This document is the outline from a presentation on spectral functions given by Pedro Fernando Morales from the Department of Mathematics at Baylor University. The presentation covers topics related to Laplace-type operators, including their eigenvalues and eigenfunctions, properties of Laplace operators, the heat kernel, and using the heat kernel to solve differential equations like the heat equation. It provides context and definitions for these concepts and notes their relationships.

1. 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 morales@baylor.edu
Athens, Ohio, 10/18/2012
Pedro Fernando Morales Math Department
Spectral Functions

2. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
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Spectral Functions

3. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
Pedro Fernando Morales Math Department
Spectral Functions

4. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
Pedro Fernando Morales Math Department
Spectral Functions

5. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
Pedro Fernando Morales Math Department
Spectral Functions

6. 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
Pedro Fernando Morales Math Department
Spectral Functions

7. 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

8. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions

9. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions

10. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions

11. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions

12. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
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Spectral Functions

13. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
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Spectral Functions

14. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
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Spectral Functions

15. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
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Spectral Functions

16. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
Pedro Fernando Morales Math Department
Spectral Functions

17. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
• Point spectrum of an operator
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Spectral Functions

18. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
• Point spectrum of an operator
• P − λI not bounded below (not injective)
Pedro Fernando Morales Math Department
Spectral Functions

19. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
• Point spectrum of an operator
• P − λI not bounded below (not injective)
• Pφ = λφ
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Spectral Functions

20. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
• Point spectrum of an operator
• P − λI not bounded below (not injective)
• Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
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21. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue?
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22. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue?
• A way to linearize a problem
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23. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue?
• A way to linearize a problem
• Measures the distortion of a system
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Spectral Functions

24. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue?
• A way to linearize a problem
• Measures the distortion of a system
• Decomposes an object into simpler pieces
Pedro Fernando Morales Math Department
Spectral Functions

25. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue?
• A way to linearize a problem
• Measures the distortion of a system
• Decomposes an object into simpler pieces
• Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions

26. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
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Spectral Functions

27. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
• Fourier expansion
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28. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
• Fourier expansion
• Characters of representations
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29. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
• Fourier expansion
• Characters of representations
• Decomposition into irreducibles
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30. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
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31. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
• M a compact manifold
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32. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
• M a compact manifold
• E a vector bundle over M
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33. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
• M a compact manifold
• E a vector bundle over M
• P : Γ(E ) → Γ(E ) a differential operator over M
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34. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
• M a compact manifold
• E a vector bundle over M
• P : Γ(E ) → Γ(E ) a differential operator over M
• With boundary conditions if ∂M = ∅
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35. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
• M a compact manifold
• E a vector bundle over M
• P : Γ(E ) → Γ(E ) a differential operator over M
• With boundary conditions if ∂M = ∅
Eigenvalue Equation
Pφ = λφ,
where φ ∈ Γ(E )
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36. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
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37. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
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38. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
There is a close relation between the shape of the manifold and
the eigenvalues (eigenfunctions) of the Laplacian
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39. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
There is a close relation between the shape of the manifold and
the eigenvalues (eigenfunctions) of the Laplacian
Pedro Fernando Morales Math Department
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40. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifold
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41. 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 M
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42. 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 M
V ∈ End(E )
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43. 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 M
V ∈ End(E )
E a connection on E
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44. 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 M
V ∈ End(E )
E a connection on E
g metric on M
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45. 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 M
V ∈ End(E )
E a connection on E
g metric on M
Laplace-type
P : Γ(E ) → Γ(E ) is a Laplace-type differential operator if P can be
written as
P = −g ij E E + V
i j
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46. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric
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47. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)
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48. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)
Real eigenvalues
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49. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)
Real eigenvalues
Bounded below
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50. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)
Real eigenvalues
Bounded below
Tend to infinity
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51. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall:
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52. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall:
Heat Equation
∆u = ut ,
for a domain D, where u = 0 at ∂D and u|t=0 = f (x) is the initial
heat distribution.
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53. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall:
Heat Equation
∆u = ut ,
for a domain D, where u = 0 at ∂D and u|t=0 = f (x) is the initial
heat distribution.
we use the heat kernel to find the heat distribution at any time:
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54. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall:
Heat Equation
∆u = ut ,
for a domain D, where u = 0 at ∂D and u|t=0 = f (x) is the initial
heat distribution.
we use the heat kernel to find the heat distribution at any time:
• Kt (t, x, y ) = ∆K (t, x, y ) , for x, y ∈ D, t > 0
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55. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall:
Heat Equation
∆u = ut ,
for a domain D, where u = 0 at ∂D and u|t=0 = f (x) is the initial
heat distribution.
we use the heat kernel to find the heat distribution at any time:
• Kt (t, x, y ) = ∆K (t, x, y ) , for x, y ∈ D, t > 0
• lim K (t, x, y ) = δ(x − y )
t→0
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56. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall:
Heat Equation
∆u = ut ,
for a domain D, where u = 0 at ∂D and u|t=0 = f (x) is the initial
heat distribution.
we use the heat kernel to find the heat distribution at any time:
• Kt (t, x, y ) = ∆K (t, x, y ) , for x, y ∈ D, t > 0
• lim K (t, x, y ) = δ(x − y )
t→0
• K (t, x, y ) = 0 for x or y in ∂D
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57. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
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58. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t, x) = K (t, x, y )f (y )dy
D
solves the heat equation.
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59. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
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60. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise, for a Laplace-type operator,
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61. 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
• (∂t − P)K (t, x, y ) = 0, C ∞ (R+ , M, M)
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62. 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
• (∂t − P)K (t, x, y ) = 0, C ∞ (R+ , M, M)
• lim K (t, x, y )f (y ) = f (x), ∀f ∈ L2 (M)
t→0 M
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63. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
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64. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t, x, y ) = e tP
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65. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t, x, y ) = e tP
= e −tλ φλ (x)φλ (y ),
λ
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66. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t, x, y ) = e tP
= e −tλ φλ (x)φλ (y ),
λ
where λ runs over the eigenvalues of P and φλ is the
corresponding eigenfunction.
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67. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
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68. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t,
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69. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t,
Asymptotic Expansion
Kt (t, x, x) ∼ bk (x)t k−d/2
k=0,1/2,1,...,
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70. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t,
Asymptotic Expansion
Kt (t, x, x) ∼ bk (x)t k−d/2
k=0,1/2,1,...,
Heat kernel coefficients
ak = bk (x)dx
M
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71. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
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72. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
• Provide geometric information about the manifold
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73. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
• Provide geometric information about the manifold
• a0 is the volume of M
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74. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
• Provide geometric information about the manifold
• a0 is the volume of M
• a1/2 is the volume of the boundary ∂M
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75. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
• Provide geometric information about the manifold
• a0 is the volume of M
• a1/2 is the volume of the boundary ∂M
• ak has curvature terms
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76. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
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77. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined in
terms of the spectrum of P)
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78. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined in
terms of the spectrum of P)
Recall: K (t) = λ e −tλ
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79. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined in
terms of the spectrum of P)
Recall: K (t) = λ e −tλ
Zeta function:
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80. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined in
terms of the spectrum of P)
Recall: K (t) = λ e −tλ
Zeta function:
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Spectral Functions

81. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined in
terms of the spectrum of P)
Recall: K (t) = λ e −tλ
Zeta function:
Pedro Fernando Morales Math Department
Spectral Functions

82. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined in
terms of the spectrum of P)
Recall: K (t) = λ e −tλ
Zeta function:
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83. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
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84. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP (s) = λ−s
λ
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85. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP (s) = λ−s
λ
e.g. λn = n gives the Riemann zeta function
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86. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP (s) = λ−s
λ
e.g. λn = n gives the Riemann zeta function
Problem:
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87. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP (s) = λ−s
λ
e.g. λn = n gives the Riemann zeta function
Problem: only defined for (s) > d/2
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88. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP (s) = λ−s
λ
e.g. λn = n gives the Riemann zeta function
Problem: only defined for (s) > d/2
All the important information lies to the left of this region!
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89. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
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90. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
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91. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Don’t take the usual meaning of convergence
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92. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Don’t take the usual meaning of convergence
Rather look at the meaning of the sum
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93. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
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94. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + · · · =
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95. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + · · · = − 1
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96. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + · · · = − 1
Special case of:
∞
rn
n=0
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97. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + · · · = − 1
Special case of:
∞
1
rn =
1−r
n=0
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98. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + · · · = − 1
Special case of:
∞
1
rn =
1−r
n=0
∞
2n
n=0
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99. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + · · · = − 1
Special case of:
∞
1
rn =
1−r
n=0
∞
1
2n = = −1
1−2
n=0
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100. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + · · · = − 1
Special case of:
∞
1
rn =
1−r
n=0
∞
1
2n = = −1
1−2
n=0
Convergent only for |r | < 1!
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101. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + · · · = − 1
Special case of:
∞
1
rn =
1−r
n=0
∞
1
2n = = −1
1−2
n=0
Convergent only for |r | < 1!
We just made an analytic continuation!!
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102. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
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103. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP (s) admits an analytic continuation to the whole complex plane,
except for simple poles at s = d/2, (d − 1)/2, . . . , 1/2, −(2n + 1)/2
for n a non-negative integer.
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104. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP (s) admits an analytic continuation to the whole complex plane,
except for simple poles at s = d/2, (d − 1)/2, . . . , 1/2, −(2n + 1)/2
for n a non-negative integer.
Residues
ad/2−s
Res ζP (s) =
Γ(s)
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105. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
• Only one pole (simple) at s = 1
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106. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
• Only one pole (simple) at s = 1
• Res ζR (1) = 1
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107. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
• Only one pole (simple) at s = 1
• Res ζR (1) = 1
a0 = Γ(d/2)
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108. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
• Only one pole (simple) at s = 1
• Res ζR (1) = 1
a0 = Γ(d/2)
ak = 0 (No other residues)
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109. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
• Only one pole (simple) at s = 1
• Res ζR (1) = 1
a0 = Γ(d/2)
ak = 0 (No other residues)
Volume of the boundary is zero
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110. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
• Only one pole (simple) at s = 1
• Res ζR (1) = 1
a0 = Γ(d/2)
ak = 0 (No other residues)
Volume of the boundary is zero
No curvature terms
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111. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
• Only one pole (simple) at s = 1
• Res ζR (1) = 1
a0 = Γ(d/2)
ak = 0 (No other residues)
Volume of the boundary is zero
No curvature terms
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112. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold:
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113. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold:
d = 1,
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114. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold:
d = 1,
√
length= π
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115. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold:
d = 1,
√
length= π
Operator:
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116. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold:
d = 1,
√
length= π
Operator:
d2
− φ = λφ
dx 2
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117. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold:
d = 1,
√
length= π
Operator:
d2
− φ = λφ
dx 2
Dirichlet boundary conditions
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118. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold:
d = 1,
√
length= π
Operator:
d2
− φ = λφ
dx 2
Dirichlet boundary conditions
eigenvalues {n2 }, n ∈ N
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119. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold:
d = 1,
√
length= π
Operator:
d2
− φ = λφ
dx 2
Dirichlet boundary conditions
eigenvalues {n2 }, n ∈ N
ζP (s) = ζR (2s)
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120. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
Pedro Fernando Morales Math Department
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121. 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 behavior
of the eigenvalues!
Pedro Fernando Morales Math Department
Spectral Functions

