Here I explore another connection between analysis and geometry by means of spectral functions. In some sense, the eigenvalues of an operator know about the geometry of the underlying space.

1. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Spectral functions in the presence of background
potentials
Pedro Morales-Almaz´n
a
Department of Mathematics
Baylor University
pedro morales@baylor.edu
April, 15th 2012
Pedro Morales-Almaz´n
a Math Department
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2. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Outline
1 Motivation
Pedro Morales-Almaz´n
a Math Department
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3. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Outline
1 Motivation
2 Heat Kernel
Pedro Morales-Almaz´n
a Math Department
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4. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Outline
1 Motivation
2 Heat Kernel
3 Zeta Function
Pedro Morales-Almaz´n
a Math Department
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5. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Outline
1 Motivation
2 Heat Kernel
3 Zeta Function
4 The Problem
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6. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Outline
1 Motivation
2 Heat Kernel
3 Zeta Function
4 The Problem
5 Zeta Revisited
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7. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Outline
1 Motivation
2 Heat Kernel
3 Zeta Function
4 The Problem
5 Zeta Revisited
6 Residues
Pedro Morales-Almaz´n
a Math Department
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8. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Outline
1 Motivation
2 Heat Kernel
3 Zeta Function
4 The Problem
5 Zeta Revisited
6 Residues
7 Applications
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9. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Geometry-Analysis Intertwine
(x 2 + y 2 − x)2 = x 2 + y 2
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10. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Geometry-Analysis Intertwine
(x 2 + y 2 − x)2 = x 2 + y 2
Cardioid
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11. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Geometry-Analysis Intertwine
(x 2 + y 2 − x)2 = x 2 + y 2
Cardioid
Torricelli’s Trumpet
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12. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Geometry-Analysis Intertwine
(x 2 + y 2 − x)2 = x 2 + y 2
Cardioid
Torricelli’s Trumpet r = 1/z, z ≥ 1
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a Math Department
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13. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
• Reidemeinster Torsion - Analytic Torsion
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14. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
• Reidemeinster Torsion - Analytic Torsion
• Gauss Bonet
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15. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
• Reidemeinster Torsion - Analytic Torsion
• Gauss Bonet
• Atiyah-Singer Index
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16. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
• Reidemeinster Torsion - Analytic Torsion
• Gauss Bonet
• Atiyah-Singer Index
• Algebraic Geometry
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17. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
• Reidemeinster Torsion - Analytic Torsion
• Gauss Bonet
• Atiyah-Singer Index
• Algebraic Geometry
Strong Connection between Geometry and Analysis
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18. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Spectral functions: Heat Kernel
The heat equation on a manifold M provides a way to analyze
heat flow given an initial heat distribution
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19. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Spectral functions: Heat Kernel
The heat equation on a manifold M provides a way to analyze
heat flow given an initial heat distribution
−∆M f (x, t) = ∂t f (x, t) f (x, 0) = g (x)
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a Math Department
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20. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Spectral functions: Heat Kernel
The heat equation on a manifold M provides a way to analyze
heat flow given an initial heat distribution
−∆M f (x, t) = ∂t f (x, t) f (x, 0) = g (x)
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21. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Heat kernel for PDO
Provides a way of solving the above PDE regardless of g (x)
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22. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Heat kernel for PDO
Provides a way of solving the above PDE regardless of g (x)
H formal inverse of P
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23. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Heat kernel for PDO
Provides a way of solving the above PDE regardless of g (x)
H formal inverse of P
f (x, t) = dy H(t, x, y )g (y )
M
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24. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
P is usually taken to be a Laplace-type operator
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25. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
P is usually taken to be a Laplace-type operator
P = −g ij E
i
E
j +V
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26. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
P is usually taken to be a Laplace-type operator
P = −g ij E
i
E
j +V
g ij is the metric on M, E is a vector bundle over M, E is a
connection on E and V ∈ End(E ).
