Magnetic Neumann Laplacian on a sharp cone

Abstract

This paper is devoted to the spectral analysis of the Laplacian with constant magnetic field on a cone of aperture $\alpha$ and Neumann boundary condition. We analyze the influence of the orientation of the magnetic field. In particular, for any orientation of the magnetic field, we prove the existence of discrete spectrum below the essential spectrum in the limit $\alpha\to 0$ and establish a full asymptotic expansion for the $n$-th eigenvalue and the $n$-th eigenfunction.