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# Spectral functions and geometric invariants

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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.

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### Spectral functions and geometric invariants

1. 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 2012Pedro Morales-Almaz´n a Math DepartmentBackground potentials
2. 2. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsOutline 1 MotivationPedro Morales-Almaz´n a Math DepartmentBackground potentials
3. 3. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsOutline 1 Motivation 2 Heat KernelPedro Morales-Almaz´n a Math DepartmentBackground potentials
4. 4. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsOutline 1 Motivation 2 Heat Kernel 3 Zeta FunctionPedro Morales-Almaz´n a Math DepartmentBackground potentials
5. 5. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsOutline 1 Motivation 2 Heat Kernel 3 Zeta Function 4 The ProblemPedro Morales-Almaz´n a Math DepartmentBackground potentials
6. 6. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsOutline 1 Motivation 2 Heat Kernel 3 Zeta Function 4 The Problem 5 Zeta RevisitedPedro Morales-Almaz´n a Math DepartmentBackground potentials
7. 7. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsOutline 1 Motivation 2 Heat Kernel 3 Zeta Function 4 The Problem 5 Zeta Revisited 6 ResiduesPedro Morales-Almaz´n a Math DepartmentBackground potentials
8. 8. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsOutline 1 Motivation 2 Heat Kernel 3 Zeta Function 4 The Problem 5 Zeta Revisited 6 Residues 7 ApplicationsPedro Morales-Almaz´n a Math DepartmentBackground potentials
9. 9. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsGeometry-Analysis Intertwine (x 2 + y 2 − x)2 = x 2 + y 2Pedro Morales-Almaz´n a Math DepartmentBackground potentials
10. 10. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsGeometry-Analysis Intertwine (x 2 + y 2 − x)2 = x 2 + y 2 CardioidPedro Morales-Almaz´n a Math DepartmentBackground potentials
11. 11. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsGeometry-Analysis Intertwine (x 2 + y 2 − x)2 = x 2 + y 2 Cardioid Torricelli’s TrumpetPedro Morales-Almaz´n a Math DepartmentBackground potentials
12. 12. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsGeometry-Analysis Intertwine (x 2 + y 2 − x)2 = x 2 + y 2 Cardioid Torricelli’s Trumpet r = 1/z, z ≥ 1Pedro Morales-Almaz´n a Math DepartmentBackground potentials
13. 13. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications • Reidemeinster Torsion - Analytic TorsionPedro Morales-Almaz´n a Math DepartmentBackground potentials
14. 14. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications • Reidemeinster Torsion - Analytic Torsion • Gauss BonetPedro Morales-Almaz´n a Math DepartmentBackground potentials
15. 15. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications • Reidemeinster Torsion - Analytic Torsion • Gauss Bonet • Atiyah-Singer IndexPedro Morales-Almaz´n a Math DepartmentBackground potentials
16. 16. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications • Reidemeinster Torsion - Analytic Torsion • Gauss Bonet • Atiyah-Singer Index • Algebraic GeometryPedro Morales-Almaz´n a Math DepartmentBackground potentials
17. 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 AnalysisPedro Morales-Almaz´n a Math DepartmentBackground potentials
18. 18. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsSpectral functions: Heat Kernel The heat equation on a manifold M provides a way to analyze heat ﬂow given an initial heat distributionPedro Morales-Almaz´n a Math DepartmentBackground potentials
19. 19. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsSpectral 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)Pedro Morales-Almaz´n a Math DepartmentBackground potentials
20. 20. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsSpectral 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)Pedro Morales-Almaz´n a Math DepartmentBackground potentials
21. 21. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsHeat kernel for PDO Provides a way of solving the above PDE regardless of g (x)Pedro Morales-Almaz´n a Math DepartmentBackground potentials
22. 22. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsHeat kernel for PDO Provides a way of solving the above PDE regardless of g (x) H formal inverse of PPedro Morales-Almaz´n a Math DepartmentBackground potentials
23. 23. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsHeat 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 ) MPedro Morales-Almaz´n a Math DepartmentBackground potentials
24. 24. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications P is usually taken to be a Laplace-type operatorPedro Morales-Almaz´n a Math DepartmentBackground potentials
25. 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 +VPedro Morales-Almaz´n a Math DepartmentBackground potentials
26. 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 DepartmentBackground potentials
27. 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 MPedro Morales-Almaz´n a Math DepartmentBackground potentials
28. 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!Pedro Morales-Almaz´n a Math DepartmentBackground potentials
29. 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,...Pedro Morales-Almaz´n a Math DepartmentBackground potentials
30. 30. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications The heat kernel coeﬃcients give geometric invariants of the manifoldPedro Morales-Almaz´n a Math DepartmentBackground potentials
31. 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 manifoldPedro Morales-Almaz´n a Math DepartmentBackground potentials
32. 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 boundaryPedro Morales-Almaz´n a Math DepartmentBackground potentials
33. 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 manifoldPedro Morales-Almaz´n a Math DepartmentBackground potentials
34. 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 derivativesPedro Morales-Almaz´n a Math DepartmentBackground potentials
35. 35. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsZeta function There is an analytic connection between the heat kernel coeﬃcients and the spectral zeta functionPedro Morales-Almaz´n a Math DepartmentBackground potentials
36. 36. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsZeta 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.Pedro Morales-Almaz´n a Math DepartmentBackground potentials
