Persistence of Anderson Localization in Schroedinger Operators With Decaying Random Potentials

Abstract

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.

Schrödinger operators with decaying randomness - Pure point spectrum

“…The results presented in this work are not too surprising considering similar result do exists in discrete case, for example [9,14]. Figotin-Germinet-Klein-Müller [8] considered the similar model with compactly supported µ with absolutely continuous single site potential. The authors showed that the random operator exhibits localization of eigenfunctions almost surely and dynamical localization below zero.…”

“…There are many works which have dealt with this question, for example [8,9,14,[27][28][29]. The results presented in this work are not too surprising considering similar result do exists in discrete case, for example [9,14].…”

Some Estimates Regarding Integrated Density of States for Random Schrödinger Operator with Decaying Random Potentials

“…In [9], Figotin-Germinet-Klein-Müller studied the Anderson Model on L 2 (R d ) with decaying random potentials given by…”