The Weyl law for the Dirichlet Laplacian

TL;DR

This paper reviews the asymptotic distribution of eigenvalues of the Dirichlet Laplacian, emphasizing spectral quantities and providing a proof via the Fourier Tauberian Theorem.

Contribution

It offers a comprehensive review and a proof of the Weyl law for the Dirichlet Laplacian using Fourier analysis techniques.

Findings

Asymptotic eigenvalue distribution follows Weyl's law

Spectral quantities are systematically introduced and analyzed

Proof based on Fourier Tauberian Theorem confirms the law

Abstract

The purpose of this paper is to review the asymptotic distribution of eigenvalues of the Dirichlet Laplacian. We introduce and recall all the relevant spectral quantities and provide a proof based on the Fourier Tauberian Theorem.