 05117 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 $\lambda^{\frac{2\eta}{12\dex}}$, for any $\eta>0$.
These estimates "interpolate" between recent results of Bourgain on
the one hand, and of SchlagShubinWolff on the other hand.
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