The Fate of Lifshitz Tails in Magnetic Fields

TL;DR

This paper studies how magnetic fields influence the low-energy behavior of the integrated density of states for Schrödinger operators with random potentials, revealing that magnetic fields preserve classical asymptotics.

Contribution

It demonstrates that magnetic fields alter the asymptotic behavior of the integrated density of states in the low-energy limit for Poisson random potentials.

Findings

Classical asymptotics hold for small energies with magnetic fields.

The integrated density of states behaves similarly across Landau levels.

Contrasts with delta-function impurity potential cases at the lowest Landau level.

Abstract

We investigate the integrated density of states of the Schr\"odinger operator in the Euclidean plane with a perpendicular constant magnetic field and a random potential. For a Poisson random potential with a non-negative algebraically decaying single-impurity potential we prove that the leading asymptotic behaviour for small energies is always given by the corresponding classical result in contrast to the case of vanishing magnetic field. We also show that the integrated density of states of the operator restricted to the eigenspace of any Landau level exhibits the same behaviour. For the lowest Landau level, this is in sharp contrast to the case of a Poisson random potential with a delta-function impurity potential.