Pedro Morales-Almaz´n
a Math Department
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27. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
P is usually taken to be a Laplace-type operator
P = −g ij E
i
E
j +V
g ij is the metric on M, E is a vector bundle over M, E is a
connection on E and V ∈ End(E ).
• g provides the geometry of M
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a Math Department
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28. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
P is usually taken to be a Laplace-type operator
P = −g ij E
i
E
j +V
g ij is the metric on M, E is a vector bundle over M, E is a
connection on E and V ∈ End(E ).
• g provides the geometry of M
• V affects the geometry!
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a Math Department
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29. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
H has the asymptotic expansion for small t
dx H(t, x, x) ∼ ak t k
M k=0,1/2,1,...
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30. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
The heat kernel coefficients give geometric invariants of the
manifold
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a Math Department
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31. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
The heat kernel coefficients give geometric invariants of the
manifold
• a0 gives the volume of the manifold
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a Math Department
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32. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
The heat kernel coefficients give geometric invariants of the
manifold
• a0 gives the volume of the manifold
• a1/2 gives the volume of the boundary
Pedro Morales-Almaz´n
a Math Department
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33. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
The heat kernel coefficients give geometric invariants of the
manifold
• a0 gives the volume of the manifold
• a1/2 gives the volume of the boundary
• a1 contains information about the curvature of the manifold
Pedro Morales-Almaz´n
a Math Department
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34. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
The heat kernel coefficients give geometric invariants of the
manifold
• a0 gives the volume of the manifold
• a1/2 gives the volume of the boundary
• a1 contains information about the curvature of the manifold
• ak contains curvature terms and their derivatives
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a Math Department
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35. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Zeta function
There is an analytic connection between the heat kernel
coefficients and the spectral zeta function
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36. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Zeta function
There is an analytic connection between the heat kernel
coefficients and the spectral zeta function
ad/2−s
Res ζP (s) =
(4π)d/2 Γ(s)
for s = d/2, (d − 1)/2, . . . , 1/2 and s = −(2l + 1)/2, for l ∈ N.
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37. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Special values at non-positive integers
Zero
ad/2
ζP (0) = − dim ker(P)
(4π)d/2
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38. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Special values at non-positive integers
Zero
ad/2
ζP (0) = − dim ker(P)
(4π)d/2
Negative integers
(−1)n n!ad/2+n
ζP (−n) =
(4π)d/2
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39. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Spectral functions: Zeta function
∞
ζP (s) = λ−s
n
n=1
where λn are the eigenvalues of P counting multiplicities.
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40. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Spectral functions: Zeta function
∞
ζP (s) = λ−s
n
n=1
where λn are the eigenvalues of P counting multiplicities.
Only defined for (s) > d/2.
Pedro Morales-Almaz´n
a Math Department
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41. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Spectral functions: Zeta function
∞
ζP (s) = λ−s
n
n=1
where λn are the eigenvalues of P counting multiplicities.
Only defined for (s) > d/2.
An analytic continuation is needed!
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a Math Department
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42. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Eigenvalue Problem
Consider an annulus of radii 0 < a < b with a smooth potential
depending only on the radius
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43. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Eigenvalue Problem
Consider an annulus of radii 0 < a < b with a smooth potential
depending only on the radius
P = −∆ + V (r ) Dirichlet BC
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44. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Let λn be the eigenvalues of P
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45. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Let λn be the eigenvalues of P
∞
ζP (s) = λ−2s
n
n=1
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46. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Let λn be the eigenvalues of P
∞
ζP (s) = λ−2s
n
n=1
Don’t have explicit knowledge of the spectrum!
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a Math Department
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47. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Let λn be the eigenvalues of P
∞
ζP (s) = λ−2s
n
n=1
Don’t have explicit knowledge of the spectrum!
Indirect definition is needed!!
