Spectral Localization by Gaussian Random Potentials in Multi-Dimensional Continuous Space

TL;DR

This paper proves that quantum particles in multi-dimensional space with certain Gaussian random potentials almost surely have a pure-point energy spectrum, especially at negative energies or weak disorder, using a multi-scale analysis approach.

Contribution

It provides a rigorous mathematical proof of spectral localization for particles in multi-dimensional Euclidean space with Gaussian random potentials, extending previous results to a broader class of potentials.

Findings

Spectrum is almost surely pure point at sufficiently negative energies.

Spectrum is pure point at negative energies with weak disorder.

Proof employs a fixed-energy multi-scale analysis technique.

Abstract

A detailed mathematical proof is given that the energy spectrum of a non-relativistic quantum particle in multi-dimensional Euclidean space under the influence of suitable random potentials has almost surely a pure-point component. The result applies in particular to a certain class of zero-mean Gaussian random potentials, which are homogeneous with respect to Euclidean translations. More precisely, for these Gaussian random potentials the spectrum is almost surely only pure point at sufficiently negative energies or, at negative energies, for sufficiently weak disorder. The proof is based on a fixed-energy multi-scale analysis which allows for different random potentials on different length scales.