 06112 A. Figotin, F. Germinet, A. Klein, P. Muller
 Persistence of Anderson localization in Schrodinger operators with decaying random potentials
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Apr 9, 06

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Abstract. We show persistence of both Anderson and dynamical localization in
Schrodinger operators with nonpositive (attractive) random
decaying potential. We consider an Andersontype Schr\"odinger
operator with a nonpositive 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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