Perturbations of continuum random Schr\"odinger operators with applications to Anderson orthogonality and the spectral shift function

TL;DR

This paper investigates how bounded, compactly supported perturbations affect multi-dimensional continuum random Schr"odinger operators, focusing on Anderson orthogonality and spectral shift functions within the localized regime, providing rigorous proofs and technical estimates.

Contribution

It establishes the occurrence of Anderson orthogonality with non-zero probability in the localized regime and links the spectral shift function to spectral projection indices, supported by key exponential decay estimates.

Findings

Anderson orthogonality occurs with positive probability in the localized regime.

Spectral shift function is identified with the index of spectral projection pairs.

Main technical estimate shows exponential decay of disorder-averaged Schatten norms.

Abstract

We study effects of a bounded and compactly supported perturbation on multi-dimensional continuum random Schr\"odinger operators in the region of complete localisation. Our main emphasis is on Anderson orthogonality for random Schr\"odinger operators. Among others, we prove that Anderson orthogonality does occur for Fermi energies in the region of complete localisation with a non-zero probability. This partially confirms recent non-rigorous findings [V. Khemani et al., Nature Phys. 11, 560-565 (2015)]. The spectral shift function plays an important role in our analysis of Anderson orthogonality. We identify it with the index of the corresponding pair of spectral projections and explore the consequences thereof. All our results rely on the main technical estimate of this paper which guarantees separate exponential decay of the disorder-averaged Schatten $p$-norm of $\chi_{a}(f(H) - f(H^{\tau})) \chi_{b}$ in $a$ and $b$. Here, $H^{\tau}$ is a perturbation of the random Schr\"odinger operator $H$, $\chi_{a}$ is the multiplication operator corresponding to the indicator function of a unit cube centred about $a\in\mathbb{R}^{d}$, and $f$ is in a suitable class of functions of bounded variation with distributional derivative supported in the region of complete localisation for $H$.

Equations

$$\omega \mapsto H_{\omega }:=H_0+V_{\omega }:=H_0+\lambda \sum _{k\in {\mathbb{Z}}^d}{\omega }_k{u}_k$$

$$0<C_{u,-}⩽\sum _{k\in {\mathbb{Z}}^d}{u}_k$$

$$\mathbb{R}\ni E\mapsto N_L(E):=\mathrm{tr}\left(\text{mathds}{1}_{(-\infty ,E]}(H_L)\right)$$

$$\mathcal{N}(E):=\underset{L\to \infty }{\lim }\frac{1}{|{\Lambda }_L|}{N}_L(E)$$

$$H^{\tau }:=H+\tau W,$$

$$\underset{\begin{array}{c}{E}^{\prime }\in U_E\\ \eta \ne 0\end{array}}{\sup }\mathbb{E}\left[\Vert {\chi }_a{R}_{E^{\prime }+\mathrm{i}\eta }(H_G){\chi }_b{\Vert }^s\right]⩽C{e}^{-\mu |a-b|}$$

$$\underset{\tau \in [0,1]}{\sup }\underset{\begin{array}{c}E\in I,\eta \ne 0\\ G\subseteq {\mathbb{R}}^d,a,b\in {\mathbb{R}}^d\end{array}}{\sup }\mathbb{E}\left[\Vert {\chi }_a{R}_{E+\mathrm{i}\eta }(H_G^{\tau }){\chi }_b{\Vert }^s\right]⩽C_s.$$

$$\Vert A+B{\Vert }_p^p⩽\Vert A{\Vert }_p^p+\Vert B{\Vert }_p^p$$

$$\operatorname{BV}(\mathbb{R}):=\big{\{}f:\mathbb{R}\to\mathbb{R}\ \text{measurable, right-continuous},\ \operatorname{TV}(f)<\infty\big{\}}.$$

$$\mathcal{F}_{I}:=\Big{\{}f\in\operatorname{BV}(\mathbb{R}):\ f\big{|}_{(-\infty,I_{-}]}\equiv\text{ const.},\,f\big{|}_{[I_{+},\infty)}\equiv 0\,\Big{\}}\subset L^{\infty}(\mathbb{R}),$$

$$\mathbb{E}\big{[}\big{\|}\chi_{a}\big{(}f(H_{G})-f(H^{\tau}_{G})\big{)}\chi_{b}\big{\|}_{p}\big{]}\leqslant C|\tau|^{s/2}\operatorname{TV}(f)\operatorname{e}^{-\mu(|a|+|b|)}.$$

$$\mathbb{E}\big{[}\big{\|}\chi_{a}\big{(}f(H_{G}^{\tau})-f(H^{\tau}_{\widetilde{G}})\big{)}\chi_{b}\big{\|}_{p}\big{]}\leqslant C\,\operatorname{TV}(f)\operatorname{e}^{-\mu[\operatorname{dist}(a,\partial{G})+\operatorname{dist}(b,\partial{G})]}.$$

$$\displaystyle\big{(}\mathbb{E}\left[\|f(H_{G})-f(H^{\tau}_{G})\|_{p}^{q}\right]\big{)}^{1/q}$$

$$f(H_G)-f(H_G^{\tau })\in {\mathcal{S}}^p$$

$$\Big{(}\mathbb{E}\left[\,\big{\|}\big{(}f(H_{L})-f(H_{L}^{\tau})\big{)}-\big{(}f(H)-f(H^{\tau})\big{)}\big{\|}_{p}^{q}\right]\Big{)}^{1/q}\leqslant C\operatorname{TV}(f)\operatorname{e}^{-\mu L}.$$

$$\lim_{n\to\infty}\operatorname{e}^{\gamma L_{n}}\big{\|}\big{(}f(H_{L_{n}})-f(H_{L_{n}}^{\tau})\big{)}-\big{(}f(H)-f(H^{\tau})\big{)}\big{\|}_{p}=0$$

$$\sup_{L>0}\mathbb{E}\big{[}\|\mathds{1}_{(-\infty,E]}({H}_{L})-\mathds{1}_{(-\infty,E]}({H}^{\tau}_{L})\|_{p}\big{]}=\infty\quad\;\Longrightarrow\quad\;E\notin\Sigma_{\mathrm{FMB}}.$$

$$\Vert \text{mathds}{1}_{(-\infty ,E]}(H_L)-\text{mathds}{1}_{(-\infty ,E]}(H_L^{\tau }){\Vert }_{2n}^{2n}$$

$$+\Vert \text{mathds}{1}_{(E,\infty )}(H_L)\text{mathds}{1}_{(-\infty ,E]}(H_L^{\tau })\text{mathds}{1}_{(E,\infty )}(H_L){\Vert }_n^n$$

$$T(E,A,B):=\text{mathds}{1}_{(-\infty ,E]}(A)-\text{mathds}{1}_{(-\infty ,E]}(B)$$

$$\Vert T(E,A,B)\Vert ⩽1.$$

$$\operatorname{tr}\big{(}f(A)-f(B)\big{)}=-\int_{\mathbb{R}}\mathrm{d}\lambda\,f^{\prime}(\lambda)\,\xi(\lambda,A,B)$$

$$\theta (E,A,B)$$

$$\displaystyle\phantom{:}=\dim\ker\big{(}T(E,A,B)-\mathds{1}\big{)}-\dim\ker\big{(}T(E,A,B)+\mathds{1}\big{)}.$$

$$T(E,H_{(L)},H_{(L)}^{\tau })\in {\mathcal{S}}^1\text{almost surely for every E∈ΣFMB}.$$

$${}{}\xi(E,H,H^{\tau})=\operatorname{tr}\big{(}T(E,H,H^{\tau})\big{)}=\theta(E,H,H^{\tau})\qquad\text{ for a.e. }E\in\Sigma_{\mathrm{FMB}}$$

$$\mathbb{E}\left[\left|\xi (E,H_L,H_L^{\tau })-\xi (E,H,H^{\tau })\right|\right]⩽C{e}^{-\mu L}\text{ for a.e. }E\in I$$

$$\underset{n\to \infty }{\lim }\xi (E,H_{L_n},H_{L_n}^{\tau })=\xi (E,H,H^{\tau })\text{ for a.e. }E\in I$$

$${}{}\operatorname{tr}\big{(}T(E,H_{(L)},H^{\tau}_{(L)})\big{)}=\theta(E,H_{(L)},H^{\tau}_{(L)})$$

$${}{}\xi(E,H_{L},H_{L}^{\tau})=\operatorname{tr}\big{(}T(E,H_{L},H_{L}^{\tau})\big{)}=\operatorname{tr}\big{(}\mathds{1}_{(-\infty,E]}(H_{L})\big{)}-\operatorname{tr}\big{(}\mathds{1}_{(-\infty,E]}(H_{L}^{\tau})\big{)}$$

$$\underset{n\to \infty }{\lim }\theta (E,H_{L_n},H_{L_n}^{\tau })=\theta (E,H,H^{\tau })\text{almost surely for every E∈ΣFMB}.$$

Full text

M.G. was supported by the DFG under grant GE 2871/1-1.

