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 Hislop and Marx (Int Math Res Not, 2018. https://doi.org/10.1093/imrn/rny156) 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 $\mathbb {Z}^d$, with $d \geqslant 1$, we treat the case of non-compactly supported probability measures with finite first moments. For random Schrödinger operators on $\mathbb {R}^d$, with $d \geqslant 1$, we prove results analogous to those in Hislop and Marx (2018) for compactly supported probability measures. The method of proof makes use of the Combes–Thomas estimate and the Helffer–Sjöstrand formula.

Appendix: Almost Analytic Extensions and the Helffer–Sjöstrand Functional Calculus

In what follows, we will rely on some aspects of the theory of almost analytic extensions of functions on the real line and the Helffer–Sjöstrand functional calculus. For reference purposes, we briefly summarize some facts here which will be of use to us; for a more detailed and pedagogical account, we refer, e.g., to [8].

Let $\tau \in \mathcal {C}^\infty$ be a fixed bump function on $\mathbb {R}$ with support in $[-2,2]$, satisfying $\tau \equiv 1$ on $[-1,1]$. Set

$$\begin{aligned} \sigma (x,y) : = \tau \left( \dfrac{y}{\langle x \rangle } \right) . \end{aligned}$$

(5.1)

Given a (complex-valued) function $f \in \mathcal {C}_c^\infty (\mathbb {R})$ and an integer $P \in \mathbb {N}$, one can define an almost analytic extension$\widetilde{f}$ of fof degreeP by

$$\begin{aligned} \widetilde{f}(x,y) = \left\{ \sum _{n=0}^P \frac{1}{n !} f^{(n)}(x) (i y)^n \right\} \sigma (x,y). \end{aligned}$$

(5.2)

Then, $\widetilde{f} \equiv f$ on $\mathbb {R}$, $\widetilde{f}$ is compactly supported on $\mathbb {C}$, and a straightforward computation shows that

$$\begin{aligned} \partial _{\overline{z}} \widetilde{f}&:= \frac{1}{2} ( \partial _x + i \partial _y ) \widetilde{f} \nonumber \\&= \frac{1}{2} \left\{ \sum _{n=0}^P \frac{1}{n !} f^{(n)}(x) (i y)^n \right\} (\sigma _x + i \sigma _y) + \frac{1}{2} \frac{1}{P!} f^{(P+1)}(x) (iy)^P \sigma . \end{aligned}$$

(5.3)

In particular, using the properties of $\sigma$, (5.3) implies that $\widetilde{f}$ is almost analytic in a neighborhood of $\mathbb {R}$, in the sense that

$$\begin{aligned} \vert \partial _{\overline{z}} \widetilde{f}(x,y) \vert \leqslant {\Vert f^{(P+1)} \Vert _\infty } ~{\vert y \vert ^P},\ ~ \vert y \vert \leqslant \langle x \rangle . \end{aligned}$$

(5.4)

One important application of almost analytic extensions of functions on $\mathbb {R}$ is an explicit representation of functions of operators via the Helffer–Sjöstrand functional calculus: For every self-adjoint operator H and $f \in \mathcal {C}_c(\mathbb {R})$, one has the representation

$$\begin{aligned} f(H) = \dfrac{1}{\pi } \int \int _\mathbb {C} \partial _{\overline{z}} \widetilde{f}(x,y) \cdot (H - z)^{-1} ~\mathrm {d} x \mathrm {d} y. \end{aligned}$$

(5.5)

It can be shown (see, e.g., [8], Chapter 2.2) that this representation is well defined in the sense that f(H) is independent of the bump function $\tau$ and the degree $P \geqslant 1$ used to define the almost analytic extension $\widetilde{f}$ in (5.2).