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Spectral Functions, The Geometric Power of Eigenvalues,

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Spectral Functions, The Geometric Power of Eigenvalues,

  1. 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, 10/18/2012Pedro Fernando Morales Math DepartmentSpectral Functions
  2. 2. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsOutline 1 IntroductionPedro Fernando Morales Math DepartmentSpectral Functions
  3. 3. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsOutline 1 Introduction 2 Laplace-type OperatorsPedro Fernando Morales Math DepartmentSpectral Functions
  4. 4. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsOutline 1 Introduction 2 Laplace-type Operators 3 Heat kernelPedro Fernando Morales Math DepartmentSpectral Functions
  5. 5. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsOutline 1 Introduction 2 Laplace-type Operators 3 Heat kernel 4 Zeta functionPedro Fernando Morales Math DepartmentSpectral Functions
  6. 6. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsOutline 1 Introduction 2 Laplace-type Operators 3 Heat kernel 4 Zeta function 5 RegularizationPedro Fernando Morales Math DepartmentSpectral Functions
  7. 7. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsOutline 1 Introduction 2 Laplace-type Operators 3 Heat kernel 4 Zeta function 5 Regularization 6 ApplicationsPedro Fernando Morales Math DepartmentSpectral Functions
  8. 8. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsPedro Fernando Morales Math DepartmentSpectral Functions
  9. 9. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsPedro Fernando Morales Math DepartmentSpectral Functions
  10. 10. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsPedro Fernando Morales Math DepartmentSpectral Functions
  11. 11. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsPedro Fernando Morales Math DepartmentSpectral Functions
  12. 12. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsEigenvaluesPedro Fernando Morales Math DepartmentSpectral Functions
  13. 13. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsEigenvaluesPedro Fernando Morales Math DepartmentSpectral Functions
  14. 14. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsEigenvaluesPedro Fernando Morales Math DepartmentSpectral Functions
  15. 15. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsEigenvaluesPedro Fernando Morales Math DepartmentSpectral Functions
  16. 16. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsEigenvalues (a little more formal)Pedro Fernando Morales Math DepartmentSpectral Functions
  17. 17. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsEigenvalues (a little more formal) • Point spectrum of an operatorPedro Fernando Morales Math DepartmentSpectral Functions
  18. 18. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsEigenvalues (a little more formal) • Point spectrum of an operator • P − λI not bounded below (not injective)Pedro Fernando Morales Math DepartmentSpectral Functions
  19. 19. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsEigenvalues (a little more formal) • Point spectrum of an operator • P − λI not bounded below (not injective) • Pφ = λφPedro Fernando Morales Math DepartmentSpectral Functions
  20. 20. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsEigenvalues (a little more formal) • Point spectrum of an operator • P − λI not bounded below (not injective) • Pφ = λφ (eigenvalue equation)Pedro Fernando Morales Math DepartmentSpectral Functions
  21. 21. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsPhilosophical question What is an eigenvalue?Pedro Fernando Morales Math DepartmentSpectral Functions
  22. 22. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsPhilosophical question What is an eigenvalue? • A way to linearize a problemPedro Fernando Morales Math DepartmentSpectral Functions
  23. 23. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsPhilosophical question What is an eigenvalue? • A way to linearize a problem • Measures the distortion of a systemPedro Fernando Morales Math DepartmentSpectral Functions
  24. 24. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsPhilosophical question What is an eigenvalue? • A way to linearize a problem • Measures the distortion of a system • Decomposes an object into simpler piecesPedro Fernando Morales Math DepartmentSpectral Functions
  25. 25. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsPhilosophical 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 methodPedro Fernando Morales Math DepartmentSpectral Functions
  26. 26. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsOther names and similar ideasPedro Fernando Morales Math DepartmentSpectral Functions
  27. 27. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsOther names and similar ideas • Fourier expansionPedro Fernando Morales Math DepartmentSpectral Functions
  28. 28. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsOther names and similar ideas • Fourier expansion • Characters of representationsPedro Fernando Morales Math DepartmentSpectral Functions
  29. 29. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsOther names and similar ideas • Fourier expansion • Characters of representations • Decomposition into irreduciblesPedro Fernando Morales Math DepartmentSpectral Functions
  30. 30. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsDifferential OperatorsPedro Fernando Morales Math DepartmentSpectral Functions
  31. 31. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsDifferential Operators • M a compact manifoldPedro Fernando Morales Math DepartmentSpectral Functions
  32. 32. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsDifferential Operators • M a compact manifold • E a vector bundle over MPedro Fernando Morales Math DepartmentSpectral Functions
  33. 33. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsDifferential Operators • M a compact manifold • E a vector bundle over M • P : Γ(E ) → Γ(E ) a differential operator over MPedro Fernando Morales Math DepartmentSpectral Functions
