Dependence of the Density of States on the Probability Distribution. Part II: Schrödinger Operators on $$\pmb {\mathbb {R}}^d$$ and Non-compactly Supported Probability Measures

Abstract

We extend our results in [15] on the quantitative continuity properties, with respect to the single-site probability measure, of the density of states measure and the integrated density of states for random Schrödinger operators. For lattice models on Z d , with d 1, we treat the case of non-compactly supported probability measures with finite first moments. For random Schrödinger operators on R d , with d 1, we prove results analogous to those in [15] for compactly supported probability measures. The method of proof makes use of the Combes-Thomas estimate and the Helffer-Sjöstrand formula.

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

“…Finally, we note that an extension of the continuity results of the present article to continuum Schrödinger operators on L 2 (R d ), as well as to discrete models with non-compactly supported single-site measures, is the subject of a follow-up paper [30], see also Remark 1.2 below.…”

“…Indeed, as outlined in section 1.3, the proof of Theorem 1.1 relies on two key steps, the first of which, the "finiterange reduction," uses polynomial approximation and thereby compactness of the support of the probability measure. The results of the present paper are extended to the situation of noncompactly supported single-site probability measures in [30]. In this paper, we will overcome this technical obstacle by working with resolvents instead of polynomial approximations.…”

“…In this paper, we will overcome this technical obstacle by working with resolvents instead of polynomial approximations. In addition, this modified approach, presented in [30], applies to continuum random Schrödinger operators. The latter will however come at a price: the singularity of resolvents near the real axis will have to be compensated by higher regularity of the functions f in Theorem 3.1.…”

On the Continuity of the Integrated Density of States in the Disorder

“…In the follow up paper [6], Hislop and Marx prove a version of their results for the continual Anderson model, which is not of the form (1.10). A modification of our argument can be applied to the continual setting as well.…”

Dependence of the density of states outer measure on the potential for deterministic Schrödinger operators on graphs with applications to ergodic and random models

“…This is the third of a series of papers [11,12] in which we explore the dependence of the density of states measure (DOSm) for Schrödinger operators on the potential. In this article, we extend and refine previous results by considering the density of states outer measure (DOSoM), defined in section 2.3.…”