Upper bounds on the density of states of single Landau levels broadened by Gaussian random potentials

TL;DR

This paper establishes upper bounds on the density of states for a single Landau level affected by Gaussian random potentials, showing the spectrum is almost surely continuous and not discrete.

Contribution

It provides rigorous upper bounds on the density of states for Landau levels under Gaussian random potentials, demonstrating absolute continuity of the spectrum.

Findings

Restricted density of states is absolutely continuous.

Explicit upper bounds on the density of states derivative.

Energy levels are almost surely not eigenvalues.

Abstract

We study a non-relativistic charged particle on the Euclidean plane R^2 subject to a perpendicular constant magnetic field and an R^2-homogeneous random potential in the approximation that the corresponding random Landau Hamiltonian on the Hilbert space L^2(R^2) is restricted to the eigenspace of a single but arbitrary Landau level. For a wide class of Gaussian random potentials we rigorously prove that the associated restricted integrated density of states is absolutely continuous with respect to the Lebesgue measure. We construct explicit upper bounds on the resulting derivative, the restricted density of states. As a consequence, any given energy is seen to be almost surely not an eigenvalue of the restricted random Landau Hamiltonian.