  34. 34. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsDifferential 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 = ∅Pedro Fernando Morales Math DepartmentSpectral Functions
  35. 35. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsDifferential 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 )Pedro Fernando Morales Math DepartmentSpectral Functions
  36. 36. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsExample (Vibrating membrane)Pedro Fernando Morales Math DepartmentSpectral Functions
  37. 37. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsPedro Fernando Morales Math DepartmentSpectral Functions
  38. 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 LaplacianPedro Fernando Morales Math DepartmentSpectral Functions
  39. 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 LaplacianPedro Fernando Morales Math DepartmentSpectral Functions
  40. 40. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsGeneralized Laplace equation M d-dimensional smooth compact Riemannian manifoldPedro Fernando Morales Math DepartmentSpectral Functions
  41. 41. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsGeneralized Laplace equation M d-dimensional smooth compact Riemannian manifold E a smooth vector bundle over MPedro Fernando Morales Math DepartmentSpectral Functions
  42. 42. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsGeneralized Laplace equation M d-dimensional smooth compact Riemannian manifold E a smooth vector bundle over M V ∈ End(E )Pedro Fernando Morales Math DepartmentSpectral Functions
  43. 43. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsGeneralized Laplace equation M d-dimensional smooth compact Riemannian manifold E a smooth vector bundle over M V ∈ End(E ) E a connection on EPedro Fernando Morales Math DepartmentSpectral Functions
  44. 44. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsGeneralized 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 MPedro Fernando Morales Math DepartmentSpectral Functions
  45. 45. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsGeneralized 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 jPedro Fernando Morales Math DepartmentSpectral Functions
  46. 46. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsProperties Laplace-type Operators SymmetricPedro Fernando Morales Math DepartmentSpectral Functions
  47. 47. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsProperties Laplace-type Operators Symmetric (Self-adjoint with suitable boundary conditions)Pedro Fernando Morales Math DepartmentSpectral Functions
  48. 48. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsProperties Laplace-type Operators Symmetric (Self-adjoint with suitable boundary conditions) Real eigenvaluesPedro Fernando Morales Math DepartmentSpectral Functions
  49. 49. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsProperties Laplace-type Operators Symmetric (Self-adjoint with suitable boundary conditions) Real eigenvalues Bounded belowPedro Fernando Morales Math DepartmentSpectral Functions
  50. 50. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsProperties Laplace-type Operators Symmetric (Self-adjoint with suitable boundary conditions) Real eigenvalues Bounded below Tend to infinityPedro Fernando Morales Math DepartmentSpectral Functions
  51. 51. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications Recall:Pedro Fernando Morales Math DepartmentSpectral Functions
  52. 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.Pedro Fernando Morales Math DepartmentSpectral Functions
  53. 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:Pedro Fernando Morales Math DepartmentSpectral Functions
  54. 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 > 0Pedro Fernando Morales Math DepartmentSpectral Functions
  55. 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→0Pedro Fernando Morales Math DepartmentSpectral Functions
  56. 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 ∂DPedro Fernando Morales Math DepartmentSpectral Functions
  57. 57. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsPedro Fernando Morales Math DepartmentSpectral Functions
  58. 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.Pedro Fernando Morales Math DepartmentSpectral Functions
  59. 59. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsHeat kernel for a Laplace-type operatorPedro Fernando Morales Math DepartmentSpectral Functions
  60. 60. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsHeat kernel for a Laplace-type operator Likewise, for a Laplace-type operator,Pedro Fernando Morales Math DepartmentSpectral Functions
  61. 61. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsHeat 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)Pedro Fernando Morales Math DepartmentSpectral Functions
  62. 62. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsHeat 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 MPedro Fernando Morales Math DepartmentSpectral Functions
  63. 63. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsRelation with eigenvaluesPedro Fernando Morales Math DepartmentSpectral Functions
  64. 64. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsRelation with eigenvalues K (t, x, y ) = e tPPedro Fernando Morales Math DepartmentSpectral Functions
  65. 65. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsRelation with eigenvalues K (t, x, y ) = e tP = e −tλ φλ (x)φλ (y ), λPedro Fernando Morales Math DepartmentSpectral Functions
  66. 66. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsRelation 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.Pedro Fernando Morales Math DepartmentSpectral Functions
  67. 67. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsHeat Kernel Asymptotic ExpansionPedro Fernando Morales Math DepartmentSpectral Functions
  68. 68. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsHeat Kernel Asymptotic Expansion For small values of t,Pedro Fernando Morales Math DepartmentSpectral Functions
  69. 69. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsHeat 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,...,Pedro Fernando Morales Math DepartmentSpectral Functions
