Quantization of the Laplacian operator on vector bundles I

TL;DR

This paper introduces a canonical quantization method for the Laplacian operator on sections of Hermitian endomorphisms of vector bundles over polarized manifolds, approximating eigenvalues and eigenspaces.

Contribution

It provides a new quantization framework for the Laplacian on vector bundles, enabling eigenvalue and eigenspace approximation when the bundle is simple.

Findings

Provides a canonical quantization of the Laplacian

Offers eigenvalue and eigenspace approximation for simple bundles

Enhances understanding of Laplacian spectral properties on vector bundles

Abstract

Let $(E,h)$ be a holomorphic Hermitian vector bundle over a polarized manifold. We provide a canonical quantization of the Laplacian operator acting on sections of the bundle of Hermitian endomorphisms of $E$. If $E$ is simple we obtain an approximation of the eigenvalues and eigenspaces of the Laplacian.