Spectral transitions for Aharonov-Bohm Laplacians on conical layers

TL;DR

This paper studies how a magnetic flux affects the spectral properties of the Laplace operator near conical surfaces, revealing a critical flux value that causes a sudden change in the spectrum and providing detailed asymptotic analysis.

Contribution

It introduces the concept of a critical magnetic flux for conical layers and derives Hardy inequalities and spectral asymptotics related to this transition.

Findings

Existence of a critical flux causing spectral transition.

Hardy-type inequality at the critical flux.

Sharp spectral asymptotics with refined estimates.

Abstract

We consider the Laplace operator in a tubular neighbourhood of a conical surface of revolution, subject to an Aharonov-Bohm magnetic field supported on the axis of symmetry and Dirichlet boundary conditions on the boundary of the domain. We show that there exists a critical total magnetic flux depending on the aperture of the conical surface for which the system undergoes an abrupt spectral transition from infinitely many eigenvalues below the essential spectrum to an empty discrete spectrum. For the critical flux we establish a Hardy-type inequality. In the regime with infinite discrete spectrum we obtain sharp spectral asymptotics with refined estimate of the remainder and investigate the dependence of the eigenvalues on the aperture of the surface and the flux of the magnetic field.