On the leading term of the eigenvalue variation for Aharonov-Bohm operators with a moving pole

TL;DR

This paper investigates how eigenvalues of Aharonov-Bohm operators change as the magnetic pole moves within a domain, identifying the leading term in their Taylor expansion as a harmonic polynomial.

Contribution

It provides a precise characterization of the leading term in the eigenvalue variation, including explicit coefficients, for Aharonov-Bohm operators with a moving pole.

Findings

The leading term is a harmonic homogeneous polynomial.

Explicit coefficients of the polynomial are determined.

The analysis applies to operators with half-integer circulation.

Abstract

We study the behavior of eigenvalues for magnetic Aharonov-Bohm operators with half-integer circulation and Dirichlet boundary conditions in a planar domain. We analyse the leading term in the Taylor expansion of the eigenvalue function as the pole moves in the interior of the domain, proving that it is a harmonic homogeneous polynomial and detecting its exact coefficients.