1. 1 Introduction

2. 2 The model

3. 3 Results

4. 3.1 Schatten-class estimates

5. 3.2 Application 1: The spectral shift function (SSF)

6. 3.3 Application 2: Anderson orthogonality

7. 4 Proofs of the results from Section 3.1

8. 4.1 Helffer–Sjöstrand formula

9. 4.2 Proof of Theorem 3.1 and Corollary 3.5

10. 4.3 Proof of Theorem 3.3 and Theorem 3.7

11. 5 Proofs of the results from Section 3.2

12. 5.1 Proof of Theorem 3.11

13. 5.2 Proof of Lemma 3.15 and Theorem 3.17

14. 6 Proofs of the results from Section 3.3

15. A Stability of fractional-moment bounds

16. B Deterministic a priori estimate for Schrödinger operators

1. Introduction

In this paper we consider an alloy-type random Schrödinger operator $H$ in multi-dimensional Euclidean space $\mathbb{R}^{d}$ and its perturbation $H^{\tau}$ by a bounded and compactly supported potential $W$ with coupling constant $\tau$. Our main goal is to investigate Anderson orthogonality for random Schrödinger operators in the region of complete localisation. These results are compiled in Section 3.3. Anderson orthogonality refers to the vanishing of the ground-state overlap (3.34) of the unperturbed and the perturbed system in the thermodynamic limit and constitutes a fundamental property of fermionic systems [And67, OT90, Mah00]. It was believed in physics for many years that this phenomenon was absent for localised systems [GBLA02]. However, the quest for an understanding of many-body localisation has led to a revival of such questions in physics, and the very recent non-rigorous studies [KNS15, DPLDS15] revealed interesting phenomena arising in this situation. They suggest:   (i)   Anderson orthogonality does occur for random Schrödinger operators when the Fermi energy is in the region of complete localisation, but with a probability strictly between 0 and 1.   (ii)   If the overlap vanishes in the thermodynamic limit, it vanishes even exponentially fast in the system size (named “statistical” Anderson orthogonality). In this paper, we provide the first partial mathematical understanding of these findings: for a Fermi energy $E$ in the region of complete localisation, we prove in Theorem 3.19 convergence of the expectations and almost-sure convergence of the finite-volume overlap to a random variable $S(E)$, which is given by a Fredholm determinant related to the difference of Fermi projections $\mathds{1}_{(-\infty,E]}(H)-\mathds{1}_{(-\infty,E]}(H^{\tau})$. Anderson orthogonality occurs if and only if this difference has an eigenvalue of modulus 1. If $S(E)\neq 0$, we prove exponential speed of convergence in the thermodynamic limit. We also identify conditions where both presence and absence of Anderson orthogonality occur with positive probability.

As an aside we obtain new criteria which offer a possibility to detect spectral phases of random Schrödinger operators other than the region of complete localisation. They are formulated in Corollaries 3.9 and  3.21.

A main tool in our analysis of Anderson orthogonality is the spectral shift function of the pair $(H,H^{\tau})$ for energies in the region of complete localisation. We obtain several new results on this subject which are compiled in Section 3.2. In Theorem 3.11 we establish convergence of the finite-volume spectral shift functions towards the infinite-volume spectral shift function and identify the latter with the index of the corresponding pair of spectral projections. We show Hölder continuity of the averaged spectral shift function in Lemma 3.15 and find a lower bound in Theorem 3.17 which establishes strict positivity of the averaged spectral shift function under appropriate conditions.

All of the aforementioned results are applications of the main technical estimates of this paper, which are summarised in Section 3.1 and are of their own interest. Theorem 3.1 states that the probabilistic expectation of fractional moments of the Schatten $p$-norm of \chi_{a}\big{(}f(H_{G})-f(H_{G}^{\tau})\big{)}\chi_{b} exhibits separate exponential decay in $a$ and $b$. This means that we have decay in both the distance of $a$ to the support of $W$ and the distance of $b$ to the support of $W$. Here, $H_{G}^{(\tau)}$ is the Dirichlet restriction of $H^{(\tau)}$ to an open region $G\subseteq\mathbb{R}^{d}$, $\chi_{a}$ is the multiplication operator corresponding to the indicator function of a unit cube centred about $a\in\mathbb{R}^{d}$ and $f$ is a function of bounded variation on $\mathbb{R}$ whose support is bounded from the right and which is constant except for energies in a compact interval inside the region of complete localisation for $H$. (A non-trivial example in this class of functions is the indicator function $f=\mathds{1}_{(-\infty,E]}$, where $E$ is in the region of complete localisation.) An analogous estimate holds for perturbations by boundary conditions and is stated in Theorems 3.3. Taken together, they allow to control macroscopic limits of differences $f(H_{L})-f(H^{\tau}_{L})$ in Schatten $p$-norms for finite-volume operators restricted to boxes of size $L$, see Theorem 3.7.

Finally, we advertise two points in our proofs whose validity is not restricted to the region of complete localisation and which may be of independent interest. Firstly, in order to express $f(K)$, $K$ a self-adjoint operator, in terms of the resolvent we formulate a suitably adapted version of the Helffer–Sjöstrand formula in Lemma 4.1 which applies to compactly supported functions of bounded variation. In order to treat such general functions $f$ we trade regularity of $f$ against boundedness of (spatially localised) resolvents. The second result is a scale-free unique continuation principle for averaged local traces of non-negative functions of infinite-volume ergodic random Schrödinger operators in Lemma 5.2. It follows from a scale-free unique continuation principle for spectral projections of finite-volume random Schrödinger operators [Kle13, NTTV15] by a suitable limit procedure. We are not aware that this consequence has been noticed elsewhere.

2. The model

We consider a random Schrödinger operator [CL90, PF92, Sto01, AW15]

[TABLE]

acting on a dense domain in the Hilbert space $L^{2}(\mathbb{R}^{d})$ for $d\in\mathbb{N}$ and $\lambda\geqslant 0$. Here, $H_{0}$ is a non-random self-adjoint operator and $\omega\mapsto V_{\omega}$ is a random alloy-type potential subject to the following assumptions.

1. The non-random operator $H_{0}:=-\Delta+V_{0}$ consists of the non-negative Laplacian $-\Delta$ on $d$-dimensional Euclidean space $\mathbb{R}^{d}$ and of a deterministic, $\mathbb{Z}^{d}$-periodic and bounded background potential $V_{0}\in L^{\infty}(\mathbb{R}^{d})$.

2. The family of canonically realised random coupling constants $\omega:=(\omega_{k})_{k\in\mathbb{Z}^{d}}\in\mathbb{R}^{\mathbb{Z}^{d}}$ is distributed independently and identically according to the Borel probability measure $\mathbb{P}:=\bigotimes_{\mathbb{Z}^{d}}P_{0}$ on $\mathbb{R}^{\mathbb{Z}^{d}}$. We write $\mathbb{E}$ for the corresponding expectation. The single-site distribution $P_{0}$ is absolutely continuous with respect to Lebesgue measure on $\mathbb{R}$ and the corresponding Lebesgue density $\rho$ obeys $0\leqslant\rho\in L^{\infty}_{c}(\mathbb{R})$ and $\operatorname{supp}(\rho)\subseteq[0,1]$.

3. The single-site potentials $u_{k}(\,\cdot\,):=u(\,\cdot\,-k)$, $k\in\mathbb{Z}^{d}$, are translates of a non-negative function $0\leqslant u\in L^{\infty}_{c}(\mathbb{R}^{d})$. Moreover, there exists a constant $C_{u,-}$ such that the covering condition

[TABLE]

holds.

The assumption $\operatorname{supp}(\rho)\subseteq[0,1]$ on the support of the single-site probability density is made for convenience only and is not stronger than requiring $\operatorname{supp}(\rho)$ to be compact. The regularity of $P_{0}$ as well as the covering condition (2.2) guarantee that the fractional-moment bounds (2.6) and the a priori bounds (2.7) below hold in fair generality for our model, see also Remarks 2.2 below.

Throughout the paper we drop the subscript $\omega$ from $H$ and other quantities when we think of these quantities as random variables (as opposed to their particular realisations). We write almost surely if an event occurs $\mathbb{P}$-almost surely, and, given an $E$-dependent statement for $E\in B\in\operatorname{Borel}(\mathbb{R})$, we write for a.e. $E\in B$, if this statement holds for Lebesgue-almost every $E\in B$.

The alloy-type random Schrödinger operator (2.1) is $\mathbb{Z}^{d}$-ergodic with respect to lattice translations. It follows that there exists a closed set $\Sigma\subset\mathbb{R}$, the non-random spectrum of $H$, such that $\Sigma=\sigma(H)$ holds almost surely [PF92]. Here, $\sigma(H)$ is the spectrum of $H$.

Given an open subset $G\subseteq\mathbb{R}^{d}$, we write $H_{G}$ for the Dirichlet restriction of $H$ to $G$. We define the random finite-volume eigenvalue counting function

[TABLE]

for $L>0$, where $\mathds{1}_{B}$ stands for the indicator function of a set $B\in\operatorname{Borel}(\mathbb{R})$, $H_{L}:=H_{\Lambda_{L}}$ and $\Lambda_{L}:=(-L/2,L/2)^{d}$ for the open cube about the origin of side-length $L$. We write $|\Lambda_{L}|=L^{d}$ for the Lebesgue volume of the latter. The regularity assumption 2 on the single-site distribution ensures a (standard) Wegner estimate [CHK07a, KM07, Ves08]. The Wegner estimate and ergodicity imply that the limit

[TABLE]

exists for all $E\in\mathbb{R}$ almost surely. The limit function $\mathcal{N}$ is called the integrated density of states of $H$. Moreover, the Wegner estimate implies that it is an absolutely continuous function under our assumptions. Its Lebesgue derivative $\mathcal{N}^{\prime}$ is called the density of states.

This paper is concerned with local perturbations of the random Schrödinger operator $H$. For a bounded and compactly supported function $W\in L^{\infty}_{c}(\mathbb{R}^{d})$, not necessarily of definite sign, we set

[TABLE]

where $\tau\in[0,1]$ is the tunable strength of the perturbation and $H^{0}=H$ is the unperturbed operator. We assume, from now on, that $W$ is fixed. The dependence of our results on the strength of the perturbation will be analysed through the dependence on $\tau$. We note that $H^{\tau}$ has the same integrated density of states as $H$.

