The spectral density function of the renormalized Bochner Laplacian on a symplectic manifold

TL;DR

This paper derives an explicit local formula for the spectral density function of the renormalized Bochner Laplacian on tensor powers of a line bundle over a compact symplectic manifold, linking it to geometric coefficients.

Contribution

It provides a new explicit local formula for the spectral density function in terms of metric and symplectic form coefficients.

Findings

Explicit local formula for spectral density function derived

Formula expressed in terms of Riemannian metric and symplectic form coefficients

Advances understanding of spectral properties of the Bochner Laplacian on symplectic manifolds

Abstract

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.