Persistence of Anderson localization in Schrödinger operators with decaying random potentials

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 $|x{|}^{-\alpha }$ 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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Cited by in corpus (3)

• One-dimensional Discrete Dirac Operators in a Decaying Random Potential I: Spectrum and Dynamics
• Dynamical Localization for the One-dimensional Continuum Anderson Model in a Decaying Random Potential
• Localization and unique continuation for non-stationary Schrödinger operators on the 2D lattice