Magnetic Schr\"{o}dinger operator with the potential supported in a curved two-dimensional strip

TL;DR

This paper studies a magnetic Schrödinger operator with a potential supported in a curved strip, showing that the magnetic field does not alter the essential spectrum and identifying conditions for the absence of discrete spectrum.

Contribution

It demonstrates that the magnetic field does not affect the essential spectrum and provides conditions under which the discrete spectrum is empty for the operator.

Findings

Magnetic field does not change the essential spectrum.

Conditions for the discrete spectrum to be empty are established.

Analysis of the operator on a curved strip with local deformation.

Abstract

We consider the magnetic Schr\"odinger operator $H=(i \nabla +A)^2- V$ with a non-negative potential $V$ supported over a strip which is a local deformation of a straight one, and the magnetic field $B:=\mathrm{rot}(A)$ is assumed to be nonzero and local. We show that the magnetic field does not change the essential spectrum of this system, and investigate a sufficient condition for the discrete spectrum of $H$ to be empty.