The weak localization for the alloy-type Anderson model on a cubic lattice

TL;DR

This paper proves Anderson localization for alloy-type random Schrödinger operators with sign-indefinite potentials on cubic lattices in the Lifshitz tails regime, specifically at low energies when the coupling is small.

Contribution

It establishes localization results for a new class of alloy-type models with sign-indefinite potentials, expanding understanding of localization phenomena.

Findings

Localization holds for small coupling constants

Results apply to energies below a certain negative threshold

Extends localization theory to sign-indefinite potentials

Abstract

We consider alloy type random Schr\"odinger operators on a cubic lattice whose randomness is generated by the sign-indefinite single-site potential. We derive Anderson localization for this class of models in the Lifshitz tails regime, i.e. when the coupling parameter $\lambda$ is small, for the energies $E \le -C \lambda^2$.