boundary are dominated by contributions that perform specular reflections

Rahav, Fishman

Bogomolny, Giraud

The integrals lead to two types of orbits:

Periodic orbits - do not hit the scatterer

Orbits with segments which start and end at r0 – diffracting orbits

jl

leads to a modified trace formula:

1 2

1 2

1 2

( )(1) (2),

,

( ) . .j jp j ik l likl iklosc p j j j

p j j j

d E A e A e A e c c

With (1)A D (2) 2A D

More diffractions: more powers of 1/ kl Higher powers of more segments Non diagonal contributions

2. Ensemble averaging of the quantization condition

20

1

( ) 1N

n n

r

z E

Approximately:

Bogomolny, Gerland, Giraud, Schmit

Properties: •LHS has poles at ‘unperturbed’ energy levels

•LHS monotonically decreasing with z

•Exactly one solution in 1,i iE E

The density of states is:2 2 2 2

0 0 0 02 2

1 1 1 1

( ) ( ) ( ) ( )( ) 1 exp 1

( ) 2 ( )

N N N Nn n n n

n k n kn n n n

r r r rdd E i

E E E E E E E E

Integrable system nE are independent random variables

The distribution of 2

0( )n r is uncorrelated with nE

One can build statistical measures, e.g. ( ) ( )d E d E And average over the unperturbed energies and wavefunction values

A kind of ensemble averageAdvantage – the integrals separate into independent farctors

Results: (simplified)

4

rectangle, periodic BC( )

ln Dirichlet BC

SP S

S S

Level repulsion 0S

1( ) CSP S e

S S Exponential falloff

Intermediate statistics

Connection to star graphs

Quantum graphs: Kottos, Smilansky

Free motion on bonds, boundary conditions on vertices

Star Graphs: Berkolaiko, Bogomolny, Keating

For star graphs, the quantization condition is

1

1tan

N

jj

l k

In the limit of infinite number of bonds with random bond lengths

The spectral statistics of star graphs are those of Seba billiard with 2

0( ) Constn r

Periodic orbit calculation of spectral statistics

Reminder: *

,

( ) cos ( ) / 2p pp p p p

p p

S SK t A A t T T

Where the lengths may be composed of several diffracting segments

What types of contributions may survive?

For the rectangular billiard:

Diagonal contributions:

The periodic orbits contribute ( ) 1K

Diffracting orbits with n segments2 1( )n nK D Sieber

Can one find diffracting orbit with the same length of a periodic orbit?

Yes. A forward diffracting orbit!

A ‘kind’ of diagonal contribution:21

( )4

K D D

Non diagonal contributions:

21 1 1

2k l kt

x l x

,l x kForThe difference in phase is small for 1t

There are many (~k) such contributions

Results:

Scatterer at the center2

4 4 62 3 41 1 1( ) 1 ( )

4 8 2 24

DK D D D O

Typical location of scatterer: 2

4 4 62 3 49 81 25( ) 1

4 128 512 1536

DK D D D O

All form factors start at 1 and exhibit a dip before going back to 1.

Intermediate statistics

Dependence on location:

For the rectangular billiard the spectral statistics depend in a complicated manner on the location of the perturbation:

Complementary explanations:

1. Degeneracies in lengths of diffracting orbits

2. The distribution of values of wavefunctions:

0 0 0( ) sin sinn r mx nya b

Differs if are rational or not0 0,x y

a b

Is such behavior typical?

The Circle billiard:

Angular momentum conservation min2

Lr r

mE

Quantum wave functions are exponentially small for minr r

20

1

( ) 1N

n n

r

z E

So for exponentially small wavefunction the

eigenvalues are almost unchanged

The spectrum of the singular circle billiard can be (approximately) divided into two components:

1. Almost unperturbed spectrum, composed of wavefunctions localized on r>r0.

2. Strongly perturbed spectrum.

How many levels are unperturbed?

1 0 0 0unpertubed

2cos 1

r r rX

R R R

Superposing the two spectra:

The statistics depend on the location of the scatterer.

Partial level repulsion?

ConclusionsConclusions

• The spectral statistics differ from known universality classes – Intermediate statistics

• Strong contribution due to diffraction – non classical

• Statistics depend on location of perturbation – non universal

• However, the statistics of different singular billiards show similarities

• The wavefunctions are not ergodic (Berkolaiko, Keating, Marklof, Winn)

Interesting open problemsInteresting open problems

• Understanding pseudointegrable systems, where the diffraction contributions are non uniform

• Resummation of the series for the form factor• Understanding singularities of form factors• Better understanding of wavefunctions• Dependence on number of scatterers