Lifshitz tails for alloy type models in a constant magnetic field

TL;DR

This paper investigates the Lifshitz tail behavior of the integrated density of states for a 2D Landau Hamiltonian with a Gaussian-decaying alloy-type potential under a constant magnetic field, revealing a specific exponential decay near Landau levels.

Contribution

It establishes the Lifshitz tail asymptotics for the Landau Hamiltonian with alloy-type perturbations, extending understanding of spectral edge behavior in magnetic quantum systems.

Findings

Lifshitz tails exhibit a $e^{- ext{log}^2|E-2bq|}$ decay at Landau levels.

The Landau level remains a band edge under the perturbation.

The decay rate depends on the Gaussian decay of the potential.

Abstract

In this note, we study Lifshitz tails for a 2D Landau Hamiltonian perturbed by a random alloy-type potential constructed with single site potentials decaying at least at a Gaussian speed. We prove that, if the Landau level stays preserved as a band edge for the perturbed Hamiltonian, at the Landau levels, the integrated density of states has a Lifshitz behavior of the type $e^{-\log^2|E-2bq|}$.