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International Mathematics Research Notices (2005) 2005:3341-3373, doi:10.1155/IMRN.2005.3341
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Copyright © 2005 Hindawi Publishing Corporation. All rights reserved.

Localization lengths for Schrödinger operators on Z2 with decaying random potentials

Thomas Chen

We study a class of Schrödinger operators on Z2 with a random potential decaying as |x|–{sigma}, 0<{sigma}≤1/2, in the limit of small disorder strength {lambda}. For the critical exponent {sigma}=1/2, we prove that the localization length of eigenfunctions is bounded below by Formula, while for 0<{sigma}<1/2, the lower bound is {lambda}–(2–{eta})/(1–2{sigma}), for any {eta}>0. These estimates "interpolate" between the lower bound {lambda}–2+{eta} due to recent work of Schlag-Shubin-Wolff for {sigma}=0, and pure a.c. spectrum for {sigma}>1/2 demonstrated in recent work of Bourgain.


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