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

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.