On the semiclassical Laplacian with magnetic field having self-intersecting zero set

TL;DR

This paper analyzes the spectral properties of a 2D magnetic Laplacian with a magnetic field that vanishes along a self-intersecting curve, revealing new decay scales and eigenvector concentration near crossing points as the semiclassical parameter approaches zero.

Contribution

It introduces a detailed spectral analysis of the magnetic Laplacian with a self-intersecting zero set, highlighting the effects of crossing points on eigenvalues and eigenfunctions.

Findings

Crossing points act as potential wells affecting eigenvalue decay.

Eigenvectors exponentially concentrate around crossing points.

A new decay scale of h^{3/2} for the lowest eigenvalues is identified.

Abstract

This paper is devoted to the spectral analysis of the Neumann realization of the 2D magnetic Laplacian with semiclassical parameter h > 0 in the case when the magnetic field vanishes along a smooth curve which crosses itself inside a bounded domain. We investigate the behavior of its eigenpairs in the limit h $\rightarrow$ 0. We show that each crossing point acts as a potential well, generating a new decay scale of h 3/2 for the lowest eigenvalues, as well as exponential concentration for eigenvectors around the set of crossing points. These properties are consequences of the nature of associated model problems in R 2 for which the zero set of the magnetic field is the union of two straight lines. In this paper we also analyze the spectrum of model problems when the angle between the two straight lines tends to 0.