Magnetic transport in a straight parabolic channel

TL;DR

This paper investigates the spectral properties of a charged particle in a parabolic channel under magnetic and potential perturbations, revealing conditions for absolute continuity and spectral gaps.

Contribution

It provides new results on the absolute continuity of the spectrum and the existence of spectral gaps under various perturbation conditions.

Findings

Absolute continuity at the spectrum's bottom under periodic perturbations.

Any finite number of spectral gaps can occur with strong confinement.

Persistence of absolutely continuous spectrum under weak localization conditions.

Abstract

We study a charged two-dimensional particle confined to a straight parabolic-potential channel and exposed to a homogeneous magnetic field under influence of a potential perturbation $W$. If $W$ is bounded and periodic along the channel, a perturbative argument yields the absolute continuity of the bottom of the spectrum. We show it can have any finite number of open gaps provided the confining potential is sufficiently strong. However, if $W$ depends on the periodic variable only, we prove by Thomas argument that the whole spectrum is absolutely continuous, irrespectively of the size of the perturbation. On the other hand, if $W$ is small and satisfies a weak localization condition in the the longitudinal direction, we prove by Mourre method that a part of the absolutely continuous spectrum persists.