Heat Kernel Asymptotics on Manifolds

Heat Kernel Asymptotics on Manifolds Ivan Avramidi New Mexico Tech Motivation Evolution Eqs (Heat transfer, Diffusion) Quantum Theory and Statistical Physics (Partition Function, Correlation Functions) Integrable Systems (KdV hiearchy) Spectral Asymptotics of Diff Operators Spectral Geometry, Isospectrality Topology of Manifolds, Index Theorems 1

Manifolds and Vector Bundles Compact n-dim manifold M (with or without smooth boundary M) with local coordinates x µ, µ = 1, 2,..., n N-dim vector space V with a positive definite inner product, ; vectors: ϕ = (ϕ A ), A = 1, 2,..., N Vector bundle V over M with fiber V Space C (V) of smooth sections of the vector bundle V; locally, ϕ = (ϕ A (x)) Inner product (ϕ, ψ) = M dvol (x) ϕ(x), ψ(x) Norm ϕ = (ϕ, ϕ) Hilbert space L 2 (V) is the completion of C (V) in L 2 -norm 2

Laplace Type Differential Operators Riemannian metric g Connection on the vector bundle V : C (V) C (T M V) Connection one-form A = A µ dx µ Smooth self-adjoint endomorphism Q of V Laplace type Operator (LTO) Locally L : C (V) C (V) L = + Q L = g 1/2 ( µ + A µ ) g 1/2 g µν ( µ + A µ ) + Q where g = det g µν, µ = x µ 3

Natural Non-Laplace Type Operators Spin-tensor vector bundle V with a canonical (Levi-Civita) connection Decomposition into irreducible components T M V = W 1 W s Projections P j : T M V W j Gradients (Stein-Weiss operators) G j = P j : C (V) C (W j ) Non-Laplace type Operators (NLTO) L = s j=1 c 2 j G j G j 4

Leading Symbol General second order partial differential operator L = a µν (x) µ ν + b µ (x) µ + c(x) with matrix-valued coefficients Leading Symbol with ξ T xm σ L (L; x, ξ) = a µν (x)ξ µ ξ ν We assume that L has a positive-definite leading symbol and is formally self-adjoint: for any ϕ, ψ C 0 (V) (Lϕ, ψ) = (ϕ, Lψ) Laplace type operator L has a positive definite scalar leading symbol σ L (L; x, ξ) = Ig µν (x)ξ µ ξ ν 5

Boundary Conditions Smooth compact Boundary Unit Normal to M Restriction of vector bundle V M N W = V M Boundary Data Boundary Operator Boundary Conditions ψ : C (V) C (W W) ψ(ϕ) = ϕ M N ϕ M B : C (W W) C (W W) Bψ(ϕ) = 0 6

Gilkey-Smith-Grubb Boundary Operators Projection Π : W W Tangential Operator (self-adjoint 1st order differential operator) Λ : C (W) C (W) Λ = 1 2 [ Γ j (ˆx) ˆ j + ˆ j Γ j (ˆx) ] + S(ˆx) with anti-self-adjoint matrix Γ j and self-adj S Gilkey-Smith-Grubb Boundary Operator B = ( Π 0 (I Π)Λ(I Π) (I Π) ) Special cases: i) Π = I, Λ = 0 Dirichlet ii) Π = 0, Λ = I Neumann iii) Γ = 0 Mixed Robin iv) Γ 0 Oblique 7

Smooth Boundary Conditions Disconnected boundary: disjoint union of compact connected components without boundary M = Σ 1 Σ m Σ i Σ j =, i j Σ 3 M Σ 1 Σ 2 Decomposition of boundary data ψ(ϕ) = ψ 1 (ϕ) ψ m (ϕ) Smooth boundary operators with different B i B = B 1 B m 8

Example: Iceberg in Water Air Water N Σ 2 Ice M M Neumann BC Σ 1 Dirichlet BC Water (thermostat) Dirichlet BC: ϕ Σ1 = 0 Air (perfect insulation) Neumann BC: N ϕ Σ2 = 0 Mixed BC (β N + γ)ϕ M = 0 with discontinuous boundary functions β Σ1 = 0, β Σ2 = 1 γ Σ1 = 1, γ Σ2 = 0 9

