Dependence of the Density of States on the Probability Distribution for Discrete Random Schrödinger Operators

Abstract

We prove that the density of states measure (DOSm) for random Schrödinger operators on Zd is weak-∗ Hölder-continuous in the probability measure. The framework we develop is general enough to extend to a wide range of discrete, random operators, including the Anderson model on the Bethe lattice, as well as random Schrödinger operators on the strip. An immediate application of our main result provides quantitive continuity estimates for the disorder dependence of the DOSm and the integrated density of states (IDS) in the weak disorder regime. These results hold for a general compactly supported single-site probability measure, without any further assumptions. The few previously available results for the disorder dependence of the IDS valid for dimensions d⩾2 assumed absolute continuity of the single-site measure and thus excluded the Bernoulli–Anderson model. As a further application of our main result, we establish quantitative continuity results for the Lyapunov exponent of random Schrödinger operators for d=1 in the probability measure with respect to the weak-∗ topology.

Publisher

Oxford University Press

Publication Date

9-1-2020

Publication Title

International Mathematics Research Notices (IMRN)

Department

Mathematics

Document Type

Article

DOI

https://dx.doi.org/10.1093/imrn/rny156

Language

English

Format

text

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