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Zeta function for perturbed surfaces of revolution

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Here we explore the Zeta function arising from a small perturbation on a surface of revolution and the effect of this on the functional determinant and in the change of the Casimir energy associated with this configuration.

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Zeta function for perturbed surfaces of revolution

  1. 1. Zeta function for perturbed surfaces of revolution Pedro Morales-Almaz´an Department of Mathematics The University of Texas at Austin pmorales@math.utexas.edu TexAMP 2016 Rice University, October 22, 2016 Pedro Morales-Almaz´an Math Department Perturbed Zeta Functions
  2. 2. Zeta Function P is a differential operator M a d dimensional manifold Spectral Zeta Function ζ(s) = λ∈σ λ=0 λ−s for (s) > d. Pedro Morales-Almaz´an Math Department Perturbed Zeta Functions
  3. 3. Surface of revolution P = ∆ the M surface of revolution y = f (x) > 0, x ∈ [a, b] Pedro Morales-Almaz´an Math Department Perturbed Zeta Functions
  4. 4. Zeta Function Zeta Function ζ(s) = ∞ k=−∞ 1 2πi γk dλ λ−2s d dλ log φk(λ; b) , for (s) > 1 and φk(λ; x) is a solution to the radial ODE with initial conditions φk(λ; a) = 0 , φk(λ; a) = 1 . Pedro Morales-Almaz´an Math Department Perturbed Zeta Functions
  5. 5. 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) Pedro Morales-Almaz´an Math Department Perturbed Zeta Functions
  6. 6. Analytic Continuation WKB asymptotics ζ(s) = sin(πs) π ∞ 0 dλ λ−2s d dλ log F(iλ) − ♣ + sin(πs) π ∞ 0 dλ λ−2s d dλ ♣ for (s) > n(♣). Pedro Morales-Almaz´an Math Department Perturbed Zeta Functions
  7. 7. 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-Almaz´an Math Department Perturbed Zeta Functions
  8. 8. Special Values: Casimir Q: How do we find a well defined quantity? A: Perturbation Pedro Morales-Almaz´an Math Department Perturbed Zeta Functions
  9. 9. Perturbed Surface of Revolution • Perturb the profile function f (x) → f (x) + g(x) • Substitute this into the previous formalism • Calculate the variation due to the perturbation d d ζ(s) =0 Pedro Morales-Almaz´an Math Department Perturbed Zeta Functions
  10. 10. Analytic Continuation WKB asymptotics ζ(s) = sin(πs) π ∞ 0 dλ λ−2s d dλ log F(iλ) − ♣ (Finite) + sin(πs) π ∞ 0 dλ λ−2s d dλ ♣ (Asymptotic) Finite: Depend on f (x) and φk(λ; b) (More complex:???) Asymptotic: Only depend on f (x) (Straightforward: Taylor Series) Pedro Morales-Almaz´an Math Department Perturbed Zeta Functions
  11. 11. Perturbation: Asymptotic terms WKB asymptotics ζ(s) = sin(πs) π ∞ 0 dλ λ−2s d dλ log F(iλ) − ♣ (Finite) + sin(πs) π ∞ 0 dλ λ−2s d dλ ♣ (Asymptotic) • ♣ only depends on f (x), hence doing f (x) → f (x) + g(x) • find terms up to O( 2) Pedro Morales-Almaz´an Math Department Perturbed Zeta Functions
  12. 12. Perturbation: Finite terms WKB asymptotics ζ(s) = sin(πs) π ∞ 0 dλ λ−2s d dλ log F(iλ) − ♣ (Finite) + sin(πs) π ∞ 0 dλ λ−2s d dλ ♣ (Asymptotic) F + f + g f + g − (f + g ) (f + g ) 1 + (f + g )2 F + 1 + f + g 2 λ2 − k2 (f + g)2 F = 0 (O( 2 )) . Pedro Morales-Almaz´an Math Department Perturbed Zeta Functions
  13. 13. Perturbation: Finite terms • F = φ + ˆφ (O( 2)) • ˆφ functional derivative • φ satisfies the original radial equation • ˆφ satisfies a non-homogeneous version of the radial equation ˆφ + f f − (f ) (f ) 1 + (f )2 ˆφ + 1 + f 2 λ2 − k2 f 2 ˆφ = G . • use variation of parameters Pedro Morales-Almaz´an Math Department Perturbed Zeta Functions
  14. 14. Perturbation: Zeta function Zeta function ˜ζ(s) = ζ(s) + ˆζ(s) (O( 2 )) Effect of the perturbation d d ˜ζ(s) =0 = ˆζ(s) Pedro Morales-Almaz´an Math Department Perturbed Zeta Functions
  15. 15. Perturbation: Casimir Energy ∆E = d d ζ∆ (−1/2) =0 = − 1 2π b a dt f (t) (f (t)2 + 1)3/2 g(t) − ζR (−2) π b a dt f (t)f (t) + 2f (t)2 + 2 f (t)3 (f (t)2 + 1)3/2 g(t) + 1 16 b a dt 2f (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 dλ λ d dλ ˆφ0(b; ıλ) φ0(b; ıλ) − 1 π ∞ 1 dλ λ d dλ ˆφ0(b; ıλ) φ0(b; ıλ) − 2 i=−1 λ−i b a dt ∂ ∂ si (t) =0 − 2 π ∞ k=1 k ∞ 0 du u d du ˆφk (b; ıuk) φk (b; ıuk) − 2 i=−1 k−i b a dt ∂ ∂ wi (t) =0 Pedro Morales-Almaz´an Math Department Perturbed Zeta Functions
  16. 16. Perturbation: Cylinder I = (c − δ, c + δ) ⊂ (a, b) , δ > 0 gδ(x, c) = χ(I) exp − (x − c) (x − c)2 − δ2 2 , Pedro Morales-Almaz´an Math Department Perturbed Zeta Functions
  17. 17. Perturbation: Cylinder Gaussian Perturbation Pedro Morales-Almaz´an Math Department Perturbed Zeta Functions
  18. 18. Perturbation: Cylinder Mixed Perturbation Pedro Morales-Almaz´an Math Department Perturbed Zeta Functions
  19. 19. Conclusions • The Casimir doesn’t get affected by perturbations made near the center • The interaction between an edge and a positive (negative) perturbation results in a negative (positive) change of the Casimir Energy • The results agree with the existing calculations for infinite cylinders Pedro Morales-Almaz´an Math Department Perturbed Zeta Functions
  20. 20. References • Thalia D Jeffres, Klaus Kirsten & Tianshi Lu (2012). Zeta function on surfaces of revolution. Journal of Physics A: Mathematical and Theoretical, 45, 345201. • M-A., P. (2016). Casimir energy for perturbed surfaces of revolution. International Journal of Modern Physics A, 31, 1650044. • Fucci, G. & M-A., P. Perturbed zeta functions on warped manifolds, Coming soon! Pedro Morales-Almaz´an Math Department Perturbed Zeta Functions
  21. 21. Questions email: pmorales@math.utexas.edu twitter: @p3d40 Pedro Morales-Almaz´an Math Department Perturbed Zeta Functions

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