On the asymptotics of a Robin eigenvalue problem

TL;DR

This paper investigates the asymptotic behavior of Robin eigenvalues as they diverge to negative infinity, establishing their relation to Dirichlet eigenpairs and providing criteria for their asymptotic characterization.

Contribution

It proves that Dirichlet eigenpairs are the only accumulation points of Robin eigenpairs and offers a full asymptotic description for these eigenvalues as the perturbation parameter approaches zero.

Findings

Robin eigenvalues diverge to -infinity as perturbation vanishes

Dirichlet eigenpairs are the only accumulation points

Full asymptotic expansion of eigenvalues provided

Abstract

The considered Robin problem can formally be seen as a small perturbation of a Dirichlet problem. However, due to the sign of the impedance value, its associated eigenvalues converge point-wise to $-\infty$ as the perturbation goes to zero. We prove that in this case, Dirichlet eigenpairs are the only accumulation points of the Robin eigenpairs with normalized eigenvectors. We then provide a criteria to select accumulating sequences of eigenvalues and eigenvectors and exhibit their full asymptotic with respect to the small parameter.