Spectral asymptotics for the semiclassical Bochner Laplacian of a line bundle with constant rank curvature

TL;DR

This paper studies the spectral behavior of the semiclassical Bochner Laplacian on line bundles with constant curvature, connecting it to magnetic Laplacians and deriving asymptotic results in the semiclassical limit.

Contribution

It provides a link between the spectral properties of the Bochner Laplacian and magnetic Laplacians, using Agmon estimates to transfer known asymptotics.

Findings

Spectral asymptotics for the Bochner Laplacian are derived from magnetic Laplacian results.

Agmon-like estimates are used to localize the spectral analysis.

The work bridges geometric analysis and spectral theory in the semiclassical regime.

Abstract

The goal of this paper is manyfold. Firstly, we want to give a short introduction to the Bochner Laplacian and explain why it acts locally as a magnetic Laplacian. Secondly, given a confining magnetic field, we use Agmon-like estimates to reduce its spectral study to magnetic Laplacians, in the semiclassical limit. Finally, we use this to translate already-known spectral asymptotics for the magnetic Laplacian to the Bochner Laplacian.