All results in this paper refer to energies in the region of complete localisation – following the terminology of [GK06]. We characterise it in terms of fractional-moment bounds [AENSS06], see also [BNSS06]. To this end we need to introduce some more notation. Given a self-adjoint operator $A$, we denote by $R_{z}(A):=(A-z)^{-1}$ its resolvent at $z\in\mathbb{C}\setminus\sigma(A)$. We write $Q_{a}:=a+[-1/2,1/2]^{d}$ for the (closed) unit cube centred at $a\in\mathbb{R}^{d}$ and $\chi_{a}:=\mathds{1}_{Q_{a}}$ for its indicator function. We denote the supremum norm on $\mathbb{R}^{d}$ by $|x|:=\sup_{j=1,\ldots,d}|x_{j}|$, where $x=(x_{1},\ldots,x_{d})\in\mathbb{R}^{d}$. Finally, $\operatorname{dist}(U,V):=\inf\{|x-y|:\ x\in U,\ y\in V\}$ stands for the distance of two subsets $U,V\subset\mathbb{R}^{d}$ with respect to the supremum norm.

Definition 2.1** (Fractional-moment bounds).**

We write $E\in\Sigma_{\mathrm{FMB}}:=\Sigma_{\mathrm{FMB}}(H)$ if there exists a neighbourhood $U:=U_{E}$ of $E$ in $\mathbb{R}$ and an exponent $0<s<1$ such that the following holds: there exist constants $C,\mu>0$ such that for all open subsets $G\subseteq\mathbb{R}^{d}$ and all $a,b\in\mathbb{R}^{d}$ the bound

[TABLE]

holds true. Here, $\|\bm{\cdot}\|$ stands for the operator norm.

Remarks 2.2**.**

• (i)

Suppose that (2.6) holds for some exponent $0<s<1$, then it holds for all exponents $0<s<1$ with constants $C$ and $\mu$ depending on $s$. The proof for discrete models in [ASFH01, Lemma B.2] extends to the continuum setting.

• (ii)

We show in Lemma A.1 that the stability of the regime of complete localisation $\Sigma_{\mathrm{FMB}}(H)=\Sigma_{\mathrm{FMB}}(H^{\tau})$ holds for perturbations $\tau W$ as considered in this paper. Hence, even though our proofs require mostly fractional moment bounds for both $H$ and $H^{\tau}$, it suffices to postulate (2.6) for the unperturbed operator $H$ only and – according to the previous remark – with a fixed exponent $s$.

• (iii)

The proof of (2.6) in [AENSS06] for alloy-type random Schrödinger operators relies – as in the discrete case – on a priori estimates for fractional moments of the resolvent. In fact, the following holds for our model: for every $0<s<1$ and every bounded interval $I\subset\mathbb{R}$ there exists a finite constant $C_{s}>0$ such that

[TABLE]

Except for the supremum in $\tau$, this is the content of Lemma 3.3 and the subsequent Remark (4) in [AENSS06]. The relevant $\tau$-dependence of the constant in that lemma enters in (3.31) upon replacing $z$ by $z-\tau W$. This leads to a polynomial $\tau$-dependence of the following norm estimates and implies uniformity in $\tau\in[0,1]$.

3. Results

The results in Subsection 3.1 are at the heart of the proofs for the applications on the spectral shift function and Anderson’s orthogonality in the following two subsections.

3.1. Schatten-class estimates

First, we fix our notation. Given $p>0$ and a compact operator $A$ on a separable Hilbert space, we say that $A\in\mathcal{S}^{p}$, the Schatten $p$-class, if $\|A\|_{p}:=\left(\operatorname{tr}(|A|^{p})\right)^{1/p}$, where $\operatorname{tr}(\boldsymbol{\cdot})$ stands for the trace. This induces a complete norm on the linear space $\mathcal{S}^{p}$ for $p\geqslant 1$, while for $0<p<1$ it induces a complete quasi-norm for which the adapted triangle inequality

[TABLE]

holds [McC67, Thm. 2.8]. We state our results for right-continuous functions $f$ on $\mathbb{R}$ which are of bounded variation

[TABLE]

Here, the total variation is given by $\operatorname{TV}(f):=\sup_{(x_{p})_{p}\in\mathcal{P}}\sum_{p}|f(x_{p+1})-f(x_{p})|$ with the supremum being taken over the set $\mathcal{P}$ of all finite partitions of $\mathbb{R}$. We write $\operatorname{BV}_{c}(\mathbb{R})$ for the subspace of all functions in $\operatorname{BV}(\mathbb{R})$ with compact support. Finally, given a bounded interval $I=[I_{-},I_{+}]\subset\mathbb{R}$, we define the vector space of functions

[TABLE]

which is a normed space with respect to the total variation.

Theorem 3.1**.**

Fix $p>0$, $0<s<1$ and let $I\subset\Sigma_{\mathrm{FMB}}$ be a compact interval. Then there exist finite constants $C,\mu>0$ such that the following holds: for all $G\subseteq\mathbb{R}^{d}$ open, $a,b\in\mathbb{R}^{d}$, $\tau\in[0,1]$ and $f\in\mathcal{F}_{I}$ we have

[TABLE]

Remark 3.2**.**

On the right hand side of (3.4) a linear $\tau$-dependence seems to be optimal. However, our proof does not allow for this.

The method we use in the proof of Theorem 3.1 can also be applied to treat a perturbation by a boundary condition.

Theorem 3.3**.**

Fix $p>0$ and let $I\subset\Sigma_{\mathrm{FMB}}$ be a compact interval. Then there exist finite constants $C,\mu>0$ such that the following holds: for all open subsets $G\subset\widetilde{G}\subseteq\mathbb{R}^{d}$ with $\operatorname{dist}(\partial\widetilde{G},\partial G)\geqslant 1$, for all $a,b\in\mathbb{R}^{d}$ such that $Q_{a}\cap{G}\neq\varnothing$ or $Q_{b}\cap{G}\neq\varnothing$, for all $\tau\in[0,1]$ and all $f\in\mathcal{F}_{I}$ we have

[TABLE]

Remark 3.4**.**

The separate exponential decay of (3.4) in $a$ and $b$ reflects that the operator $f(H_{G})-f(H_{G}^{\tau})$ is exponentially small away from the support of $H_{G}-H_{G}^{\tau}$. An analogous comment applies to (3.5).

A consequence of Theorem 3.1 is

Corollary 3.5**.**

Fix $p,q>0$ and $\tau\in[0,1]$. Let $I\subset\Sigma_{\mathrm{FMB}}$ be a compact interval. Then there exists a finite constant $C_{\tau}>0$ such that for all $G\subseteq\mathbb{R}^{d}$ open and $f\in\mathcal{F}_{I}$

[TABLE]

The constant $C_{\tau}$ vanishes algebraically in $\tau$ as $\tau\downarrow 0$. In particular, for $f\in\mathcal{F}_{I}$

[TABLE]

for any $p>0$ almost surely.

Remark 3.6**.**

To obtain trace class properties as in (3.7) for the pair $(-\Delta,-\Delta+V)$, where $V$ is a short-range scattering potential, $f$ needs to be in an appropriate Besov space [FP15]. This implies that $f$ has to be more regular than being continuous. In the region of complete localisation for random Schrödinger operators, even discontinuous functions $f$ are allowed in (3.7), as we traded regularity of the function $f$ for regularity of the resolvent (2.6).

Theorem 3.1 and Theorem 3.3 together imply

Theorem 3.7**.**

Fix $p,q>0$ and let $I\subset\Sigma_{\mathrm{FMB}}$ be a compact interval. Then there exist finite constants $C,\mu>0$ such that for all $f\in\mathcal{F}_{I}$, $\tau\in[0,1]$ and $L>0$

[TABLE]

Moreover, given a sequence $(L_{n})_{n\in\mathbb{N}}\subset(0,\infty)$ of lengths with $L_{n}/\ln n\to\infty$ as $n\to\infty$, then for every $\gamma\in(0,\mu)$ the convergence

[TABLE]

holds almost surely.

Remark 3.8**.**

If $(X_{n})_{n\in\mathbb{N}}\subset(0,\infty)$ is a sequence of random variables with summable expectation, $(\mathbb{E}[X_{n}])_{n\in\mathbb{N}}\in\ell^{1}(\mathbb{N})$, then $\lim_{n\to\infty}X_{n}=0$ holds almost surely. This elementary fact shows that (3.9) follows directly from (3.8).

Corollary 3.5 or Theorem 3.7 offer a way to detect spectral phases of random Schrödinger operators other than the region of complete localisation.

Corollary 3.9**.**

Let $E\in\Sigma$ and $p>0$, then

[TABLE]

Remark 3.10**.**

Corollary 3.9 provides a reasonable condition to conclude that $E\notin\Sigma_{\mathrm{FMB}}$ because it is met in the case of absolutely continuous spectrum and non-trivial scattering. This can be seen as follows: assume $W\geqslant 0$ and observe that the finite-volume spectral shift operator $T(E,{H}_{L},{H}^{\tau}_{L})$ is finite rank for all $E\in\mathbb{R}$, hence even super trace class. An iteration of (6.6) then implies the representation

[TABLE]

for every $n\in\mathbb{N}$. Now, suppose we knew there exists a spectral interval $J\subset\mathbb{R}$ such that $H$ has absolutely continuous spectrum in $J$ almost surely. It follows from (3.10) and [GKMO16, Thm. 3.4] that, almost surely given any $p>0$, $\|\mathds{1}_{(-\infty,E]}({H}_{L})-\mathds{1}_{(-\infty,E]}({H}^{\tau}_{L})\|_{p}$ diverges at least logarithmically in $L$ for a.e. $E\in J$ at which there is non-trivial scattering, i.e.  where the quantity in (3.45) below is strictly positive.

