Semiclassical spectral analysis of the Bochner-Schr\"odinger operator on symplectic manifolds of bounded geometry

TL;DR

This paper analyzes the spectral properties of the Bochner-Schr"odinger operator on symplectic manifolds with bounded geometry, providing asymptotic descriptions, gap existence conditions, and kernel behavior insights.

Contribution

It offers a novel semiclassical spectral analysis of the operator, including spectrum asymptotics, gap conditions, and kernel expansions on symplectic manifolds.

Findings

Spectrum asymptotics in terms of model operators

Existence of spectral gaps under curvature conditions

Complete asymptotic expansion of spectral projection kernels

Abstract

We study the Bochner-Schr\"odinger operator $H_{p}=\frac 1p\Delta^{L^p\otimes E}+V$ on high tensor powers of a positive line bundle $L$ on a symplectic manifold of bounded geometry. First, we give a rough asymptotic description of its spectrum in terms of the spectra of the model operators. This allows us to prove the existence of gaps in the spectrum under some conditions on the curvature of the line bundle. Then we consider the spectral projection of such an operator corresponding to an interval whose extreme points are not in the spectrum and study asymptotic behavior of its kernel. First, we establish the off-diagonal exponential estimate. Then we state a complete asymptotic expansion in a fixed neighborhood of the diagonal.