The multiformity of Lifshits tails caused by random Landau Hamiltonians with repulsive impurity potentials of different decay at infinity

TL;DR

This paper investigates how different decay rates of impurity potentials affect the low-energy behavior of the integrated density of states in a quantum system with magnetic fields, revealing diverse Lifshits tail phenomena.

Contribution

It generalizes and unifies previous results by analyzing Lifshits tails for various decay types of impurity potentials in magnetic Landau Hamiltonians.

Findings

Lifshits tails depend on the decay rate of impurity potentials.

Different decay types lead to distinct low-energy asymptotics.

The work extends previous models to a broader class of potentials.

Abstract

For a charged quantum particle in the Euclidean plane subject to a perpendicular constant magnetic field and repulsive impurities, randomly distributed according to Poisson's law, we determine the leading low-energy fall-off of the integrated density of states in case the single-impurity potential has either super-Gaussian or regular sub-Gaussian long-distance decay. The forms of the resulting so-called magnetic Lifshits tails reflect the great variety of these decays. On the whole, we summarize, unify, and generalize results in previous works of K. Broderix, L. Erd\H{o}s, D.\ Hundertmark, W. Kirsch, and ourselves.