A note on Berezin-Toeplitz quantization of the Laplace operator

TL;DR

This paper introduces a self-adjoint operator on endomorphisms of holomorphic sections of a line bundle over a Hodge manifold, which approximates the Laplace operator via Berezin-Toeplitz quantization as the line bundle's power increases.

Contribution

It presents a novel operator that approximates the Laplace operator in the context of Berezin-Toeplitz quantization on Hodge manifolds, with convergence properties as the line bundle's power grows.

Findings

Operator approximates Laplace operator with increasing accuracy

Error tends to zero as polarization line bundle's power increases

Provides a new tool for quantization on Hodge manifolds

Abstract

Given a Hodge manifold, it is introduced a self-adjoint operator on the space of endomorphisms of the global holomorphic sections of the polarization line bundle. Such operator is shown to approximate the Laplace operator on functions when composed with Berezin-Toeplitz quantization map and its adjoint up to an error which tends to zero when taking higher powers of the polarization line bundle.