05-117 Thomas Chen

Localization lengths for Schr\"odinger operators on Z^2 with decaying random potentials (104K, AMS Latex, 26 pages, 1 Figure.) Mar 25, 05

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Abstract. We study a class of Schr\"odinger operators on $\Z^2$ 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 $2^{\lambda^{-\frac14+\eta}}$, while for $0<\dex<\frac12$, the lower bound is $\lambda^{-\frac{2-\eta}{1-2\dex}}$, for any $\eta>0$. These estimates "interpolate" between recent results of Bourgain on the one hand, and of Schlag-Shubin-Wolff on the other hand.

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