Discrete spectrum of the magnetic Laplacian on almost flat magnetic barriers

TL;DR

This paper demonstrates that slight irregularities in magnetic barriers can induce bound states in the magnetic Laplacian, revealing new spectral properties related to almost flat magnetic step fields.

Contribution

It introduces a novel effective operator framework for magnetic Laplacians with step fields on nearly flat barriers, highlighting the impact of non-smoothness on eigenvalues.

Findings

Existence of bound states below the essential spectrum due to non-smoothness.

Construction of an effective operator for magnetic step fields.

Illustrative example demonstrating the emergence of eigenvalues.

Abstract

The magnetic Laplacian with a step magnetic field has been intensively studied during the last years. We adapt the construction introduced by Bonnaillie-No\"el, Fournais, Kachmar and Raymond to prove the existence of bound states of a new effective operator involving a magnetic step field on a domain with an almost flat magnetic barrier. This result emphasizes the fact that even a small non-smoothness of the discontinuity region can cause the appearance of eigenvalues below the essential spectrum. We also give an example where this effective operator arises.