Discontinuous Boundary Value Problem Boundary decomposition: disjoint union M = Σ 1 Σ 2 Σ 0 with Σ 1, Σ 2 smooth compact submanifolds of co-dim 1 with same boundary Σ 1 = Σ 2 = Σ 0 and Σ 0 a smooth compact submanifold of codim 2 without boundary Σ 0 M Σ 2 Σ 1 Boundary Data ψ(ϕ) = ψ 1 (ϕ) ψ 2 (ϕ) Discontinuous boundary operator (on Σ 0 ) B = B 1 B 2 Zaremba BC ϕ Σ1 = 0 ( N + S)ϕ Σ2 = 0 with S an endomorphism of W 10

Ellipticity Ellipticity Invertibility (locally) An operator L is (strongly) elliptic if: leading symbol σ L (L; x, ξ) is elliptic, (the matrix [σ L (L; x, ξ) λi] is invertible for any x in the interior of M, ξ 0, λ C R + ) satisfies Lopatinski-Shapiro condition on the boundary M (existence of a unique local solution in a neighborhood of the boundary vanishing at infinity) Proposition. A formally self-adjoint second order PDO with Gilkey-Smith boundary conditions is elliptic and self-adjoint provided the leading symbol (first-order part) of the tangential operator Λ is sufficiently small The ellipticity for the Zaremba BC is more subtle problem (one needs an additional condition on the co-dim 2 submanifold Σ 0 ) 11

Spectral Theorem Eigenvalues, λ k, and eigenvectors, ϕ k, Lϕ k = λ k ϕ k, ϕ k = 1 Theorem. For a second-order self-adjoint elliptic operator on a compact manifold with positive definite leading symbol: i) eigenvalues λ k have finite multiplicities and form an unbounded increasing real sequence λ 1 λ 2 λ k λ k+1 ii) eigenvectors ϕ k are smooth sections that form an orthonormal basis in L 2 (V) (ϕ k, ϕ n ) = δ kn 12

Spectral Geometry Analysis determines (induces) Geometry A second-order elliptic partial differential operator on a manifold determines the geometry of the manifold A Laplace type operator determines Riemannian geometry A non-laplace type operator determines a collection of Finsler geometries M. Kac (1966): Can one hear the shape of a drum? No What geometric information can be extracted from the spectrum of a differential operator on a manifold? Large λ n Local structure Small λ n Global structure 13

Heat Kernel Heat Equation Initial Condition Boundary Condition ( ) t + L ϕ = 0 ϕ(0, x) = φ 0 (x) Bψ(ϕ) = 0 Operator solution Kernel form ϕ = exp( tl)φ 0 ϕ(t, x) = M dvol (y)u(t x, y)φ 0 (y) Fundamental Solution Heat Kernel ( ) t + L U(t x, y) = 0 U(0 + x, y) = δ(x, y) Bψ x [U(t x, y)] = 0 14

Heat Trace Heat Semigroup (bounded trace-class for t > 0) exp( tl) : L 2 (V) L 2 (V) Heat Kernel U(t x, x ) = k=1 e tλ kϕ k (x) ϕ k (x ) Heat Trace converges for t > 0 Tr exp( tl) = e tλ k k=1 Important relation (global local) Tr exp( tl) = M dvol (x)tr U(t x, x) 15

Zeta-Function and Determinant Zeta Function ζ(s, λ) = Tr (L λ) s = k=1 converges for Re s >> 0, Re λ << 0 (λ k λ) s Relation to the heat trace ζ(s, λ) = 1 Γ(s) 0 dt t s 1 e tλ Tr exp( tl) Analytic continuation Determinant = meromorphic function of s ζ(s, λ) is analytic at s = 0 ζ (0, λ) = log Det (L λ) 16

