A 3D-Schroedinger operator under magnetic steps with semiclassical applications

TL;DR

This paper investigates a Schrödinger operator with a discontinuous magnetic field in a half-space, exploring spectral properties and semiclassical localization relevant to superconductivity applications.

Contribution

It introduces a new model for the Schrödinger operator with magnetic steps and analyzes its spectral infimum, linking it to semiclassical problems in bounded domains.

Findings

Spectral infimum can be an eigenvalue under certain magnetic conditions.

Localization of ground states near magnetic discontinuities is characterized.

Provides conditions for eigenvalues in a magnetic step setting.

Abstract

We define a Schr\"odinger operator on the half-space with a discontinuous magnetic field having a piecewise-constant strength and a uniform direction. Motivated by applications in the theory of superconductivity, we study the infimum of the spectrum of the operator. We give sufficient conditions on the strength and the direction of the magnetic field such that the aforementioned infimum is an eigenvalue of a reduced model operator on the half-plane. We use the Schr\"odinger operator on the half-space to study a new semiclassical problem in bounded domains of the space, considering a magnetic Neumann Laplacian with a piecewise-constant magnetic field. We then make precise the localization of the semiclassical ground state near specific points at the discontinuity jump of the magnetic field.