The Spectral Density Function of the Renormalized Bochner Laplacian on a Symplectic Manifold
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Journal of Mathematical Sciences, Vol. 251, No. 5, December, 2020
THE SPECTRAL DENSITY FUNCTION OF THE RENORMALIZED BOCHNER LAPLACIAN ON A SYMPLECTIC MANIFOLD Yu. A. Kordyukov Institute of Mathematics, UFRC RAS 112, Chernyshevskii St., Ufa 450008, Russia Novosibirsk State University 1, Pirogova St., Novosibirsk 630090, Russia [email protected]
UDC 517.9
We consider the renormalized Bochner Laplacian acting on tensor powers of a positive line bundle on a compact symplectic manifold. We derive an explicit local formula for the spectral density function in terms of coefficients of the Riemannian metric and symplectic form. Bibliography: 15 titles.
1
Introduction
1.1. Preliminaries. Let (X, B) be a compact symplectic manifold of dimension 2n. We assume that there exists a Hermitian line bundle (L, hL ) on X with a Hermitian connection ∇L : C ∞ (X, L) → C ∞ (X, T ∗ X ⊗ L), which satisfies the pre-quantization condition: iRL = B,
(1.1)
where RL = (∇L )2 is the curvature of the connection. Thus, [B] ∈ H 2 (X, 2πZ). Let g be a Riemannian metric on X. For any p ∈ N we denote by Lp := L⊗p the pth tensor p power of L. We consider the induced Bochner Laplacian ΔL acting on C ∞ (X, Lp ) by p ∗ p p ΔL = ∇L ∇L , (1.2) p
where ∇L : C ∞ (X, Lp ) → C ∞ (X, T ∗ X ⊗ Lp ) is the connection on Lp induced by ∇L , and p p (∇L )∗ : C ∞ (X, T ∗ X ⊗ Lp ) → C ∞ (X, Lp ) is the formal adjoint of ∇L . In the case where (L, hL ) is the trivial Hermitian line bundle, the Hermitian connection ∇L can be written as ∇L = d − iA with some real-valued 1-form A, and we have RL = −idA,
B = dA.
Thus, B can be considered as a magnetic 2-form and A as the associated magnetic potential. p odinger operator The Bochner Laplacian ΔL is related with the semiclassical magnetic Schr¨ = (id + A)∗ (id + A), HA
>0
Translated from Problemy Matematicheskogo Analiza 106, 2020, pp. 125-138. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2515-0696
696
by the formula p
, ΔL = −2 HA
1 = , p
p ∈ N.
Let B ∈ End(T X) be a skew-adjoint endomorphism such that u, v ∈ T X.
B(u, v) = g(Bu, v),
(1.3)
The renormalized Bochner Laplacian Δp is a second order differential operator acting on C ∞ (X, Lp ) by p Δp = ΔL − pτ, where τ is a smooth function on X given by τ (x) =
1 Tr[(B(x)∗ B(x))1/2 ], 2
x ∈ X.
(1.4)
This operator was introduced in [1]. An almost complex structure J ∈ End(T X) compatible with B and g is defined by J = B(B ∗ B)−1/2 .
(1.5)
We put μ0 =
inf
u∈Tx X,x∈X
Bx (u, J(x)u) . |u|2g
(1.6)
We denote by σ(Δp ) the spectrum of Δp in L2 (X, Lp ). By [2, Corollary 1.2], there exists a constant CL > 0 such that for any p σ(Δp ) ⊂ [−CL , CL ] ∪ [2pμ0 − CL , +∞). We consider the finite-dimensional vector subspace Hp ⊂ L2 (X, Lp ) spanned by the eigensections of Δp corresponding to eigenvalues in [−CL , CL ]. Its dimension dp grows polynomially as p → ∞ (p) [2, Corollary 1.2]. We denote by λj , j = 1, 2, . . . , dp , the eigenvalues of Δp in [−CL , CL ] taken with multiplicities. The spectral density function is a function ρ ∈
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