Lifshitz tails and localization in the three-dimensional Anderson model

Abstract

Consider the three-dimensional Anderson model with a zero mean and bounded independent, identically distributed random potential. Let $\lambda$ be the coupling constant measuring the strength of the disorder, and let $\sigma (E)$ be the self-energy of the model at energy $E$. For any $\epsilon >0$ and sufficiently small $\lambda$, we derive almost-sure localization in the band $E\le -\sigma (0)-{\lambda }^{4-\epsilon }$. In this energy region, we show that the typical correlation length ${\xi }_E$ behaves roughly as $O((|E|-\sigma (E){)}^{-1/2})$, completing the argument outlined in the preprint of T. Spencer [18]