  70. 70. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsHeat 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 MPedro Fernando Morales Math DepartmentSpectral Functions
  71. 71. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsPropertiesPedro Fernando Morales Math DepartmentSpectral Functions
  72. 72. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsProperties • Provide geometric information about the manifoldPedro Fernando Morales Math DepartmentSpectral Functions
  73. 73. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsProperties • Provide geometric information about the manifold • a0 is the volume of MPedro Fernando Morales Math DepartmentSpectral Functions
  74. 74. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsProperties • Provide geometric information about the manifold • a0 is the volume of M • a1/2 is the volume of the boundary ∂MPedro Fernando Morales Math DepartmentSpectral Functions
  75. 75. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsProperties • Provide geometric information about the manifold • a0 is the volume of M • a1/2 is the volume of the boundary ∂M • ak has curvature termsPedro Fernando Morales Math DepartmentSpectral Functions
  76. 76. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsSpectral FunctionsPedro Fernando Morales Math DepartmentSpectral Functions
  77. 77. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsSpectral Functions The heat kernel is an example of an spectral function (defined in terms of the spectrum of P)Pedro Fernando Morales Math DepartmentSpectral Functions
  78. 78. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsSpectral Functions The heat kernel is an example of an spectral function (defined in terms of the spectrum of P) Recall: K (t) = λ e −tλPedro Fernando Morales Math DepartmentSpectral Functions
  79. 79. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsSpectral 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 DepartmentSpectral Functions
  80. 80. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsSpectral 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 DepartmentSpectral Functions
  81. 81. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsSpectral 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 DepartmentSpectral Functions
  82. 82. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsSpectral 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 DepartmentSpectral Functions
  83. 83. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsZeta functionPedro Fernando Morales Math DepartmentSpectral Functions
  84. 84. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsZeta function Zeta function The zeta function associated with the operator P is defined by ζP (s) = λ−s λPedro Fernando Morales Math DepartmentSpectral Functions
  85. 85. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsZeta 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 functionPedro Fernando Morales Math DepartmentSpectral Functions
  86. 86. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsZeta 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:Pedro Fernando Morales Math DepartmentSpectral Functions
  87. 87. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsZeta 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/2Pedro Fernando Morales Math DepartmentSpectral Functions
  88. 88. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsZeta 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!Pedro Fernando Morales Math DepartmentSpectral Functions
  89. 89. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsRegularizationPedro Fernando Morales Math DepartmentSpectral Functions
  90. 90. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsRegularization Is a method of making sense of a divergent expressionPedro Fernando Morales Math DepartmentSpectral Functions
  91. 91. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsRegularization Is a method of making sense of a divergent expression Don’t take the usual meaning of convergencePedro Fernando Morales Math DepartmentSpectral Functions
  92. 92. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsRegularization 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 sumPedro Fernando Morales Math DepartmentSpectral Functions
  93. 93. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsPedro Fernando Morales Math DepartmentSpectral Functions
  94. 94. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications Example 1 + 2 + 4 + 8 + 16 + · · · =Pedro Fernando Morales Math DepartmentSpectral Functions
  95. 95. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications Example 1 + 2 + 4 + 8 + 16 + · · · = − 1Pedro Fernando Morales Math DepartmentSpectral Functions
  96. 96. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications Example 1 + 2 + 4 + 8 + 16 + · · · = − 1 Special case of: ∞ rn n=0Pedro Fernando Morales Math DepartmentSpectral Functions
  97. 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=0Pedro Fernando Morales Math DepartmentSpectral Functions
  98. 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=0Pedro Fernando Morales Math DepartmentSpectral Functions
  99. 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=0Pedro Fernando Morales Math DepartmentSpectral Functions
  100. 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!Pedro Fernando Morales Math DepartmentSpectral Functions
  101. 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!!Pedro Fernando Morales Math DepartmentSpectral Functions
  102. 102. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsAnalytic continuationPedro Fernando Morales Math DepartmentSpectral Functions
  103. 103. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsAnalytic 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.Pedro Fernando Morales Math DepartmentSpectral Functions
  104. 104. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsAnalytic 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)Pedro Fernando Morales Math DepartmentSpectral Functions
  105. 105. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsRiemann Zeta • Only one pole (simple) at s = 1Pedro Fernando Morales Math DepartmentSpectral Functions
