Spectral Boundary of Positive Random Potential in a Strong Magnetic Field

TL;DR

This paper analyzes the spectral boundary behavior of positive random potentials in strong magnetic fields, deriving explicit formulas for the density of states near the spectral edge in two and three dimensions.

Contribution

It provides a novel semiclassical analysis of the Lifshitz tail for positive delta-function scatterers under strong magnetic fields, extending understanding in both 2D and 3D cases.

Findings

Density of states in 2D scales as e^{f-2} near the spectral boundary.

In 3D, the density of states follows a complex exponential involving f and energy.

Explicit formulas for the Lifshitz tail behavior in strong magnetic fields.

Abstract

We consider the problem of randomly distributed positive delta-function scatterers in a strong magnetic field and study the behavior of density of states close to the spectral boundary at $E=\hbar\omega_{c}/2$ in both two and three dimensions. Starting from dimensionally reduced expression of Brezin et al. and using the semiclassical approximation we show that the density of states in the Lifshitz tail at small energies is proportio- nal to $e^{f-2}$ in two dimensions and to $\exp(-3.14f\ln(3.14f/\pi e)/ \sqrt(2me))$ in three dimensions, where $e$ is the energy and $f$ is the density of scatterers in natural units.