Weak Disorder Localization and Lifshitz Tails

 This paper is devoted to the study of localization of discrete random Schrödinger Hamiltonians in the weak disorder regime. Consider an i.i.d. Anderson model and assume that its left spectral edge is 0. Let γ be the coupling constant measuring the strength of the disorder. For γ small, we prove a Lifshitz tail type estimate and use it to derive localization in a band starting at 0 going up to a distance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} of the average of the potential. In this energy region, we show that the localization length at energy E is bounded from above by a constant times the square root of the distance between E and the average of the potential.