Semiclassical trace formula for the Bochner-Schr\"odinger operator

TL;DR

This paper derives a detailed asymptotic expansion for the Schwartz kernel and trace of functions of the semiclassical Bochner-Schr"odinger operator on line bundles over manifolds, as the tensor power parameter grows large.

Contribution

It provides the first complete asymptotic expansion for the Schwartz kernel and trace of spectral functions of the Bochner-Schr"odinger operator in the semiclassical limit.

Findings

Asymptotic expansion of the Schwartz kernel on the diagonal in powers of p^{-1}.

Complete asymptotic expansion for the trace of spectral functions when the manifold is compact.

Results applicable to operators on tensor powers of line bundles over manifolds of bounded geometry.

Abstract

We study the semiclassical Bochner-Schr\"odinger operator $H_{p}=\frac{1}{p^2}\Delta^{L^p\otimes E}+V$ on tensor powers $L^p$ of a Hermitian line bundle $L$ twisted by a Hermitian vector bundle $E$ on a Riemannian manifold of bounded geometry. For any function $\varphi\in C^\infty_c(\mathbb R)$, we consider the bounded linear operator $\varphi(H_p)$ in $L^2(X,L^p\otimes E)$ defined by the spectral theorem. We prove that its smooth Schwartz kernel on the diagonal admits a complete asymptotic expansion in powers of $p^{-1}$ in the semiclassical limit $p\to \infty$. In particular, when the manifold is compact, we get a complete asymptotic expansion for the trace of $\varphi(H_p)$.