37. 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/2Pedro Morales-Almaz´n a Math DepartmentBackground potentials
38. 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/2Pedro Morales-Almaz´n a Math DepartmentBackground potentials
39. 39. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsSpectral functions: Zeta function ∞ ζP (s) = λ−s n n=1 where λn are the eigenvalues of P counting multiplicities.Pedro Morales-Almaz´n a Math DepartmentBackground potentials
40. 40. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsSpectral 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 DepartmentBackground potentials
41. 41. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsSpectral 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!Pedro Morales-Almaz´n a Math DepartmentBackground potentials
42. 42. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsEigenvalue Problem Consider an annulus of radii 0 < a < b with a smooth potential depending only on the radiusPedro Morales-Almaz´n a Math DepartmentBackground potentials
43. 43. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsEigenvalue Problem Consider an annulus of radii 0 < a < b with a smooth potential depending only on the radius P = −∆ + V (r ) Dirichlet BCPedro Morales-Almaz´n a Math DepartmentBackground potentials
44. 44. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications Let λn be the eigenvalues of PPedro Morales-Almaz´n a Math DepartmentBackground potentials
45. 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=1Pedro Morales-Almaz´n a Math DepartmentBackground potentials
46. 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!Pedro Morales-Almaz´n a Math DepartmentBackground potentials
47. 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!!Pedro Morales-Almaz´n a Math DepartmentBackground potentials
48. 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 EPPedro Morales-Almaz´n a Math DepartmentBackground potentials
49. 49. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications Remark Important points are not included in the convergence region!Pedro Morales-Almaz´n a Math DepartmentBackground potentials
50. 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 !Pedro Morales-Almaz´n a Math DepartmentBackground potentials
51. 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 ∞Pedro Morales-Almaz´n a Math DepartmentBackground potentials
52. 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 DepartmentBackground potentials
53. 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)Pedro Morales-Almaz´n a Math DepartmentBackground potentials
54. 54. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsWKB Asymptotics N−2 b log Rk (b; ıλ) = log A + + dtSik (t)ξ −i + O(ξ −N+1 ) i=−1 aPedro Morales-Almaz´n a Math DepartmentBackground potentials
55. 55. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsAnalytic Continuation The zeta function can then be written as ζP (s) =Pedro Morales-Almaz´n a Math DepartmentBackground potentials
56. 56. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsAnalytic Continuation The zeta function can then be written as ∞ ζP (s) = 0Pedro Morales-Almaz´n a Math DepartmentBackground potentials
57. 57. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsAnalytic Continuation The zeta function can then be written as ∞ ζP (s) = ♣ 0Pedro Morales-Almaz´n a Math DepartmentBackground potentials
58. 58. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsAnalytic Continuation The zeta function can then be written as ∞ ζP (s) = ♣ − asymptotic terms 0Pedro Morales-Almaz´n a Math DepartmentBackground potentials
59. 59. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsAnalytic Continuation The zeta function can then be written as ∞ ζP (s) = ♣ − asymptotic terms 0 ∞ + asymptotic terms 0Pedro Morales-Almaz´n a Math DepartmentBackground potentials
60. 60. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsAnalytic Continuation The zeta function can then be written as ∞ ζP (s) = ♣ − asymptotic terms 0 ∞ + asymptotic terms 0Pedro Morales-Almaz´n a Math DepartmentBackground potentials
61. 61. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications N−2 ζP (s) = Z (s) + Ai (s) i=−1Pedro Morales-Almaz´n a Math DepartmentBackground potentials
62. 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 partPedro Morales-Almaz´n a Math DepartmentBackground potentials
63. 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 −iPedro Morales-Almaz´n a Math DepartmentBackground potentials
64. 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/2Pedro Morales-Almaz´n a Math DepartmentBackground potentials
65. 65. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsResidues Residue at s = 1 1 2 Res ζ(s)|s=1 = b − a2 ∝ Vol(M) 4Pedro Morales-Almaz´n a Math DepartmentBackground potentials
66. 66. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsResidues 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) 4Pedro Morales-Almaz´n a Math DepartmentBackground potentials
67. 67. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications Value at s = 0 b 1 ζ(0) = − rdr V (r ) 2 aPedro Morales-Almaz´n a Math DepartmentBackground potentials
68. 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)) 8Pedro Morales-Almaz´n a Math DepartmentBackground potentials
69. 69. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues Applications • All heat kernel coeﬃcients can be obtain with this procedure!Pedro Morales-Almaz´n a Math DepartmentBackground potentials
70. 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 eigenvaluesPedro Morales-Almaz´n a Math DepartmentBackground potentials
71. 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 informationPedro Morales-Almaz´n a Math DepartmentBackground potentials
72. 72. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsApplications • Casimir eﬀectPedro Morales-Almaz´n a Math DepartmentBackground potentials
73. 73. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsApplications • Casimir eﬀect • One-loop eﬀective actionPedro Morales-Almaz´n a Math DepartmentBackground potentials
74. 74. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsApplications • Casimir eﬀect • One-loop eﬀective action • Geometric invariantsPedro Morales-Almaz´n a Math DepartmentBackground potentials
75. 75. Motivation Heat Kernel Zeta Function The Problem Zeta Revisited Residues ApplicationsQuestions?Pedro Morales-Almaz´n a Math DepartmentBackground potentials