Magnetic Neumann problems with Aharonov-Bohm potentials: boundary asymptotics of eigenvalues and splitting phenomena

TL;DR

This paper analyzes the asymptotic behavior of eigenvalues for a magnetic Schrödinger operator with Aharonov-Bohm potential under Neumann boundary conditions, revealing convergence and splitting phenomena as the pole approaches the boundary.

Contribution

It provides a detailed boundary asymptotic analysis of eigenvalues and describes eigenvalue splitting phenomena in the presence of Aharonov-Bohm potentials.

Findings

Eigenvalues converge to Neumann Laplacian eigenvalues as the pole approaches the boundary.

The eigenvalue variation exhibits a logarithmic vanishing rate.

Multiple eigenvalues split into separate branches near the boundary.

Abstract

We study a planar magnetic Schr\"odinger operator with an Aharonov-Bohm vector potential, under Neumann boundary conditions. Through a gauge transformation, the corresponding eigenvalue problem can be formulated in terms of the Laplacian on a fractured domain, where the fracture lies along the segment connecting the pole to its projection on the boundary. As the pole approaches the boundary, we prove that the eigenvalues converge to those of the Neumann Laplacian and the variation exhibits a logarithmic vanishing rate. In the case of multiple eigenvalues, when the pole approaches a fixed point of the boundary, we observe a splitting phenomenon, with the largest branch separating from the others.