The Coherent State Transform and an Application to the Hyperbolic Laplacian

TL;DR

This paper reviews the theory of overfilling families and uses coherent states to establish Weyl asymptotics for Euclidean and hyperbolic Laplacians with Dirichlet boundary conditions.

Contribution

It introduces a novel application of the coherent state transform to derive Weyl asymptotics for Laplacians on different geometries.

Findings

Weyl asymptotics are proven for the Euclidean Laplacian.

Weyl asymptotics are proven for the hyperbolic Laplacian.

The method applies to domains with Dirichlet boundary conditions.

Abstract

We review some basic ideas from the theory of overfilling families and use the special case of coherent states to prove Weyl asymptotics for the euclidean Laplacian and the hyperbolic Laplacian on a domain with Dirichlet boundary conditions.