Sharp asymptotic estimates for eigenvalues of Aharonov-Bohm operators with varying poles

TL;DR

This paper derives precise asymptotic estimates for eigenvalues of Aharonov-Bohm operators with moving poles, confirming conjectures through advanced monotonicity and blow-up analysis techniques.

Contribution

It provides the first sharp asymptotic formulas for eigenvalues as the pole approaches a nodal line, advancing understanding of magnetic eigenvalue behavior.

Findings

Eigenvalues exhibit sharp asymptotics as the pole approaches a zero of the eigenfunction.

Theoretical confirmation of previously conjectured eigenvalue behaviors.

Application of Almgren-type monotonicity and blow-up analysis to magnetic operators.

Abstract

We investigate the behavior of eigenvalues for a magnetic Aharonov-Bohm operator with half-integer circulation and Dirichlet boundary conditions in a planar domain. We provide sharp asymptotics for eigenvalues as the pole is moving in the interior of the domain, approaching a zero of an eigenfunction of the limiting problem along a nodal line. As a consequence, we verify theoretically some conjectures arising from numerical evidences in preexisting literature. The proof relies on an Almgren-type monotonicity argument for magnetic operators together with a sharp blow-up analysis.