Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators

TL;DR

This paper investigates how the eigenvalues of certain differential operators change when small sets are removed, establishing precise relations and applying these results to the spectral analysis of Aharonov-Bohm operators with moving poles.

Contribution

It provides a sharp relation between eigenfunction vanishing order and eigenvalue variation, and applies this to analyze eigenvalue asymptotics of Aharonov-Bohm operators with colliding poles.

Findings

Eigenvalue variation relates to eigenfunction vanishing order.

Asymptotic behavior of Aharonov-Bohm eigenvalues with moving poles.

Spectral stability under removal of small capacity sets.

Abstract

We first establish a sharp relation between the order of vanishing of a Dirichlet eigenfunction at a point and the leading term of the asymptotic expansion of the Dirichlet eigenvalue variation, as a removed compact set concentrates at that point. Then we apply this spectral stability result to the study of the asymptotic behaviour of eigenvalues of Aharonov-Bohm operators with two colliding poles moving on an axis of symmetry of the domain.