On the magnetic laplacian with a piecewise constant magnetic field in $\mathbb{R}^3_+$

TL;DR

This paper studies the magnetic Laplacian with a piecewise constant magnetic field in three-dimensional half-space, providing insights into spectral properties relevant for understanding superconductors under discontinuous magnetic fields.

Contribution

It reviews recent results on the spectral behavior of the magnetic Laplacian with piecewise constant magnetic fields in 3D half-space, relevant for superconductor models.

Findings

Analysis of the spectrum above the infimum.

Insights into the threshold for the normal state in superconductors.

Understanding the effect of discontinuous magnetic fields on spectral properties.

Abstract

We consider the Neumann realization of the magnetic laplacian in $\mathbb{R}^3_+$, in the case in which the magnetic field has a piecewise constant strength and a uniform direction. This operator is expected to be an effective model in studying the threshold to the normal state for a 3D superconductor exposed to a discontinuous magnetic field. We review some recent results above the infimum of the spectrum of the aforementioned operator.