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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

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