Magnetic Neumann Laplacian on a sharp cone

TL;DR

This paper investigates the spectral properties of the magnetic Laplacian with Neumann boundary conditions on a sharp cone, analyzing how magnetic field orientation affects eigenvalues and eigenfunctions, especially as the cone becomes very narrow.

Contribution

It provides a detailed spectral analysis including asymptotic expansions of eigenvalues and eigenfunctions for the magnetic Laplacian on a cone, considering various magnetic field orientations.

Findings

Existence of discrete spectrum below the essential spectrum as cone narrows

Full asymptotic expansion for eigenvalues and eigenfunctions

Influence of magnetic field orientation on spectral properties

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.