TY - JOUR

T1 - Dependence of the density of states on the probability distribution for discrete random schrödinger operators

AU - Hislop, Peter D.

AU - Marx, Christoph A.

N1 - Publisher Copyright:
© The Author(s) 2018.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/9/1

Y1 - 2020/9/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85075348086&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85075348086&partnerID=8YFLogxK

U2 - 10.1093/imrn/rny156

DO - 10.1093/imrn/rny156

M3 - Article

AN - SCOPUS:85075348086

SN - 1073-7928

VL - 2020

SP - 5279

EP - 5341

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 17

ER -