3.2. Application 1: The spectral shift function (SSF)

In this subsection we apply the previous findings to prove convergence of the finite-volume spectral shift functions for energies in $\Sigma_{\mathrm{FMB}}$. Moreover, we show that the infinite-volume spectral shift function, the trace of the spectral shift operator and the index of the corresponding pair of Fermi projections coincide. In addition, we establish Hölder continuity and positivity of the disorder-averaged spectral shift function in $\Sigma_{\mathrm{FMB}}$.

Let $A$ and $B$ be self-adjoint operators in a Hilbert space. For any given $E\in\mathbb{R}$, we call

[TABLE]

spectral shift operator. Being the difference of two projections, we recall that

[TABLE]

Now, we assume further that $A$ and $B$ are bounded from below and that $\operatorname{e}^{-A}-\operatorname{e}^{-B}\in\mathcal{S}^{1}$. Then there exists a unique function $\xi:=\xi(\,\boldsymbol{\cdot}\,,A,B)\in L_{\text{loc}}^{1}(\mathbb{R})$, the spectral shift function (SSF), such that the equality

[TABLE]

holds for all test functions $f\in C^{\infty}(\mathbb{R})$ with $\lim_{\lambda\to\infty}f(\lambda)=0$ and $\operatorname{supp}(f^{\prime})$ compact. We refer to, e.g., [Yaf92] for details and further properties. Finally, given two self-adjoint projections $P$ and $Q$ such that $\pm 1\notin\sigma_{\text{ess}}(P-Q)$, the essential spectrum of $P-Q$, we define their index according to [ASS94, Prop. 3.1] as $\operatorname{index}(P,Q):=\dim\ker(P-Q-1)-\dim\ker(P-Q+1)$. Further background on the index can be found in [ASS94]. In the particular case of the spectral projections $P=\mathds{1}_{(-\infty,E]}(A)$ and $Q=\mathds{1}_{(-\infty,E]}(B)$, we write

[TABLE]

For $L>0$ and $A=H_{(L)}$ and $B=H_{(L)}^{\tau}$, we recall from Corollary 3.5 that

[TABLE]

Hence, the index $\theta(E,H_{(L)},H^{\tau}_{(L)})$ is well defined almost surely for every $E\in\Sigma_{\mathrm{FMB}}$ and every $L>0$. On the other hand, we infer from [HKNSV06, Thm. 1] that $\operatorname{e}^{-H_{(L)}}-\operatorname{e}^{-H^{\tau}_{(L)}}\in\mathcal{S}^{1}$ holds almost surely for all $L>0$. The SSF $\xi(\,\boldsymbol{\cdot}\,,H_{(L)},H^{\tau}_{(L)})$ is therefore well-defined almost surely as a function in $L_{\text{loc}}^{1}(\mathbb{R})$.

For energies outside the essential spectrum of a pair of operators, it is known that all three quantities, the SSF, the index and the trace of the spectral shift operator, are well-defined and coincide, see [Pus09, Prop. 2.1]. This extends to the region of complete localisation:

Theorem 3.11**.**

Let $\tau\in[0,1]$. Then,

• (i)

the SSF, the trace of the shift operator and the index coincide, i.e.

[TABLE]

holds almost surely.

• (ii)

given a compact interval $I\subset\Sigma_{\mathrm{FMB}}$, there exist constants $C,\mu>0$ such that the bound

[TABLE]

holds for all $L>0$. Moreover, given a sequence of lengths with $L_{n}/\ln n\to\infty$ as $n\to\infty$, the convergence

[TABLE]

holds almost surely.

Remarks 3.12**.**

• (i)

It follows from [ASS94, Thm. 4.1] that (3.16) implies

[TABLE]

almost surely for every $E\in\Sigma_{\mathrm{FMB}}$ and $L>0$. The news in Theorem 1 is the left equality. The right equality shows that all quantities are integer-valued.

• (ii)

In general, one cannot expect the SSF and the index to coincide as soon as they are both well defined. For example, if $E$ lies within the absolutely continuous spectrum of two operators, both the index and the SSF may be well defined. But in this case, the SSF cannot be expected to be integer-valued, whereas the index is by definition. For the very same reason, the convergence (3.19) cannot hold in general. The finite-volume spectral shift function is given by

[TABLE]

for all $E\in\mathbb{R}$ and, thus, integer-valued by definition. The infinite-volume quantity need not be. However, for general Schrödinger operators $-\Delta+V_{0}$ and perturbations by bounded, compactly supported potentials, vague convergence of the finite-volume SSF is known [HM10].

• (iii)

Theorem 2 is formulated in terms of the SSF. It holds for the index as well, even without having to exclude a Lebesgue null set of exceptional energies. Under the assumptions of (3.19) we obtain

[TABLE]

This follows directly from (3.20) and applying Theorem 3.7 to the function $f=\mathds{1}_{(-\infty,E]}$.

An immediate consequence of Theorem 2 is

Corollary 3.13**.**

Let $\tau\in[0,1]$. Given a sequence of lengths with $L_{n}/\ln n\to\infty$ as $n\to\infty$, then

[TABLE]

holds almost surely.

Remark 3.14**.**

Bounds like (3.23) are unknown for general deterministic continuum Schrödinger operators in $d\geqslant 2$. In the case of the pair $-\Delta$ and $-\Delta+W$ with $0\leqslant W\in L^{\infty}_{c}(\mathbb{R}^{d})$, a partial converse is shown in [Kir87]: for any diverging sequence $(L_{n})_{n\in\mathbb{N}}$ there exists a dense subset $I\subseteq(0,\infty)$ such that for all $E\in I$

[TABLE]

In contrast, $L^{p}$-bounds for finite- or infinite-volume SSF of deterministic continuum Schrödinger operators are known, see for example [HKNSV06, CHN01, HS02]. Moreover, general upper bounds on the SSF are proved in [Sob93] assuming that a trace-class limiting absorption principle holds. In the case of random Schrödinger operators and for a particular choice of the perturbation, [CHK07b] show that the disorder-averaged finite- and infinite-volume SSF are locally bounded uniformly in the system size.

The next lemma implies Hölder continuity of the averaged trace of the shift operator in the region of complete localisation.

Lemma 3.15**.**

Let $I\subset\Sigma_{\mathrm{FMB}}$ be a compact interval and $\alpha\in(0,1)$. Then there exists a finite constant $C>0$ such that for all $L>0$, all $\tau\in[0,1]$ and all $E,E^{\prime}\in I$

[TABLE]

Remark 3.16**.**

Choosing representatives of the SSF as in Theorem 1 and (3.21), we infer from Lemma 3.15 that for all $E,E^{\prime}\in I$

[TABLE]

Lemma 3.15 will be used in the proof of the next theorem which establishes pointwise positivity of the averaged SSF in the region of complete localisation for sufficiently positive perturbations. This will also be useful in the following section.

Theorem 3.17**.**

Assume that the Lebesgue density $\rho$ of the single-site distribution, which is specified in 2, satisfies $\rho_{-}:=\operatorname*{ess\,inf}_{x\in[0,1]}\rho(x)>0.$

Let $\tau\in(0,1]$ and assume that the perturbation $W$ and the bump function $u_{0}$ satisfy

[TABLE]

with constants $c,C,\delta>0$ and some $x\in\Lambda_{1}$. Then

[TABLE]

where $\mathcal{N}^{\prime}$ is the density of states of $H$.

Remark 3.18**.**

It is proven in [DGHKM17] that under the assumptions of Theorem 3.17 we have

[TABLE]

for every compact interval I\subset\Sigma_{\mathrm{FMB}}\cap\operatorname{int}\big{(}\sigma(H_{0})+[0,C_{u,-}]\big{)}, where $C_{u,-}$ is the constant from (2.2) and $\text{int}(A)$ stands for the interior of a set $A\in\operatorname{Borel}(\mathbb{R})$. This implies that

[TABLE]

3.3. Application 2: Anderson orthogonality

We apply the Schatten-class estimates from Section 3.1 to analyse Anderson orthogonality in the region of complete localisation. Let $\tau>0$ be fixed for now. The finite-volume operators $H_{L}$ and $H_{L}^{\tau}$ have discrete spectrum, and we denote their eigenvalues by $\lambda_{1}^{L}\leqslant\lambda_{2}^{L}\leqslant\cdots$ and $\mu_{1}^{L}\leqslant\mu_{2}^{L}\leqslant\cdots$, respectively, ordered by magnitude and repeated according to their multiplicities. The corresponding orthonormal bases of eigenfunctions are $(\varphi_{k}^{L})_{k\in\mathbb{N}}$ and $(\psi_{k}^{L})_{k\in\mathbb{N}}$, respectively. For $N\in\mathbb{N}$ we consider two non-interacting $N$-Fermion systems in the box $\Lambda_{L}$ with single-particle Schrödinger operators $H_{L}$ and $H^{\tau}_{L}$. The corresponding $N$-particle operators are

[TABLE]

They act on the totally antisymmetrised tensor product $\mathcal{H}_{N,L}:=\bigwedge_{j=1}^{N}L^{2}(\Lambda_{L})$ with standard scalar product $\langle\,\boldsymbol{\cdot}\,,\,\boldsymbol{\cdot}\,\rangle_{\mathcal{H}_{N,L}}$. The respective ground states of $\bm{H}_{N,L}$ and $\bm{H}^{\tau}_{N,L}$ are then given by the totally antisymmetrised and normalised tensor products $\bm{\Phi}_{N,L}:=\varphi_{1}^{L}\wedge\dotsm\wedge\varphi_{N}^{L}$ and $\bm{\Psi}_{N,L}:=\psi_{1}^{L}\wedge\dotsm\wedge\psi_{N}^{L}$. The modulus of their scalar product is given by

[TABLE]

where the equality follows from the Leibniz formula for determinants, and the scalar product appearing in the matrix entries is the one on the single-particle Hilbert space $L^{2}(\Lambda_{L})$. We note that $S_{N,L}$ depends on the coupling constant $\tau$, but we suppress it in our notation. Our subsequent analysis relies on rewriting $S_{N,L}$ in terms of a Fredholm determinant. We recall that, given an operator $K\in\mathcal{S}^{1}$ with eigenvalues $(b_{n})_{n\in\mathbb{N}}$, listed according to their algebraic multiplicities, the Fredholm determinant $\det(\mathds{1}-K):=\prod_{n\in\mathbb{N}}(1-b_{n})$ is well defined by the trace-class assumption on $K$ [RS78, Sect. XIII.17].

