Density of States of an Electron in a Gaussian Random Potential for (4-epsilon)-dimensional Space

TL;DR

This paper calculates the density of states for an electron in a Gaussian random potential in a space of dimension 4 minus epsilon, considering the entire energy range and including effects near the mobility edge, using perturbation theory.

Contribution

It introduces a perturbative approach to compute the density of states in non-integer dimensions close to four, accounting for all orders of 1/epsilon in the expansion.

Findings

Derived expressions for density of states near the mobility edge.

Identified the significance of all powers of 1/epsilon for large N.

Provided a systematic expansion in 1/epsilon for the density of states.

Abstract

The density of states for the Schroedinger equation with a Gaussian random potential is calculated in a space of dimension d=4-epsilon in the entire energy range including the vicinity of a mobility edge. Leading terms in 1/epsilon are taken into account for N \sim 1 (N is an order of perturbation theory) while all powers of 1/epsilon are essential for N>>1 with calculation of the expansion coefficients in the leading order in N.