Resolvents of Bochner Laplacians in the semiclassical limit

TL;DR

The paper introduces a new class of pseudodifferential operators to analyze the spectral properties of Bochner Laplacians on line bundles over compact manifolds in the semiclassical limit, revealing detailed spectral information.

Contribution

It defines Heisenberg semiclassical pseudodifferential operators and applies them to study resolvents and spectral projections of Bochner Laplacians with nondegenerate curvature.

Findings

Construction of Heisenberg semiclassical pseudodifferential operators.

Analysis of resolvents and spectral projections of Bochner Laplacians.

Insights into generalized Bergman kernels in the semiclassical limit.

Abstract

We introduce a new class of pseudodifferential operators, called Heisenberg semiclassical pseudodifferential operators, to study the space of sections of a power of a line bundle on a compact manifold, in the limit where the power is large. This class contains the Bochner Laplacian associated with a connection of the line bundle, and when the curvature is nondegenerate, its resolvent and some associated spectral projections, including generalized Bergman kernels.