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a Math Department
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48. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Integral Representation
∞
sin(πs) ∂
ζP (s) = dλ λ−2s log R0 (b; ıλ)
π 0 ∂λ
∞ ∞
2 sin(πs) ∂
dλ λ−2s log Rk (b; ıλ)
π ∂λ
k=1 0
(s) is big enough, and Rk (b; ıλ) is BC for the EP
Pedro Morales-Almaz´n
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49. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Remark Important points are not included in the convergence
region!
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50. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Remark Important points are not included in the convergence
region!
Regularization is needed !
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51. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Remark Important points are not included in the convergence
region!
Regularization is needed !
Problem: Bad behavior coming from ∞
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52. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Remark Important points are not included in the convergence
region!
Regularization is needed !
Problem: Bad behavior coming from ∞
Solution: Get rid of the ∞
Pedro Morales-Almaz´n
a Math Department
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53. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Remark Important points are not included in the convergence
region!
Regularization is needed !
Problem: Bad behavior coming from ∞
Solution: Get rid of the ∞
We use WKB to find the asymptotic terms of Rk (b; ıλ)
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54. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
WKB Asymptotics
N−2 b
log Rk (b; ıλ) = log A + +
dtSik (t)ξ −i + O(ξ −N+1 )
i=−1 a
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55. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Analytic Continuation
The zeta function can then be written as
ζP (s) =
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56. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Analytic Continuation
The zeta function can then be written as
∞
ζP (s) =
0
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57. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Analytic Continuation
The zeta function can then be written as
∞
ζP (s) = ♣
0
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58. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Analytic Continuation
The zeta function can then be written as
∞
ζP (s) = ♣ − asymptotic terms
0
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59. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Analytic Continuation
The zeta function can then be written as
∞
ζP (s) = ♣ − asymptotic terms
0
∞
+ asymptotic terms
0
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60. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Analytic Continuation
The zeta function can then be written as
∞
ζP (s) = ♣ − asymptotic terms
0
∞
+ asymptotic terms
0
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61. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
N−2
ζP (s) = Z (s) + Ai (s)
i=−1
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62. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
N−2
ζP (s) = Z (s) + Ai (s)
i=−1
Z (s) is the finite part
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63. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
N−2
ζP (s) = Z (s) + Ai (s)
i=−1
Z (s) is the finite part
Ai (s) is the contribution of ξ −i
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64. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
N−2
ζP (s) = Z (s) + Ai (s)
i=−1
Z (s) is the finite part
Ai (s) is the contribution of ξ −i
ζP (s) is now defined form (s) > d/2 − N/2
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65. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Residues
Residue at s = 1
1 2
Res ζ(s)|s=1 = b − a2 ∝ Vol(M)
4
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66. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Residues
Residue at s = 1
1 2
Res ζ(s)|s=1 = b − a2 ∝ Vol(M)
4
Residue at s = 1/2
1
Res ζ(s)|s=1/2 = − (a + b) ∝ Vol(∂M)
4
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67. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Value at s = 0
b
1
ζ(0) = − rdr V (r )
2 a
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68. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Value at s = 0
b
1
ζ(0) = − rdr V (r )
2 a
Residue s = −1/2
1
Res ζ(s)|s=−1/2 = − (bV (b) + aV (a))
8
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69. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
• All heat kernel coefficients can be obtain with this procedure!
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a Math Department
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70. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
• All heat kernel coefficients can be obtain with this procedure!
• Geometric information encoded in the eigenvalues
Pedro Morales-Almaz´n
a Math Department
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71. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
• All heat kernel coefficients can be obtain with this procedure!
• Geometric information encoded in the eigenvalues
• Big eigenvalue behavior has the information
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72. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Applications
• Casimir effect
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73. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Applications
• Casimir effect
• One-loop effective action
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74. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Applications
• Casimir effect
• One-loop effective action
• Geometric invariants
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a Math Department
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75. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications
Questions?
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a Math Department
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