Localization for random Schr\"odinger operators with low density potentials

TL;DR

This paper demonstrates that low-density random potentials in Schr"odinger operators lead to Lifschitz behavior and localization near the spectrum's bottom, applicable to both discrete and continuous models with Bernoulli or Poisson potentials.

Contribution

It establishes Lifschitz behavior and localization for low-density random Schr"odinger operators, extending results to continuous models and specific potential distributions.

Findings

Lifschitz behavior occurs near the spectrum's bottom at low densities.

Localization is proven for energies within a certain low-density zone.

Results apply to both discrete and continuous Schr"odinger operators with Bernoulli or Poisson potentials.

Abstract

We prove that, for a density of disorder $\rho$ small enough, a certain class of discrete random Schr\"odinger operators on $\Z^d$ with diluted potentials exhibits a Lifschitz behaviour from the bottom of the spectrum up to energies at a distance of the order $\rho^\alpha$ from the bottom of the spectrum, with $\alpha>2(d+1)/d$. This leads to localization for the energies in this zone for these low density models. The same results hold for operators on the continuous, and in particular, with Bernoulli or Poisson random potential.