Sharp boundary behavior of eigenvalues for Aharonov-Bohm operators with varying poles

TL;DR

This paper studies how eigenvalues of Aharonov-Bohm operators change sharply as the pole approaches a boundary point, linking the convergence rate to the nodal lines of the limiting eigenfunction.

Contribution

It establishes a precise relation between eigenvalue convergence rates and nodal line counts for Aharonov-Bohm operators near boundary points.

Findings

Eigenvalues converge at a rate determined by nodal lines of the limiting eigenfunction.

Constructs a limit profile depending on the pole's approach direction.

Uses Almgren-type monotonicity to analyze magnetic operators.

Abstract

In this paper, we investigate the behavior of the eigenvalues of a magnetic Aharonov-Bohm operator with half-integer circulation and Dirichlet boundary conditions in a bounded planar domain. We establish a sharp relation between the rate of convergence of the eigenvalues as the singular pole is approaching a boundary point and the number of nodal lines of the eigenfunction of the limiting problem, i.e. of the Dirichlet Laplacian, ending at that point. The proof relies on the construction of a limit profile depending on the direction along which the pole is moving, and on an Almgren-type monotonicity argument for magnetic operators.