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

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

2005-03-28

arxiv.org/abs/math-ph/0503064v3

Int. Math. Res. Notices, 2005:54, 3341-3373 (2005).

Physics

Mathematical Physics

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

Scientific paper

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 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.

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