Spectral and dynamical properties of random models with nonlocal and singular interactions

Abstract

We give a spectral and dynamical description of certain models of random Schrödinger operators on L 2 (R d ) for which a modified version of the fractional moment method of Aizenman and Molchanov [3] can be applied. One family of models includes Schrödinger operators with random nonlocal interactions constructed from multidimensional wavelet bases. The second family includes Schrödinger operators with random singular interactions randomly located on sublattices of Z d , for d = 1, 2, 3. We prove that these models are amenable to Aizenman-Molchanov-type analysis of the Green's function, thereby eliminating the use of multiscale analysis. The basic technical result is an estimate on the expectation of fractional moments of the Green's function. Among our results, we prove a Wegner estimate, Hölder continuity of the integrated density of states, and spectral and Hilbert-Schmidt dynamical localization at negative energies.

Transport in the random Kronig-Penney model

“…We are not aware of any publication in which this is actually proved for model (1), but the general methods 15 can be applied for this model (details will be given elsewhere, see, however, Ref. 6 where localization at the bottom of the bottom of the spectrum is proved).…”

“…Formula (6) shows that the density of states conserves a one-dimensional van Hove singularity at E l (even though the model is random). This is due to the fact that the random potential cannot move any eigenvalue from below E l to above E l and vice versa 11 (this follows from a basic SturmLiouville arguement, see Sec.…”

An introduction to the mathematics of Anderson localization

“…Here we will follow the works [2] and [12], where these questions were settled. Earlier work in [26] extended certain aspects of the fractional moment method to continuum models, but still relied on finite-rank perturbation arguments by, for example, considering continuum models with random point interactions.…”

Absolutely continuous spectrum and spectral transition for some continuous random operators

“…In this paper we consider a two classes of random potentials and show the absence of point spectrum for the corresponding random Schrödinger operators for large energies. We are motivated by the models considered by Rodnianski-Schlag [18] and those by Hislop-Kirsch-Krishna [10].…”