Localization lengths for Schroedinger operators on Z^2 with decaying random potentials

Abstract

We study a class of Schrödinger operators on ${\mathbb{Z}}^2$ with a random potential decaying as $|x|^{-\dex}$, $0<\dex\leq\frac12$, in the limit of small disorder strength $\lambda$. For the critical exponent $\dex=\frac12$, we prove that the localization length of eigenfunctions is bounded below by $2^{{\lambda }^{-\frac{1}{4}+\eta }}$, while for $0<\dex<\frac12$, the lower bound is $λ^{-\frac{2-η}{1-2\dex}}$, for any $\eta >0$. These estimates "interpolate" between the lower bound ${\lambda }^{-2+\eta }$ due to recent work of Schlag-Shubin-Wolff for $\dex=0$, and pure a.c. spectrum for $\dex>\frac12$ demonstrated in recent work of Bourgain.

AMS Latex, 26 pages, 1 Figure. Final version. To appear in Int. Math. Res. Notices