For fixed $E\in\mathbb{R}$, which will be referred to as the Fermi energy in this context, we define the $L$-dependent particle number

[TABLE]

The limit $L\to\infty$ is a realisation of the thermodynamic limit with particle density given by the integrated density of states $\lim_{L\to\infty}N_{L}(E)/|\Lambda_{L}|=\mathcal{N}(E)$ of the Hamiltonian $H$. We define the finite-volume ground-state overlap

[TABLE]

Following physics terminology, we say that Anderson orthogonality occurs at an energy $E$, if $S_{L}(E)$ vanishes as $L\to\infty$. Despite of the relevance in physics, mathematical results of this phenomenon are only rather recent and refer only to scattering systems: if the Fermi energy belongs to the absolutely continuous spectrum of the operator $H$, then there is algebraic decay of $S_{L}(E)$ as $L\to\infty$ [KOS14, GKM14, GKMO16, KOS15, Geb15].

We define the infinite-volume ground-state overlap as

[TABLE]

We note that if $E\in\Sigma_{\mathrm{FMB}}$, then $T(E,H,H^{\tau})\in\mathcal{S}^{p}$ almost surely for any $p>0$ by Corollary 3.5.

Theorem 3.19**.**

• (i)

For all $E\in\Sigma_{\mathrm{FMB}}\cap\operatorname{int}(\Sigma)$ the convergence

[TABLE]

holds. If, in addition, $L_{n}/\ln n\to\infty$ as $n\to\infty$, then the pointwise convergence

[TABLE]

holds almost surely.

• (ii)

Let $E\in\Sigma_{\mathrm{FMB}}$. Then

[TABLE]

holds almost surely.

• (iii)

Let $I\subset\Sigma_{\mathrm{FMB}}$ be a compact interval. Then we have

[TABLE]

• (iv)

Suppose that the assumptions of Theorem 3.17 are fulfilled. Then

[TABLE]

where $\mathcal{N}^{\prime}$ is the density of states of $H$.

Remarks 3.20**.**

• (i)

The set of energies for which (3.40) holds is not empty for our model, see Remark 3.18.

• (ii)

Theorem 3 shows that there is no relic of Anderson orthogonality when switching off the perturbation. A similar phenomenon is expected to occur in the large-disorder limit.

• (iii)

Theorem 4 shows that Anderson orthogonality is not absent almost surely in the region of complete localisation, contrary to established wisdom in the physics literature [GBLA02]. Only very recently, physical reasoning and numerical evidence was put forward that Anderson orthogonality does occur in the region of complete localisation [KNS15, DPLDS15].

• (iv)

For sign-definite perturbations $W$, the condition (3.38) for Anderson orthogonality to occur in the region of complete localisation is equivalent to a non-vanishing index,

[TABLE]

almost surely for every $E\in\Sigma_{\mathrm{FMB}}$. This follows from the fact that for a sign-definite perturbation, $1$ and $-1$ cannot both be eigenvalues of $T(E,H,H^{\tau})$ and therefore 1\in\sigma\big{(}T(E,H,H^{\tau})^{2}\big{)}\iff\theta(E,H,H^{\tau})\neq 0.

• (v)

Let $I\subset\Sigma_{\mathrm{FMB}}$ be compact. Then, Theorem 3.19 3 and 4 imply that there exists $\tau_{I}>0$ such that for all $0<\tau<\tau_{I}$ and a.e. $E\in I\cap\,\{E^{\prime}\in\mathbb{R}:\mathcal{N}^{\prime}(E^{\prime})\}>0$

[TABLE]

Thus, $S(E)$ is not almost surely constant, and both presence and absence of Anderson orthogonality occur with positive probability for energies in $\Sigma_{\mathrm{FMB}}$.

• (vi)

Let $E\in\Sigma_{\mathrm{FMB}}\cap\operatorname{int}(\Sigma)$ such that $\theta(E,H,H^{\tau})=0$. In this case our proof of Theorem 1 provides exponential speed of convergence of the overlap in (3.37): we conclude from (Proof of Theorem 3.19.) and (3.9) that, given a sequence of lengths with $L_{n}/\ln n\to\infty$ as $n\to\infty$, there exists a constant $c>0$ such that

[TABLE]

holds almost surely. According to the physical arguments in [DPLDS15], exponential speed of convergence occurs with positive probability for $E\in\Sigma_{\mathrm{FMB}}$ even if $\theta(E,H,H^{\tau})\neq 0$. More precisely, it is argued there that $\mathbb{E}[\ln S_{L}(E)]\sim-L$ for large $L$. This phenomenon is named statistical Anderson orthogonality in [DPLDS15] to distinguish it from the usual algebraic decay for energies in the scattering regime, see also the next remark.

Similarly to Corollary 3.9, Theorem 1 and 3 offer a way to detect spectral phases of random Schrödinger operators other than the region of complete localisation.

Corollary 3.21**.**

Let $E\in\operatorname{int}(\Sigma)$. Assume there exists a null sequence of coupling constants $(\tau_{k})_{k\in\mathbb{N}}\subset(0,1)$ and a sequence of lengths $(L_{n})_{n\in\mathbb{N}}$ with $L_{n}/\ln n\to\infty$ as $n\to\infty$ such that the $\tau_{k}$-dependent overlap obeys $\lim_{n\to\infty}S_{L_{n}}(E)=0$ almost surely for every $k\in\mathbb{N}$, then $E\notin\Sigma_{\mathrm{FMB}}$.

Remark 3.22**.**

The assumption of Corollary 3.21 is reasonable. This can be seen as follows: assume $W\geqslant 0$ and suppose we knew there exists a spectral interval $J\subset\Sigma$ such that $H$ has absolutely continuous spectrum in $J$ almost surely. Let $(L_{n})_{n\in\mathbb{N}}$ be a sequence of lengths such that $L_{n}/\operatorname{e}^{n^{\alpha}}\to\infty$ as $n\to\infty$ for some $\alpha>1$. Then, Theorem 2.2 in [GKMO16], see also [GKM14], applies realisationwise to the operators $H$ and $H^{\tau}$ for any $\tau\in(0,1]$, and we infer that almost surely and for a.e. energy $E\in J$

[TABLE]

The decay exponent

[TABLE]

relates to the energy-dependent scattering matrix $\mathbb{S}_{E}$ and is strictly positive if the perturbation $W$ causes non-trivial scattering at energy $E$. We remark that the super-exponential growth of the length scales $(L_{n})_{n\in\mathbb{N}}$ avoids the necessity of passing to a subsequence in (3.44) as is done in [GKMO16, Thm. 2.2]. This is so because it implies a sufficiently fast $L^{1}$-convergence in the proof of [GKMO16, Lemma 3.3(i)] so that subsequences can be avoided in that lemma.

4. Proofs of the results from Section 3.1

4.1. Helffer–Sjöstrand formula

In this section we prove a spatially localised variant of the Helffer–Sjöstrand formula which allows for jump discontinuities.

We consider functions $f\in\operatorname{BV}_{c}(\mathbb{R})$. Given such $f$, we choose a cutoff function $\Xi\in C_{c}^{\infty}(\mathbb{R}^{2})$ with $\Xi\equiv 1$ in a neighbourhood $N_{f}$ of $\operatorname{supp}(f)\times\{0\}\subset\mathbb{R}^{2}$. We then define the complex Borel measure $\zeta_{f}$ on $\mathbb{R}^{2}$ by

[TABLE]

where $\mathrm{d}f$ denotes Lebesgue-Stieltjes integration with respect to $f$. We write $|\zeta_{f}|$ for the total variation measure of $\zeta_{f}$.

Lemma 4.1**.**

Let $f\in\operatorname{BV}_{c}(\mathbb{R})$ and $\zeta_{f}$ as in (4.1). Let $K$ be a self-adjoint operator and let $A$, $B$ be bounded operators on the Hilbert space $\mathcal{H}$. If

[TABLE]

then

[TABLE]

holds, where the right-hand side is a Bochner integral with respect to the operator norm.

Remark 4.2**.**

Lemma 4.1 can be extended to appropriate Besov spaces, $B_{p,q}^{s}$, $1\leqslant p,q\leqslant\infty$ and $0<s<1$, by using Dynkin’s characterization of Besov spaces in terms of quasi-analytic extensions [Dyn83].

Before we prove the lemma, we apply it to spatially localised functions of random Schrödinger operators which obey the a priori estimate (2.7).

Corollary 4.3**.**

Let $G\subseteq\mathbb{R}^{d}$ be open, $a,b\in G$ and let $H^{\tau}_{G}$ be defined as in (2.1) and (2.5). Then, for $f\in\operatorname{BV}_{c}(\mathbb{R})$, the equality

[TABLE]

holds almost surely.

• Proof.

