On spectral properties of translationally invariant magnetic Schr\"odinger operators

TL;DR

This paper investigates the spectral properties of translationally invariant magnetic Schrödinger operators, showing the spectrum is absolutely continuous and analyzing the long-time behavior of quantum states, revealing localization and directional propagation characteristics.

Contribution

It provides a detailed spectral analysis of magnetic Schrödinger operators with translational invariance and characterizes the long-term quantum dynamics in such fields.

Findings

Spectrum of the operator is absolutely continuous.

Quantum particles remain localized orthogonal to the potential's direction.

Propagation along the potential's direction is governed by group velocities.

Abstract

We consider a class of translationally invariant magnetic fields such that the corresponding potential has a constant direction. Our goal is to study basic spectral properties of the Schr\"odinger operator ${\bf H}$ with such a potential. In particular, we show that the spectrum of ${\bf H}$ is absolutely continuous and we find its location. Then we study the long-time behaviour of solutions $\exp(-i {\bf H} t)f$ of the time dependent Schr\"odinger equation. It turnes out that a quantum particle remains localized in the plane orthogonal to the direction of the potential. Its propagation in this direction is determined by group velocities. It is to a some extent similar to a evolution of a one-dimensional free particle but "exits" to $+\infty$ and $-\infty$ might be essentially different.