Bifurcation of double eigenvalues for Aharonov-Bohm operators with a moving pole

TL;DR

This paper investigates how double eigenvalues of Aharonov-Bohm operators change as the pole moves near the origin, revealing bifurcation phenomena especially in symmetric domains like the disk.

Contribution

It provides a detailed analysis of eigenvalue bifurcation for Aharonov-Bohm operators with moving poles, including explicit results for symmetric domains such as the disk.

Findings

Bifurcation occurs when the pole moves along certain straight lines near the origin.

In symmetric domains, eigenvalues can be double at the center and bifurcate into two branches nearby.

Explicit characterization of bifurcation behavior in the disk case.

Abstract

We study double eigenvalues of Aharonov-Bohm operators with Dirichlet boundary conditions in planar domains containing the origin. We focus on the behavior of double eigenvalues when the potential's circulation is a fixed half-integer number and the operator's pole is moving on straight lines in a neighborhood of the origin. We prove that bifurcation occurs if the pole is moving along straight lines in a certain number of cones with positive measure. More precise information is given for symmetric domains; in particular, in the special case of the disk, any eigenvalue is double if the pole is located at the centre, but there exists a whole neighborhood where it bifurcates into two distinct branches.