We verify that (4.2) holds almost surely. There exists $\delta>0$ (independent of $f$) such that $\operatorname{supp}(\Xi)\subset\mathbb{R}\times[-\delta,\delta]$. For fixed $0<s<1$, we use the deterministic norm bound $\|R_{z}(H^{\tau}_{G})\|^{1-s}\leqslant 1/|\text{Im}z|^{1-s}$, $z\in\mathbb{C}\setminus\mathbb{R}$, and estimate

[TABLE]

where $C_{s}:=\sup_{{(E,\eta)\in\operatorname{supp}(\zeta_{f}),\eta\neq 0}}\mathbb{E}\left[\|\chi_{a}R_{E+\mathrm{i}\eta}(H^{\tau}_{G})\chi_{b}\|^{s}\right]<\infty$ is finite by the a priori estimate (2.7). The claim then follows from

[TABLE]

• Proof of Lemma 4.1.

We note that assumption (4.2) implies that the right-hand side of (4.3) is well defined as a Bochner integral with respect to the operator norm. For $\varepsilon>0$, we split the integration into the set $\mathcal{C}_{\varepsilon}:=\mathbb{R}\times[-\varepsilon,\varepsilon]$ and its complement in $\mathbb{R}^{2}$. Using dominated convergence and the integrability assumption (4.2), we see that the integral over $\mathcal{C}_{\varepsilon}$ vanishes as $\varepsilon\to 0$. Therefore, we obtain that

[TABLE]

in the operator-norm topology with

[TABLE]

for $\lambda\in\mathbb{R}$.

The second part of the proof deals with the new aspect that discontinuity points of $f$ challenge the convergence of $f_{\varepsilon}$ as $\varepsilon\downarrow 0$. Let $\varphi,\psi\in\mathcal{H}$ and define the complex spectral measure \mu_{\varphi,\psi}:=\langle\varphi,A\mathds{1}_{\raisebox{-1.20552pt}{\bm{\cdot}}}(K)B\psi\rangle of $K$. The functional calculus, (4.8) and dominated convergence imply

[TABLE]

We prove below that the set of discontinuity points of $f$ is a $\mu_{\varphi,\psi}$-null set. Using this, another application of dominated convergence in (4.9) yields $\lim_{\varepsilon\downarrow 0}Af_{\varepsilon}(K)B=Af(K)B$ weakly, and the lemma follows.

It remains to prove that $f$ is continuous $\mu_{\varphi,\psi}$-almost everywhere. Without loss of generality, we assume $\|\varphi\|=\|\psi\|=1$. Since $f\in\operatorname{BV}_{c}(\mathbb{R})$, it is right-continuous and has left limits at all points. Hence, the set $\mathcal{U}:=\{\lambda\in\mathbb{R}:f\text{ not continuous in }\lambda\}$ consists of jump discontinuities only. Moreover, $\mathcal{U}$ is countable so that $\mu_{\varphi,\psi}(\mathcal{U})=\sum_{\lambda\in\mathcal{U}}\mu_{\varphi,\psi}(\{\lambda\})$. We fix an arbitrary $\lambda\in\mathcal{U}$ and set $\delta f_{\lambda}:=\lim_{\varepsilon\downarrow 0}[f(\lambda)-f(\lambda-\varepsilon)]\neq 0$. We choose $y_{0}>0$ small enough such that $\Xi(\lambda,y)=1$ whenever $|y|\leqslant y_{0}$. Assumption (4.2) implies

[TABLE]

where $h(y):=\int_{\mathbb{R}}\mathrm{d}\mu_{\varphi,\psi}(\lambda^{\prime})\,y/(\lambda^{\prime}-\lambda-\mathrm{i}y)$. Dominated convergence implies that $\mathbb{R}\ni y\mapsto h(y)$ is continuous and that $\lim_{y\to 0}h(y)=\mathrm{i}\,\mu_{\varphi,\psi}(\{\lambda\})$. Now, we assume that $\mu_{\varphi,\psi}(\{\lambda\})\neq 0$. Then there exists $0<y_{1}\leqslant y_{0}$ such that $|h(y)|\geqslant|\mu_{\varphi,\psi}(\{\lambda\})|/2$ whenever $|y|\leqslant y_{1}$, and we conclude that (Proof of Lemma 4.1.) yields a contradiction. Therefore we must have $\mu_{\varphi,\psi}(\{\lambda\})=0$, and $\mathcal{U}$ is a null set. This implies the desired continuity of $f$. ∎

4.2. Proof of Theorem 3.1 and Corollary 3.5

We introduce a lower bound of the spectra of all considered random Schrödinger operators,

[TABLE]

Here, the inequality holds almost surely for all $\tau\in[0,1]$ and all $G\subseteq\mathbb{R}^{d}$ open.

We begin with an operator-norm version of Theorem 3.1. Given a function $f\in\operatorname{BV}(\mathbb{R})$, we use the notation $\operatorname{supp}(f^{\prime})$ for the support of the complex measure defined by Lebesgue-Stieltjes integration with respect to $f$.

Theorem 4.4**.**

Fix $0<s<1$ and a compact set $S\subset\Sigma_{\mathrm{FMB}}$. Then there exist finite constants $C,\mu>0$ such that for all functions $f\in\operatorname{BV}_{c}(\mathbb{R})$ with $\operatorname{supp}(f^{\prime})\subseteq S$, for all $\tau\in[0,1]$, all open $G\subseteq\mathbb{R}^{d}$ and all $a,b\in\mathbb{R}^{d}$ we have

[TABLE]

• Proof.

We fix $f\in\operatorname{BV}_{c}(\mathbb{R})$ such that $\operatorname{supp}(f^{\prime})\subseteq S$. Let $S_{-}:=\inf S$, $S_{+}:=\sup S$ and observe $\operatorname{supp}(f)\subseteq[S_{-},S_{+}]$. We choose a cutoff function $\Xi=\Xi_{f}$ subject to

1. $\Xi\in C_{c}^{\infty}(\mathbb{R}^{2})$ with $0\leqslant\Xi\leqslant 1$ and $\|\partial_{x}\Xi\|_{\infty},\|\partial_{y}\Xi\|_{\infty}\leqslant 3$,

2. $\operatorname{supp}(\Xi)\subseteq[S_{-}-1,S_{+}+1]\times[-1,1]$,

3. $\Xi\equiv 1$ on $[S_{-}-1/2,S_{+}+1/2]\times[-1/2,1/2]$.

Now fix $G\subseteq\mathbb{R}^{d}$ open and $a,b\in\mathbb{R}^{d}$. Let $\zeta_{h}$ be the complex Borel measure defined in (4.1). We apply Corollary 4.3 to the operators $f(H_{G})$ and $f(H^{\tau}_{G})$, choose some $0<s<1$ and use the norm bound $\|R_{z}(H_{G}^{(\tau)})\|^{1-s/2}\leqslant 1/|\text{Im}z|^{1-s/2}$. This gives the estimate

[TABLE]

We introduce the subset of lattice points

[TABLE]

needed as centres to cover $\operatorname{supp}(W)$ by closed unit cubes and deduce from the resolvent equation and the Cauchy-Schwarz inequality

[TABLE]

We infer from (4.1), 4.2 and 4.2 that

[TABLE]

On the set $Z_{2}$ we estimate the right-hand side of (Proof.) by the Combes-Thomas estimate stated in [GK03, Cor. 1] for deterministic Schrödinger operators. Even though this result is formulated for Schrödinger operators on $L^{2}(\mathbb{R}^{d})$, the argument extends to Schrödinger operators on $L^{2}(G)$ for arbitrary $G\subseteq\mathbb{R}^{d}$ open – see also [She14]. Thus, there exist constants $C_{0},\mu_{0}>0$, which are independent of $G\subseteq\mathbb{R}^{d}$, $\tau\in[0,1]$ and $\omega\in\Omega$, such that

[TABLE]

for all $z\in\mathbb{C}$ subject to $\operatorname{dist}(z,\sigma(H_{G}^{\tau}))\geqslant 1/2$. For $(x,y)\in Z_{2}$ we have

[TABLE]

and therefore we obtain finite constants $C_{1}>0$ and $\mu_{1}:=2s\mu_{0}/C_{S}>0$ such that

[TABLE]

On the set $Z_{1}$ we use the fractional-moment bounds (2.6) for $H$ and $H^{\tau}$, which can be applied because of $\operatorname{supp}(f^{\prime})\subseteq\Sigma_{\mathrm{FMB}}$ and Remark 2. Hence, there exist finite constants $C_{2}>0$ and $\mu_{2}>0$ such that

[TABLE]

Collecting the estimates in (Proof.), (Proof.), (4.19) and (4.20), we obtain finite constants $C_{3}>0$ and $\mu:=\min\{\mu_{1},\mu_{2}\}/2>0$, which depend on $s$ and $S$ but are independent of $G$ and $\tau\in[0,1]$ such that

[TABLE]

To obtain the last inequality, we applied (4.6) and the estimate

[TABLE]

• proof of Theorem 3.1.

We set $I_{-}:=\min I$, $I_{+}:=\max I$ and assume without restriction that $I_{+}\geqslant E_{0}$, the lower bound (4.2) for the spectrum of $H_{G}^{\tau}$ (otherwise the statement is trivial). We fix $G\subseteq\mathbb{R}^{d}$ open and $a,b\in\mathbb{R}^{d}$. Since $\|A\|_{p}\leqslant\|A\|_{1}$ for any Schatten norm with $p\geqslant 1$, we restrict ourselves to $0<p\leqslant 1$.

