Quantitative spectral stability for Aharonov-Bohm operators with many coalescing poles

TL;DR

This paper analyzes how eigenvalues of Aharonov-Bohm operators behave as multiple poles coalesce, providing asymptotic expansions and stability insights, especially for configurations with an odd number of poles and two-pole cases.

Contribution

It introduces a gauge transformation linking the problem to Laplacian eigenvalues with cracks and derives precise asymptotics for eigenvalues in coalescing pole scenarios.

Findings

Asymptotic expansion of eigenvalues related to pole configuration

Identification of eigenvalue behavior for odd number of poles

Analysis of two-pole case and eigenvalue variation signs

Abstract

The behavior of simple eigenvalues of Aharonov-Bohm operators with many coalescing poles is discussed. In the case of half-integer circulation, a gauge transformation makes the problem equivalent to an eigenvalue problem for the Laplacian in a domain with straight cracks, laying along the moving directions of poles. For this problem, we obtain an asymptotic expansion for eigenvalues, in which the dominant term is related to the minimum of an energy functional associated with the configuration of poles and defined on a space of functions suitably jumping through the cracks. Concerning configurations with an odd number of poles, an accurate blow-up analysis identifies the exact asymptotic behaviour of eigenvalues and the sign of the variation in some cases. An application to the special case of two poles is also discussed.