Semi-classical analysis of the Laplace operator with Robin boundary conditions

TL;DR

This paper derives a detailed asymptotic expansion for Laplacian eigenvalues on bounded domains with Robin boundary conditions using semi-classical analysis, exploring how boundary condition functions influence the spectrum.

Contribution

It introduces a semi-classical framework for analyzing Laplacian eigenvalues with Robin boundary conditions, including three regimes of boundary function dependence.

Findings

Two-term asymptotic expansion for eigenvalue sums

Identification of three regimes based on boundary function dependence

Framework applicable to various boundary conditions

Abstract

We prove a two-term asymptotic expansion of eigenvalue sums of the Laplacian on a bounded domain with Neumann, or more generally, Robin boundary conditions. We formulate and prove the asymptotics in terms of semi-classical analysis. In this reformulation it is natural to allow the function describing the boundary conditions to depend on the semi-classical parameter and we identify and analyze three different regimes for this dependence.