Lifshitz Tails in Constant Magnetic Fields

TL;DR

This paper studies the asymptotic behavior of the integrated density of states for a 2D Landau Hamiltonian with random potential, revealing different decay regimes near spectral edges, including Landau levels.

Contribution

It provides new asymptotic formulas for Lifshitz tails in magnetic fields, considering various decay types of the potential and the position relative to Landau levels.

Findings

Different asymptotic behaviors near Landau levels and away from them.

Formulation of Lifshitz tail asymptotics for various potential decay types.

Extension of known results to magnetic field scenarios with rational flux.

Abstract

We consider the 2D Landau Hamiltonian $H$ perturbed by a random alloy-type potential, and investigate the Lifshitz tails, i.e. the asymptotic behavior of the corresponding integrated density of states (IDS) near the edges in the spectrum of $H$. If a given edge coincides with a Landau level, we obtain different asymptotic formulae for power-like, exponential sub-Gaussian, and super-Gaussian decay of the one-site potential. If the edge is away from the Landau levels, we impose a rational-flux assumption on the magnetic field, consider compactly supported one-site potentials, and formulate a theorem which is analogous to a result obtained in the case of a vanishing magnetic field.