Persistence of Anderson localization in Schrödinger operators with decaying random potentials

• Autores: Alexander Figotin, François Germinet, Abel Klein, Peter Müller
• Localización: Arkiv för matematik, ISSN 0004-2080, Vol. 45, Nº 1, 2007, págs. 15-30
• Idioma: inglés
• DOI: 10.1007/s11512-006-0039-0
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• Resumen

  • 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|-a at infinity, we determine the number of bound states below a given energy E<0, asymptotically as a?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 a; (b) dynamical localization holds uniformly in a.