Abstract
We show persistence of both Anderson and dynamical localization in Schrödinger operators with non-positive (attractive) random decaying potential. We consider an Anderson-type Schrödinger 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|-α at infinity, we determine the number of bound states below a given energy E<0, asymptotically as α↓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 α; (b) dynamical localization holds uniformly in α.
Citation
Alexander Figotin. François Germinet. Abel Klein. Peter Müller. "Persistence of Anderson localization in Schrödinger operators with decaying random potentials." Ark. Mat. 45 (1) 15 - 30, April 2007. https://doi.org/10.1007/s11512-006-0039-0
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