On Aharonov-Bohm operators with multiple colliding poles of any circulation

TL;DR

This paper investigates the spectral stability of Aharonov-Bohm operators with multiple colliding poles of any circulation, deriving asymptotic eigenvalue behavior and providing sharp estimates through a detailed analytical framework.

Contribution

It introduces a new formulation of the eigenvalue problem for colliding poles with arbitrary circulation and characterizes the dominant asymptotic term in eigenvalue variations.

Findings

Asymptotic expansion of eigenvalues near pole collisions

Characterization of the dominant term via an energy functional

Sharp eigenvalue variation estimates in specific configurations

Abstract

This paper deals with quantitative spectral stability for Aharonov-Bohm operators with many colliding poles of whichever circulation. An equivalent formulation of the eigenvalue problem is derived as a system of two equations with real coefficients, coupled through prescribed jumps of the unknowns and their normal derivatives across the segments joining the poles with the collision point. Under the assumption that the sum of all circulations is not integer, the dominant term in the asymptotic expansion for eigenvalues is characterized in terms of the minimum of an energy functional associated with the configuration of poles. Estimates of the order of vanishing of the eigenvalue variation are then deduced from a blow-up analysis, yielding sharp asymptotics in some particular examples.