Heat Kernel Asymptotics Spectral Asymptotics of λ n as n are described by short-time asymptotics of the heat kernel as t 0 [Minakshisundaram-Pleijel, Greiner, Seeley] Tr exp( tl) (4πt) n 2 k=0 t k 2A k + log t k=0 t k 2H k Spectral Invariants A k = H k = M a k + Σ 0 h k Σ 1 b (1) k + Σ 2 b (2) k + Σ 0 c k 17

General Properties: Interior coefficients are local invariants of intrinsic interior geometry only Co-dimension 1 boundary coefficients b k are local invariants of both the intrinsic geometry of M and extrinsic geometry of the boundary M in M Co-dimension 2 boundary coefficients c k are local invariants of the intrinsic geometry of M, the extrinsic geometry of the boundary M in M as well as the extrinsic geometry of the co-dimension 2 submanifold Σ 0 in the boundary M There are no odd-order interior coefficients a 2k+1 = 0 18

For manifolds without boundary there are no boundary coefficients b (1) k = b (2) k = h k = 0 For manifolds with boundary and smooth boundary conditions there are no log-coefficients and only one type of boundary coefficients, h k = 0, and either b (1) k = 0 or b (2) k = 0 For smooth manifolds with smooth boundary there are no log-coefficients h k = 0 (even for non-smooth local boundary conditions) [Seeley (2001)]

Known Results for Spectral Invariants Laplace type Operators Interior Coefficients a 0, a 2, a 4, a 6, a 8 Co-dim 1 Coefficients (smooth BC) b 1, b 2, b 3, b 4, b 5 b 1, b 2, b 3 (Dirichlet, Neumann) (Oblique) Co-dim 2 Coefficients (non-smooth BC) c 2 (Zaremba) Non-Laplace type Operators Interior coefficients: a 0, a 2 Boundary coefficients: b 1 (Dirichlet) 19

Isospectrality and Integrability Isospectral Deformation L L(τ) Lax evolution equations L = [L, K], τ K = K Integrals of motion τ Tr exp( tl) = τ A k = 0 Example Schrödinger Operator L(τ) = 2 x + u(x, τ) : C (S 1 ) C (S 1 ) KdV hierarchy (inf-dim Hamiltonian system) τ u = x δa k (u) δu(x) Korteweg-de Vries equation K = 4 3 x 3(u x + x u) ( τ L = ) τ u = 3 xu + 6u x u ( = [L, K] ) 20

Geometric Framework Decomposition of the manifold M = M int M1 bnd M2 bnd M0 bnd where Mi bnd is a narrow strip near Σ i of width ε > 0 and M int is the interior of M on a finite distance > ε from M Co-dim 1 Geometry Σ 1, Σ 2 Co-dim 2 Geometry Σ 0 normal N extrinsic curvature K local coordinates (r, ˆx) 2-dim normal bundle {N, n} two extrinsic curvtr s K, L local polar coord s (ρ, θ, ˆx) 21

Construction of the Parametrix different approximations in different domains glue together in a smooth way control the remainder in asymptotic expansion as t 0 and its dependence on ε compute asymptotic expansion as t 0 in each domain take the limit ε 0 Local analysis near diagonal as t 0 and x x 22

Interior Parametrix Fixed point x 0 M int Scaling: coordinates x µ x µ 0 + ε(xµ x µ 0 ) x µ x µ 0 + ε(x µ x µ 0 ) t ε 2 t derivatives x µ 1 ε Power series x µ, L L ε Asymptotic expansion k=0 t 1 ε 2 ε k 2 L k t U int ε k=0 ε 2 n+k U int k 23

Interior Heat Invariants Recursive differential equations ( t + L 0 )U int k Initial conditions = k j=1 L j U int k j U int k (0; x, x ) = 0 Homogeneity Uk int (t; x 0, x 0 ) = t (k n)/2 Uk int (1; x 0, x 0 ) Spectral Invariants a k (x) = (4π) n/2 tr Uk int (1; x, x) 24