Let $f\in\mathcal{F}_{I}$. As we want to apply Theorem 4.4, which requires functions of compact support, we introduce $h:=f\mathds{1}_{[E_{0}-1,\infty)}$. Then we have $h\in\text{BV}_{c}(\mathbb{R})$, $\operatorname{TV}(h)\leqslant 2\operatorname{TV}(f)$, $f(H^{\tau}_{G})=h(H^{\tau}_{G})$ for every $\tau\in[0,1]$ and $\operatorname{supp}(h^{\prime})\subseteq I\cup\{E_{0}-1\}\subset\Sigma_{\mathrm{FMB}}$. The last inclusion holds by the definition of $E_{0}$ and the Combes-Thomas estimate [GK03, Cor. 1], which extends to finite-volume operators.

We obtain for any $0<r<p$

[TABLE]

The adapted triangle inequality (3.1) and the deterministic a priori estimate from Lemma B.1 imply

[TABLE]

for every $0<p^{\prime}\leqslant 1$, where $C_{p^{\prime}}$ depends also on $I_{+}$, but is uniform in the disorder, $G$ and $\tau\in[0,1]$, and independent of $h$. We apply (proof of Theorem 3.1.) with $p^{\prime}=p-r$ to estimate the expectation of (4.23) by

[TABLE]

Here, we introduced $s_{1}:=r/p<1$ and also used Jensen’s inequality.

Now, we choose $0<s_{2}<1$ and apply Theorem 4.4 with $S=I\cup\{E_{0}-1\}$ to the expectation on the right-hand side of (proof of Theorem 3.1.). This yields finite constants $C_{1},\mu_{1}>0$ (which depend on $s_{2}$ and $I$) such that

[TABLE]

where we used

[TABLE]

Inserting (4.26) into (proof of Theorem 3.1.) and observing

[TABLE]

we obtain

[TABLE]

with a suitable finite constant $C>0$. Since $s_{1},s_{2}\in(0,1)$ are both arbitrary, the claim follows with $s:=s_{1}s_{2}$. ∎

• Proof of Corollary 3.5.

The (quasi-) norm estimates $\mathbb{E}\left[|X|^{q_{1}}\right]^{1/q_{1}}\leqslant\mathbb{E}\left[|X|^{q_{2}}\right]^{1/q_{2}}\quad\text{ and }\quad\|A\|_{p_{1}}\leqslant\|A\|_{p_{2}}$ for random variables $X$ and Schatten-class operators $A$ hold for all $0<p_{2}\leqslant p_{1}<\infty$ and all $0<q_{1}\leqslant q_{2}<\infty$. Thus, we assume without loss of generality that $p\leqslant 1$, $q\geqslant 1$ such that $k:=q/p\in\mathbb{N}$.

For $f\in\mathcal{F}_{I}$ and $G\subseteq\mathbb{R}^{d}$ open we abbreviate $T_{f}:=f(H_{G})-f(H^{\tau}_{G})$. Because of the adapted triangle inequality (3.1) for $p\leqslant 1$ we estimate $\|T_{f}\|_{p}^{p}\leqslant\sum_{a,b\in\mathbb{Z}^{d}}\|\chi_{a}T_{f}\chi_{b}\|_{p}^{p}$. A $k$-fold application of this inequality yields

[TABLE]

Next, we apply Hölder’s inequality to the expectation in (4.30) and obtain

[TABLE]

Now, we choose $0<s<1$. Theorem 3.1 implies the existence of finite constants $C,\mu>0$ (depending only on $p,I,s$) such that \mathbb{E}\big{[}\|\chi_{a_{l}}T_{f}\chi_{b_{l}}\|_{p}\big{]}\leqslant C\tau^{s/2}\operatorname{TV}(f)\operatorname{e}^{-\mu(|a_{l}|+|b_{l}|)} for all $a_{l},b_{l}\in\mathbb{Z}^{d}$. The deterministic a priori estimate from Lemma B.1 provides the existence of a finite constant $C_{1}$ (depending only on $p,I,s$, but not on $\omega$) such that $\|\chi_{a_{l}}T_{f}\chi_{b_{l}}\|_{p}^{pk-1}\leqslant C_{1}\operatorname{TV}(f)^{pk-1}$. We thus obtain

[TABLE]

for all $a_{l},b_{l}\in\mathbb{Z}^{d}$. Estimating the right-hand side of (4.31) by (4.32) and using $q=pk$, we arrive at $\mathbb{E}\left[\|T_{f}\|_{p}^{q}\right]\leqslant C_{2}\tau^{s/2}\operatorname{TV}(f)^{q}$, where the finite constant $C_{2}$ depends on $p,I,s$. In particular, the constant $C_{\tau}$ in the statement vanishes algebraically as $\tau\downarrow 0$. ∎

4.3. Proof of Theorem 3.3 and Theorem 3.7

• Proof of Theorem 3.3.

We will follow the strategy in the proofs of Theorems 4.4 and 3.1 and use the notation introduced there. As it is done there, we assume without restriction that $I_{+}\geqslant E_{0}$ and that $0<p\leqslant 1$. Moreover, we restrict ourselves to the case $b\in\mathbb{R}^{d}$ with $Q_{b}\cap G\neq\varnothing$.

Let $a\in\mathbb{R}^{d}$, $\tau\in[0,1]$, $f\in\mathcal{F}_{I}$ and define again the truncation $h:=f\mathds{1}_{[E_{0}-1,\infty)}$. We write $\zeta_{h}$ for the complex measure defined as in (4.1) with a cutoff function $\Xi=\Xi_{h}$ that satisfies 4.2 – 4.2, where $f$ is replaced by $h$. Proceeding along the lines of (proof of Theorem 3.1.) and (Proof.), we obtain for any $s,s^{\prime}\in(0,1)$

[TABLE]

with finite constants $C_{1}=C_{1,p,s^{\prime},I_{+}}>0$ and $C_{2}=C_{2,p,s,s^{\prime},I_{+}}>0$.

Case 1. We assume $\operatorname{dist}(a,\partial G)>1$ and $\operatorname{dist}(b,\partial G)>1$. Hence, we have $Q_{b}\subset G$ in this case. We apply the geometric resolvent inequality, see e.g. [Sto01, Lemma 2.5.2], to the operator norm in the last line of (Proof of Theorem 3.3.). Even though it is only stated for boxes there, the key estimate, [Sto01, Lemma 2.5.3], covers our setup. Hereby we use the assumption $\operatorname{dist}(\partial G,\partial\widetilde{G})>1$. We obtain in analogy to [DGHKM17, Eq. (4.9)]

[TABLE]

where $\Lambda_{2}(c):=c+\Lambda_{2}$ and

[TABLE]

denotes the set of lattice points needed as centres to cover a strip of width $1$ around the boundary $\partial G$ of $G\subset\mathbb{R}^{d}$ by unit cubes. The constant $C_{3}$ is uniform in $z\in\mathbb{C}$ on each compact subset of $\mathbb{C}$. It is also uniform in $a\in\mathbb{R}^{d}$ and $b\in G$ with $\operatorname{dist}(b,\partial G)>1$. Thus, the expextation in the last line of (Proof of Theorem 3.3.) is seen to be bounded from above by

[TABLE]

with a constant $C_{4}=C_{4,s,I_{+}}$. We decompose the support of $\zeta_{h}$ as in (4.16) and treat the product of the expectations on $Z_{1}$ with the fractional-moment estimate as in (4.20) and on $Z_{2}$ with the Combes-Thomas estimate as in and (4.19) (with $H_{G}$ replaced by $H^{\tau}_{\widetilde{G}}$ in both estimates). The remaining integral is estimated as in (4.6). We then arrive at

[TABLE]

as an upper bound for (4.36) with finite constants $C_{5},C_{6},\mu>0$, all depending only on $s$ and $I$. In the last step we used (4.27), the estimate

[TABLE]

and that the sum on the right-hand side of (4.38) is bounded from above uniformly in $a,b\in\mathbb{R}^{d}$ by, e.g., the Cauchy-Schwarz inequality. Now, the claim follows upon inserting (Proof of Theorem 3.3.) as an upper bound for the expectation into the last line of (Proof of Theorem 3.3.) and observing (4.28). This finishes Case 1.

Case 2. We assume $\operatorname{dist}(a,\partial G)\leqslant 1$ or $\operatorname{dist}(b,\partial G)\leqslant 1$. Hence, we have

[TABLE]

in this case. We estimate the operator norm on the right-hand side of (Proof of Theorem 3.3.) by the triangle inequality and each of the resulting two terms by the fractional-moment estimate (2.6), see also Remark 2. The remaining integral is again estimated by (4.6) and (4.27), and we obtain the upper bound $C_{7}\operatorname{e}^{-\mu|a-b|}\operatorname{TV}(f)$ for the expectation in the last line of (Proof of Theorem 3.3.) with finite constants $C_{7},\mu>0$ depending only on $s$ and $I$. Now, the claim follows upon observing (4.28) and (4.39). ∎

• Proof of Theorem 3.7.

