Regularity of the Density of States of Random Schrödinger Operators

Abstract

present here. Even for these models the random potentials need to satisfy a complete covering condition. The Anderson model on the lattice for which regularity results were known earlier also satisfies the complete covering condition.

The density of states and local eigenvalue statistics for random band matrices of fixed width

“…These authors proved regularity properties of the density of states for random Schrödinger operators (lattice and continuum) in the localization regime. The proof presented here applies to the random Schrödinger operators on a class of infinite graphs treated by in [13] and extends the results of [13] to probability measures with unbounded support. The method also applies to fixed bandwidth RBM for d = 2, 3 provided certain localization bounds are known.…”

“…The limit function, known to be the semicircle law with a band-width dependent error [4,12,11,15], is identified as the intensity of the limiting Poisson point process. The proof of these results for the density of states relies on a new result that simplifies and extends some of the ideas used by Dolai, Krishna, and Mallick [13]. These authors proved regularity properties of the density of states for random Schrödinger operators (lattice and continuum) in the localization regime.…”

“…Applications to random Schrödinger operators on infinite graphs. A theorem similar to Theorem 1.1 applies to the local eigenvalue statistics and the DOSf for discrete random Schrödinger operators on a wide variety of infinite graphs G, including the lattice Z d , as described in [13] for energies in the complete localization regime Σ CL . The family of graphs G are infinite graphs with a metric d G and with the property that for any…”

“…The regularity part of the following theorem was proven in [13,Theorem 3.4] for compactly-supported probability measures with an additional technical assumption that the k th -derivative of the probability density is Hölder continuous due to a constraint in [13,Theorem 2.2]. The proof of the Poisson statistics for lattices Z d was given in [16].…”

“…The techniques developed in sections 4 and 5 applied to RSO yield the following theorem removing the technical constraint on the probability density and the necessity that the probability density have compact support. We also remark that the random perturbations need not be rank one but may be of uniform finite rank as treated in [13] and [8, chapters 6-7] on the Wegner N-orbital model. We summarize this in the following theorem stated for the rank one case.…”