Interior Parametrix for LTO Semi-classical Ansatz U int (t x, x ) (4πt) n 2 exp k=0 ( t k 2 α k (x, x ) d2 (x, x ) 4t Heat Equation Differential Recursion Relations for α k (x, x ) covariant Taylor Expansion near x Diagonal Values (x x ) a k = tr α k (x, x) ) Remark: very effective algorithm (there is a code for Mathematica) 25

Dirichlet Parametrix for LTO i) Fix a point ˆx 0 on Σ 1 ii) Choose normal coordinates ˆx on Σ 1 iii) Replace M bnd 1 Σ 1 R + iv) Scaling ˆx ˆx 0 + ε(ˆx ˆx 0 ) r εr t ε 2 t v) Expansion as ε 0 U bnd,(1) k=0 ε 2 n+k U bnd,(1) k vi) Diagonal Values U bnd,(1) k,diag (t r) = t(k n)/2 exp ( +Polynomial r2 t ) Y (1) k ( r t ) 26

vii) Integration over M bnd 1 M bnd 1 Σ 1 ε 0 dr viii) Limit ε 0 b (1) k in terms of 0 dξ e ξ2 ξ n Y (1) k (ξ) Remarks: Need to scale the BC and volume element Heat kernel behaves like distribution near boundary This is the origin of boundary integrals and shift in power of t Neumann parametrix is done similarly 27

Zaremba Parametrix for LTO i) Fix a point ˆx 0 on Σ 0 Σ 0 M Σ 2 Σ 1 Σ 0 y Σ 1 ˆx 0 Σ 2 ˆx N y r x r ii) Local coordinates (r, y, ˆx) or (ρ, θ, ˆx) ˆx normal coordinates on Σ 0 r normal geodesic distance to M y signed normal geodesic distance to Σ 0 (ρ, θ) polar coordinates in normal bundle iii) Replace M bnd 0 Σ 0 R R + 28

iv) Scaling ˆx ˆx 0 + ε(ˆx ˆx 0 ) r εr or ρ ερ y εy θ θ t ε 2 t Remark: Global analysis in θ (solve a one-dim mixed boundary value problem) vi) Expansion in ε vii) Differential Recursion Relations viii) Homogeneity ix) Heat Kernel Diagonal x) Integration over M bnd 0 ε dρ ρ π/2 dθ M bnd 0 Σ 0 0 π/2 viii) Limit ε 0 gives the co-dim 2 coefficients c k 29

Remarks Need to scale the BC and volume element Heat kernel behaves like distribution near co-dim 2 submanfld Σ 0 This is the origin of co-dim 2 coefficients c k and shift in power of t 30

Spectral Invariants for Zaremba LTO A 0 = N vol (M) A 1 = A 2 = π 2 N [vol (Σ 2) vol (Σ 1 )] M + N 3 ( ) N 6 R tr Q M K + 2 where N = dim V and α(s) = 1 Σ 2 tr S + α(s) π 4 N vol (Σ 0) for s 7 for any finite s Here s is a parameter of an additional boundary condition at Σ 0 31

Spectral Invariants for Oblique LTO A 1 = π 2 M dvol (ˆx) tr [ I 2Π + 2β(Γ) ] β = R n 1 dˆξ π (n 1)/2 exp{ (gij I + Γ i Γ j )ˆξ iˆξ j } Strong ellipticity { Convergence of the integral over ζ Classical case: Γ = 0 β = I 32

Spectral Invariants of NLTO Interior invariant A 0 = R n dξ π n/2tr exp[ σ L(L; x, ξ)] Auxiliary functions Φ(λ, ˆx, ˆξ) = dω 2π [σ L(L; 0, ˆx, ω, ˆξ ) λi] 1 Ψ(ˆx, ˆξ) = w+i w i dλ 2πi e λ { tr [Φ(λ, ˆx, ˆξ)] 1 λ Φ(λ, ˆx, ˆξ) Boundary coefficient (for Dirichlet BC) A 1 = π M dˆx R n 1 dˆξ π (n 1)/2Ψ(ˆx, ˆξ) } 33

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