zeta function for perturbed surfaces of revolution

Zeta function for perturbed surfaces of revolution

Pedro Morales-Almazan

Department of MathematicsThe University of Texas at Austin

[email protected]

TexAMP 2016Rice University, October 22, 2016

Pedro Morales-Almazan Math Department

Perturbed Zeta Functions

mailto:[email protected]

Zeta Function

P is a differential operatorM a d dimensional manifold

Spectral Zeta Function

ζ(s) =∑λ∈σλ 6=0

λ−s

for <(s) > d .



Surface of revolution

P = ∆ theM surface of revolution y = f (x) > 0, x ∈ [a, b]



Zeta Function

Zeta Function

ζ(s) =∞∑

k=−∞

1

2πi

∫γk

dλλ−2s d

dλlog φk(λ; b) ,

for <(s) > 1 and φk(λ; x) is a solution to the radial ODE withinitial conditions φk(λ; a) = 0 , φ′k(λ; a) = 1 .



Analytic Continuation

Plan

• Extend ζ(s) to the entire complex plane

• Integral representation is good for small λ (converge)

• Integral representation is bad for big λ (divergence)

• Analytic continuation (subtract the behavior for big λ)




WKB asymptotics

ζ(s) =sin(πs)

π

∫ ∞0

dλλ−2s d

dλlog F (iλ)−♣

+sin(πs)

π

∫ ∞0

dλλ−2s d

dλ♣

for <(s) > n(♣).



Special values

Functional Determinant

Formally defined as exp(−ζ ′(0))Well defined since

Res ζ(0) = 0

Casimir Energy

The vacuum energy can be found by limh→0 ζ(−1/2 + h)Not well defined!

Res ζ(−1/2) = − 1

256

(f −1(a)f ′2(a)

(1 + f ′2(a))+

f −1(b)f ′2(b)

(1 + f ′2(b))

)− 1

32

(f ′′(a)

(1 + f ′2(a))2+

f ′′(b)

(1 + f ′2(b))2

)Pedro Morales-Almazan Math Department


Special Values: Casimir

Q: How do we find a well defined quantity? A: Perturbation



Perturbed Surface of Revolution

• Perturb the profile function

f (x) 7→ f (x) + εg(x)

• Substitute this into the previous formalism

• Calculate the variation due to the perturbation

d

dεζ(s)

∣∣∣ε=0




WKB asymptotics

ζ(s) =sin(πs)

π

∫ ∞0

dλλ−2s d

dλlog F (iλ)−♣ (Finite)

+sin(πs)

π

∫ ∞0

dλλ−2s d

dλ♣ (Asymptotic)

Finite: Depend on f (x) and φk(λ; b) (More complex:???)Asymptotic: Only depend on f (x) (Straightforward: Taylor Series)



Perturbation: Asymptotic terms

WKB asymptotics

ζ(s) =sin(πs)

π

∫ ∞0

dλλ−2s d


+sin(πs)

π

∫ ∞0

dλλ−2s d

dλ♣ (Asymptotic)

• ♣ only depends on f (x), hence doing f (x) 7→ f (x) + εg(x)

• find terms up to O(ε2)



Perturbation: Finite terms

WKB asymptotics

ζ(s) =sin(πs)

π

∫ ∞0

dλλ−2s d


+sin(πs)

π

∫ ∞0

dλλ−2s d

dλ♣ (Asymptotic)

F ′′ +

(f ′ + εg ′

f + εg− (f ′ + εg ′) (f ′′ + εg ′′)

1 + (f ′ + εg ′)2

)F ′

+(

1 +(f ′ + εg ′

)2)(

λ2 − k2

(f + εg)2

)F = 0 (O(ε2)) .



Perturbation: Finite terms

• F = φ+ εφ (O(ε2))

• φ functional derivative

• φ satisfies the original radial equation

• φ satisfies a non-homogeneous version of the radial equation

φ′′ +

(f ′

f− (f ′) (f ′′)

1 + (f ′)2

)φ′

+(

1 +(f ′)2)(

λ2 − k2

f 2

)φ = G .

• use variation of parameters



Perturbation: Zeta function

Zeta function

ζ(s) = ζ(s) + εζ(s) (O(ε2))

Effect of the perturbation

d

dεζ(s)

∣∣∣ε=0

= ζ(s)



Perturbation: Casimir Energy

∆E =d

dεζ∆ε(−1/2)

∣∣∣∣ε=0

= − 1

2π

∫ b

a

dtf ′′(t)

(f ′(t)2 + 1)3/2g(t)

−ζ′R(−2)

π

∫ b

a

dtf (t)f ′′(t) + 2f ′(t)2 + 2

f (t)3 (f ′(t)2 + 1)3/2g(t)

+1

16

∫ b

a

dt2f ′(t)3

(f ′(t)2 + 1

)+ f (t)f ′(t)

(5f ′(t)2 − 3

)f ′′(t)

f (t)3 (f ′(t)2 + 1)5 g(t)

− 1

π

∫ 1

0

dλλd

dλ

φ0(b; ıλ)

φ0(b; ıλ)

− 1

π

∫ ∞1

dλλd

dλ

(φ0(b; ıλ)

φ0(b; ıλ)−

2∑i=−1

λ−i

∫ b

a

dt∂

∂εsi (t)

∣∣∣∣ε=0

)

− 2

π

∞∑k=1

k

∫ ∞0

du ud

du

(φk(b; ıuk)

φk(b; ıuk)−

2∑i=−1

k−i

∫ b

a

dt∂

∂εwi (t)

∣∣∣∣ε=0

)



Perturbation: Cylinder

I = (c − δ, c + δ) ⊂ (a, b) , δ > 0

gδ(x , c) = χ(I ) exp

(−(

(x − c)

(x − c)2 − δ2

)2),



Perturbation: Cylinder Gaussian Perturbation



Perturbation: Cylinder Mixed Perturbation



Conclusions

• The Casimir doesn’t get affected by perturbations made nearthe center

• The interaction between an edge and a positive (negative)perturbation results in a negative (positive) change of theCasimir Energy

• The results agree with the existing calculations for infinitecylinders



References

• Thalia D Jeffres, Klaus Kirsten & Tianshi Lu (2012). Zetafunction on surfaces of revolution. Journal of Physics A:Mathematical and Theoretical, 45, 345201.

• M-A., P. (2016). Casimir energy for perturbed surfaces ofrevolution. International Journal of Modern Physics A, 31,1650044.

• Fucci, G. & M-A., P. Perturbed zeta functions on warpedmanifolds, Coming soon!



Questions

email: [email protected]: @p3d40



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zeta function for perturbed surfaces of revolution

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