Spectral stability under removal of small segments

TL;DR

This paper provides sharp asymptotic estimates for how eigenvalues of the Laplacian change when small segments are removed from a domain, extending previous results to non-simple eigenvalues.

Contribution

It advances the understanding of spectral stability by deriving precise asymptotics for eigenvalue variations under segment removal, including non-simple eigenvalues.

Findings

Sharp asymptotic estimates for simple eigenvalues.

Extension of results to non-simple eigenvalues.

Analysis of eigenvalue variation when segments are tangent to nodal lines.

Abstract

In the present paper we deepen the works of L. Abatangelo, V. Felli, L. Hillairet and C. Lena on the asymptotic estimates of the eigenvalue variation under removal of segments from the domain in R2. We get a sharp asymptotic estimate when the eigenvalue is simple and the removed segment is tangent to a nodal line of the associated eigenfunction. Moreover, we extend their results to the case when the eigenvalue is not simple.