Eigenvalue variation under moving mixed Dirichlet-Neumann boundary conditions and applications

TL;DR

This paper investigates how eigenvalues of elliptic operators change under moving mixed boundary conditions, providing explicit formulas for simple eigenvalues and exploring implications for Aharonov-Bohm eigenvalues with coalescing poles.

Contribution

It offers explicit asymptotic formulas for eigenvalue variations under boundary condition changes and applies these results to Aharonov-Bohm eigenvalues with coalescing poles.

Findings

Explicit constants for eigenvalue asymptotics with simple eigenvalues.

Insights into eigenvalue behavior in Aharonov-Bohm settings with singularities.

Enhanced understanding of spectral stability under boundary condition variations.

Abstract

We deal with the sharp asymptotic behaviour of eigenvalues of elliptic operators with varying mixed Dirichlet-Neumann boundary conditions. In case of simple eigenvalues, we compute explicitly the constant appearing in front of the expansion's leading term. This allows inferring some remarkable consequences for Aharonov-Bohm eigenvalues when the singular part of the operator has two coalescing poles.