Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain

TL;DR

This paper provides a new proof for the two-term spectral asymptotics of the Dirichlet Laplacian on bounded domains, including the main and second terms, under weak boundary smoothness conditions.

Contribution

It introduces a novel proof technique that captures both the leading and second-order terms in the spectral asymptotics for the Dirichlet Laplacian.

Findings

Derived the first and second terms of the spectral asymptotics

Extended validity to domains with weak boundary smoothness

Confirmed the surface area as a key geometric term in the second asymptotic

Abstract

Let -\Delta denote the Dirichlet Laplace operator on a bounded open set in \mathbb{R}^d. We study the sum of the negative eigenvalues of the operator -h^2 \Delta - 1 in the semiclassical limit h \to 0+. We give a new proof that yields not only the first term of the asymptotic formula but also the second term involving the surface area of the boundary of the set. The proof is valid under weak smoothness assumptions on the boundary.