122. 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 behavior
of the eigenvalues!
Weyl’s law
Let N(λ) be the number of eigenvalues less than λ, then
1
N(λ) ∼ Vol(M)λd/2 ,
(4π)d/2 Γ(d/2)
where d = dim(M).
Pedro Fernando Morales Math Department
Spectral Functions

123. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Pedro Fernando Morales Math Department
Spectral Functions

124. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional Determinant
Pedro Fernando Morales Math Department
Spectral Functions

125. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional Determinant
Spectral dimension
Pedro Fernando Morales Math Department
Spectral Functions

126. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional Determinant
Spectral dimension
Physics:
Pedro Fernando Morales Math Department
Spectral Functions

127. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional Determinant
Spectral dimension
Physics:
Casimir Energy: λn eigenvalues of the Hamiltonian,
∞
ECas = λn
n=1
Pedro Fernando Morales Math Department
Spectral Functions

128. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional Determinant
Spectral dimension
Physics:
Casimir Energy: λn eigenvalues of the Hamiltonian,
∞
ECas = λn
n=1
One-Loop Effective Action (Functional Determinant)
Pedro Fernando Morales Math Department
Spectral Functions

129. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional Determinant
Spectral dimension
Physics:
Casimir Energy: λn eigenvalues of the Hamiltonian,
∞
ECas = λn
n=1
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients:
ad/2−z = Γ(z) Res ζP (z)
for z = d/2, (d − 1)/2, . . . , 1/2, −(2n + 1)/2, n ∈ N
Pedro Fernando Morales Math Department
Spectral Functions

130. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuum
fluctuations
Pedro Fernando Morales Math Department
Spectral Functions

131. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important?
• Believed to explain the stability of an electron
Pedro Fernando Morales Math Department
Spectral Functions

132. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important?
• Believed to explain the stability of an electron
• Very sensitive to the geometry of the space (Quantum and
Comsmological implications)
Pedro Fernando Morales Math Department
Spectral Functions

133. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important?
• Believed to explain the stability of an electron
• Very sensitive to the geometry of the space (Quantum and
Comsmological implications)
• Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions

134. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
• Eigenvalues know a lot of the geometry of a system!
Pedro Fernando Morales Math Department
Spectral Functions

135. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
• Eigenvalues know a lot of the geometry of a system!
• Describe the dynamics in a geometric object
Pedro Fernando Morales Math Department
Spectral Functions

136. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
• Eigenvalues know a lot of the geometry of a system!
• Describe the dynamics in a geometric object
• New information appear when regularizing expressions
Pedro Fernando Morales Math Department
Spectral Functions

137. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
• Eigenvalues know a lot of the geometry of a system!
• Describe the dynamics in a geometric object
• New information appear when regularizing expressions
• Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions

138. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions?
Thank you!
Pedro Fernando Morales Math Department
Spectral Functions