  106. 106. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsRiemann Zeta • Only one pole (simple) at s = 1 • Res ζR (1) = 1Pedro Fernando Morales Math DepartmentSpectral Functions
  107. 107. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsRiemann Zeta • Only one pole (simple) at s = 1 • Res ζR (1) = 1 a0 = Γ(d/2)Pedro Fernando Morales Math DepartmentSpectral Functions
  108. 108. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsRiemann Zeta • Only one pole (simple) at s = 1 • Res ζR (1) = 1 a0 = Γ(d/2) ak = 0 (No other residues)Pedro Fernando Morales Math DepartmentSpectral Functions
  109. 109. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsRiemann 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 zeroPedro Fernando Morales Math DepartmentSpectral Functions
  110. 110. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsRiemann 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 termsPedro Fernando Morales Math DepartmentSpectral Functions
  111. 111. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsRiemann 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 termsPedro Fernando Morales Math DepartmentSpectral Functions
  112. 112. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications Manifold:Pedro Fernando Morales Math DepartmentSpectral Functions
  113. 113. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications Manifold: d = 1,Pedro Fernando Morales Math DepartmentSpectral Functions
  114. 114. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications Manifold: d = 1, √ length= πPedro Fernando Morales Math DepartmentSpectral Functions
  115. 115. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications Manifold: d = 1, √ length= π Operator:Pedro Fernando Morales Math DepartmentSpectral Functions
  116. 116. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications Manifold: d = 1, √ length= π Operator: d2 − φ = λφ dx 2Pedro Fernando Morales Math DepartmentSpectral Functions
  117. 117. Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications Manifold: d = 1, √ length= π Operator: d2 − φ = λφ dx 2 Dirichlet boundary conditionsPedro Fernando Morales Math DepartmentSpectral Functions
  118. 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 ∈ NPedro Fernando Morales Math DepartmentSpectral Functions
  119. 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)Pedro Fernando Morales Math DepartmentSpectral Functions
  120. 120. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsMeromorphic structure Convergence problems(poles) come from the large λ behaviorPedro Fernando Morales Math DepartmentSpectral Functions
  121. 121. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsMeromorphic 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 DepartmentSpectral Functions
  122. 122. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsMeromorphic 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 DepartmentSpectral Functions
  123. 123. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsApplications Zeta Regularized TracePedro Fernando Morales Math DepartmentSpectral Functions
  124. 124. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsApplications Zeta Regularized Trace Functional DeterminantPedro Fernando Morales Math DepartmentSpectral Functions
  125. 125. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsApplications Zeta Regularized Trace Functional Determinant Spectral dimensionPedro Fernando Morales Math DepartmentSpectral Functions
  126. 126. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsApplications Zeta Regularized Trace Functional Determinant Spectral dimension Physics:Pedro Fernando Morales Math DepartmentSpectral Functions
  127. 127. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsApplications Zeta Regularized Trace Functional Determinant Spectral dimension Physics: Casimir Energy: λn eigenvalues of the Hamiltonian, ∞ ECas = λn n=1Pedro Fernando Morales Math DepartmentSpectral Functions
  128. 128. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsApplications 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 DepartmentSpectral Functions
  129. 129. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsApplications 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 ∈ NPedro Fernando Morales Math DepartmentSpectral Functions
  130. 130. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsCasimir Effect Is a quantum field effect that arises when considering vacuum fluctuationsPedro Fernando Morales Math DepartmentSpectral Functions
  131. 131. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsWhy is so important? • Believed to explain the stability of an electronPedro Fernando Morales Math DepartmentSpectral Functions
  132. 132. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsWhy 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 DepartmentSpectral Functions
  133. 133. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsWhy 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 energyPedro Fernando Morales Math DepartmentSpectral Functions
  134. 134. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsConclusions • Eigenvalues know a lot of the geometry of a system!Pedro Fernando Morales Math DepartmentSpectral Functions
  135. 135. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsConclusions • Eigenvalues know a lot of the geometry of a system! • Describe the dynamics in a geometric objectPedro Fernando Morales Math DepartmentSpectral Functions
  136. 136. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsConclusions • Eigenvalues know a lot of the geometry of a system! • Describe the dynamics in a geometric object • New information appear when regularizing expressionsPedro Fernando Morales Math DepartmentSpectral Functions
  137. 137. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsConclusions • 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 DepartmentSpectral Functions
  138. 138. Introduction Laplace-type Operators Heat kernel Zeta function Regularization ApplicationsQuestions? Thank you!Pedro Fernando Morales Math DepartmentSpectral Functions

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