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Magnetic transport due to a translationally invariant potential obstacle

Pavel Exner, David Spitzkopf

Abstract

We consider a two-dimensional system in which a charged particle is exposed to a homogeneous magnetic field perpendicular to the plane and a potential that is translationally invariant in one dimension. We derive several conditions on such a perturbation under which the Landau levels change into an absolutely continuous spectrum.

Magnetic transport due to a translationally invariant potential obstacle

Abstract

We consider a two-dimensional system in which a charged particle is exposed to a homogeneous magnetic field perpendicular to the plane and a potential that is translationally invariant in one dimension. We derive several conditions on such a perturbation under which the Landau levels change into an absolutely continuous spectrum.

Paper Structure

This paper contains 5 sections, 9 theorems, 22 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The model
  3. Some simple properties
  4. More on the absolute continuity
  5. Properties of the spectrum and numerical examples

Key Result

Proposition 2.1

The spectrum of $h_v(p)$ is purely discrete and simple consisting of eigenvalues $\epsilon_n(p),\: n=0,1,2,\dots\,$.

Figures (4)

  • Figure 1: Dispersion curves for approximants of a repulsive $\delta$-interaction; the inset shows the potential shape.
  • Figure 2: Dispersion curves for approximants of an attractive $\delta$-interaction.
  • Figure 3: Dispersion curves for flat-bottom potential well.
  • Figure 4: Dispersion curves for sign-changing potential.

Theorems & Definitions (22)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Example 3.3
  • ...and 12 more