Multiple Aharonov--Bohm eigenvalues: the case of the first eigenvalue on the disk

TL;DR

This paper proves that the first eigenvalue of Aharonov--Bohm operators in a disk is simple when the pole is not at the origin, clarifying the eigenvalue's behavior depending on pole location.

Contribution

It demonstrates that the first eigenvalue is simple for poles away from the origin, resolving a known double eigenvalue case for the disk.

Findings

First eigenvalue is simple when pole is not at the origin

Eigenvalue multiplicity depends on pole position

Clarifies eigenvalue behavior for Aharonov--Bohm operators

Abstract

It is known that the first eigenvalue for Aharonov--Bohm operators with half-integer circulation in the unit disk is double if the potential's pole is located at the origin. We prove that in fact it is simple as the pole $a\neq 0$.