06-112 A. Figotin, F. Germinet, A. Klein, P. Muller

Persistence of Anderson localization in Schrodinger operators with decaying random potentials (370K, pdf) Apr 9, 06

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Abstract. We show persistence of both Anderson and dynamical localization in Schrodinger operators with non-positive (attractive) random decaying potential. We consider an Anderson-type Schr\"odinger operator with a non-positive ergodic random potential, and multiply the random potential by a decaying envelope function. If the envelope function decays slower than $|x|^{-2}$ at infinity, we prove that the operator has infinitely many eigenvalues below zero. For envelopes decaying as $|x|^{-\alpha}$ at infinity, we determine the number of bound states below a given energy $E<0$, asymptotically as $\alpha\downarrow 0$. To show that bound states located at the bottom of the spectrum are related to the phenomenon of Anderson localization in the corresponding ergodic model, we prove: ~(a)~ these states are exponentially localized with a localization length that is uniform in the decay exponent $\alpha$; (b)~ dynamical localization holds uniformly in $\alpha$.

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