On multiple eigenvalues for Aharonov-Bohm operators in planar domains

TL;DR

This paper investigates the conditions under which multiple eigenvalues occur for Aharonov-Bohm operators in planar domains, providing theoretical validation for numerical simulations and analyzing the structure of parameter sets that preserve eigenvalue multiplicity.

Contribution

It offers new theoretical insights into the structure of pole-circulation configurations that maintain double eigenvalue multiplicity in Aharonov-Bohm operators.

Findings

Set of pole-circulation pairs can be isolated points under certain conditions

Theoretical results confirm previous numerical simulations

Provides sufficient conditions for eigenvalue multiplicity stability

Abstract

We study multiple eigenvalues of a magnetic Aharonov-Bohm operator with Dirichlet boundary conditions in a planar domain. In particular, we study the structure of the set of the couples position of the pole-circulation which keep fixed the multiplicity of a double eigenvalue of the operator with the pole at the origin and half-integer circulation. We provide sufficient conditions for which this set is made of an isolated point. The result confirms and validates a lot of numerical simulations available in preexisting literature.