Persistence of Anderson localization in Schr\"odinger operators with decaying random potentials

TL;DR

This paper demonstrates that Anderson and dynamical localization persist in Schrödinger operators with decaying random potentials, showing the existence of infinitely many bound states and uniform localization properties under certain decay conditions.

Contribution

It establishes the persistence of Anderson localization in Schrödinger operators with non-positive, decaying random potentials, including asymptotic bounds on bound states and uniform localization results.

Findings

Infinitely many eigenvalues below zero for slow decay potentials

Exponential localization of bound states with uniform localization length

Dynamical localization holds uniformly across decay exponents

Abstract

We show persistence of both Anderson and dynamical localization in Schr\"odinger 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 $\alpha$.