As in the proof of Corollary 3.5 we assume without loss of generality that $p\leqslant 1$, $q\geqslant 1$ and we define and $k:=q/p\in\mathbb{N}$. Let $f\in\mathcal{F}_{I}$. The argument leading to (4.31) in the proof of Corollary 3.5 implies in the present context the estimate

[TABLE]

where T_{f,L}:=\big{(}f(H_{L})-f(H^{\tau}_{L})\big{)}-\big{(}f(H)-f(H^{\tau})\big{)}. Below, we will decompose $T_{f,L}=T_{f,L}^{(1)}-T_{f,L}^{(2)}$ in two different ways. The first way is with the terms

[TABLE]

and the second is with the terms

[TABLE]

In any case, the adapted triangle inequality (3.1) and Minkowski’s inequality on $L^{k}(\Omega,\mathbb{P})$ imply \mathbb{E}\big{[}\|\chi_{a_{l}}T_{f,L}\chi_{b_{l}}\|_{p}^{pk}\big{]}^{1/k}\leqslant\sum_{j=1}^{2}\mathbb{E}\big{[}\|\chi_{a_{l}}T_{f,L}^{(j)}\chi_{b_{l}}\|_{p}^{pk}\big{]}^{1/k}. Moreover, for either choice of the decomposition, the deterministic a priori bound from Lemma B.1 implies $\|\chi_{a_{l}}T_{f,L}^{(j)}\chi_{b_{l}}\|_{p}^{pk-1}\leqslant C_{1}\operatorname{TV}(f)^{pk-1}$ with a finite constant $C_{1}=C_{1,p,I}>0$ that does not depend on $\omega$ but only on the lower bound $E_{0}$ from (4.2) for the spectra. We conclude that

[TABLE]

Next we split the summation over each pair $(a_{l},b_{l})\in\mathbb{Z}^{d}\times\mathbb{Z}^{d}$ into two parts: the box $\Lambda^{2}_{L/2}:=(\Lambda_{L/2}\times\Lambda_{L/2})\cap(\mathbb{Z}^{{d}}\times\mathbb{Z}^{d})$ and its complement $\Lambda^{2,c}_{L/2}:=(\ Z^{{d}}\times\mathbb{Z}^{d})\setminus\Lambda^{2}_{L/2}$. For $(a_{l},b_{l})\in\Lambda^{2}_{L/2}$ we compare infinite- and finite-volume operators, that is, we choose the decomposition (4.41). Theorem 3.3 then provides finite constants $C_{2}=C_{2,p,I}>0$ and $\mu_{2}=\mu_{2,p,I}>0$ such that

[TABLE]

For $(a_{l},b_{l})\in\Lambda^{2,c}_{L/2}$ we compare unperturbed and perturbed operators, that is, we choose the decomposition (4.42). We fix $0<s<1$. Theorem 3.1 then provides finite constants $C_{3}=C_{3,p,s,I}>0$ and $\mu_{3}=\mu_{3,p,s,I}>0$ such that

[TABLE]

where $C_{3}^{\prime}=C^{\prime}_{3,p,s,I}>0$ is another finite constant. We conclude from (4.43), (Proof of Theorem 3.7.) and (Proof of Theorem 3.7.) that

[TABLE]

with finite constants $C=C_{p,q,s,I}>0$ and $\mu=\mu_{p,q,s,I}>0$. This proves (3.8).

The almost-sure convergence (3.9) for a super-logarithmically growing sequence of lengths follows from (3.8) with Remark 3.8. ∎

5. Proofs of the results from Section 3.2

5.1. Proof of Theorem 3.11

• Proof of Theorem 3.11.

We fix $\tau\in[0,1]$. Let $I\subset\Sigma_{\mathrm{FMB}}$ be a compact interval and $E\in I$. Theorem 2 follows from Theorem 1 and Theorem 3.7 applied to the Fermi function $f=\mathds{1}_{(-\infty,E]}$.

It remains to prove the left equality in Theorem 1, because the right equality follows already from Remark 1. Let $I\subset\Sigma_{\mathrm{FMB}}$ be again a compact interval. Then, Corollary 3.5 and Fubini’s Theorem imply \mathbb{E}\Big{[}\int_{I}\mathrm{d}E\,\big{\|}T(E,H,H^{\tau})\big{\|}_{1}\Big{]}<\infty. Hence, we have $\|T(\,\boldsymbol{\cdot}\,,H,H^{\tau})\|_{1}\in L^{1}(I)$ almost surely. Thus, the left inequality in Theorem 1 follows from Lemma 5.1 below, and the proof is complete. ∎

It remains to prove the following deterministic lemma.

Lemma 5.1**.**

Let $A$ and $B$ be two self-adjoint operators in a Hilbert space $\mathcal{H}$ which are bounded from below. We assume that $\operatorname{e}^{-A}-\operatorname{e}^{-B}\in\mathcal{S}^{1}$ and that, for some open interval $I\subset\mathbb{R}$, the mapping $I\ni E\mapsto\|T(E,A,B)\|_{1}$ is an $L^{1}(I)$-function. Then the SSF and the trace of the shift operator coincide, i.e.

[TABLE]

• Proof of Lemma 5.1.

The assumption $\operatorname{e}^{-A}-\operatorname{e}^{-B}\in\mathcal{S}^{1}$ in the lemma is only needed for the spectral shift function to be well defined according to (3.14).

We show that the function E\mapsto\operatorname{tr}\big{(}T(E,A,B)\big{)} satisfies (3.14) for every $f\in C^{\infty}(\mathbb{R})$ with $\operatorname{supp}(f^{\prime})\subseteq I$ and $\lim_{\lambda\to\infty}f(\lambda)=0$.

Let $f\in C^{\infty}(\mathbb{R})$ be such a function. By assumption we have $\mathbb{R}\ni E\mapsto|f^{\prime}(E)|\|T(E,A,B)\|_{1}\in L^{1}(\mathbb{R})$. Hence, $\int_{\mathbb{R}}\mathrm{d}E\,f^{\prime}(E)\,T(E,A,B)$ is a trace-norm convergent Bochner integral. Moreover, the identity

[TABLE]

holds, as we argue below. Since the mapping $\mathcal{S}^{1}\ni B\mapsto\operatorname{tr}(B)$ is a bounded linear functional on $\mathcal{S}^{1}$, it interchanges with the Bochner integral, and (5.2) implies the assertion

[TABLE]

It remains to prove (5.2). We fix $\phi\in\mathcal{H}$ and compute, using Fubini’s theorem,

[TABLE]

where we introduced the spectral measure $\mu^{A}_{\phi,\phi}:=\langle\phi,\mathds{1}_{\,\boldsymbol{\cdot}\,}(A)\phi\rangle$. The analogous computation for $B$ gives

[TABLE]

where we used again the continuity of the functional $\mathcal{S}^{1}\ni B\mapsto\langle\varphi,B\varphi\rangle$, which therefore interchanges with the Bochner integral. The operator equality (5.2) follows by polarisation. ∎

5.2. Proof of Lemma 3.15 and Theorem 3.17

• Proof of Lemma 3.15.

We fix $E,E^{\prime}\in I$ arbitrary with $E<E^{\prime}$ and define $J:=[E,E^{\prime}]$. The estimate (proof of Theorem 3.1.) with $f=h=\mathds{1}_{J}$ implies

[TABLE]

where $\theta\in(0,1)$ and the constant $C_{\theta}$ is independent of $L$ and $\tau\in[0,1]$. We will apply the Helffer–Sjöstrand formula to the operator $\mathds{1}_{J}(H_{(L)}^{(\tau)})$ and need an appropriate cutoff function for this purpose. Let $g\in C_{c}^{\infty}(\mathbb{R})$ be a smooth indicator function such that $g(x):=1$ for $x\in[-1,1]$, $g(x):=0$ for $x\in\mathbb{R}\backslash[-2,2]$, $\|g\|_{\infty}\leqslant 1$ and $\|g^{\prime}\|_{\infty}\leqslant 2$. We define the centre $J_{c}:=(E+E^{\prime})/2$ of the interval $J$ and the cutoff function $\Xi_{J}\in C^{\infty}_{c}(\mathbb{R}^{2})$ by

[TABLE]

Let $\zeta_{\mathds{1}_{J}}$ be the complex measure defined in (4.1) with the cutoff function (5.7). We note that $\operatorname{supp}(\zeta_{\mathds{1}_{J}})\subseteq J\times[-2|J|,2|J|]$. We now apply Lemma 4.1 to the difference in (Proof of Lemma 3.15.) and obtain for all $a,b\in\mathbb{Z}^{d}$

[TABLE]

where we abbreviated $z:=x+\mathrm{i}y$, $R_{z}:=R_{z}(H_{(L)})$ and $R^{\tau}_{z}:=R_{z}(H^{\tau}_{(L)})$. Let $p,q\in(1,\infty)$ with $1/p+1/q=1$ and let $\delta\in(0,1/p)$. Then, we estimate with the help of the resolvent equation

[TABLE]

where $\Gamma_{W}$ was defined in (4.14). For $(x,y)\in\operatorname{supp}(\zeta_{\mathds{1}_{J}})$ we have $x\in J\subseteq I\subset\Sigma_{\mathrm{FMB}}$ so that each expectation can be bounded with the localisation estimate from Lemma A.1, which is uniform in $\tau\in[0,1]$, $L>0$ and $J\subseteq I$. Treating the resulting $c$-sum of exponentials as in (4.22) and throwing away the decay in $|a-b|$, we arrive at

[TABLE]

with finite constants $C_{1},\mu_{1}>0$ that depend only on $p$, $\delta$ and $I$. This estimate needs to be inserted into (5.8), so the following integral is relevant

[TABLE]

To obtain the bound, we used $\delta<1/p$ and applied (4.6) with the particular choice (5.7) for the cutoff function $\Xi_{J}$. The constant $C_{2}>0$ depends only on $p$, $\delta$ and $I$. Collecting the estimates (Proof of Lemma 3.15.), (5.8), (5.10) and (5.11), we conclude that

[TABLE]

with a finite constant $C_{3}>0$ that depends on $\theta$, $p$, $\delta$ and $I$, but not on $J\subseteq I$, $L>0$ or $\tau\in[0,1]$. The lemma follows, because the exponent $\theta(p^{-1}-\delta)$ may take on any value in $(0,1)$. ∎

Our proof of Theorem 3.17 uses the scale-free unique continuation principle of [Kle13, NTTV15] applied to averaged local traces of non-negative functions of infinite-volume ergodic random Schrödinger operators.

Lemma 5.2**.**

Let $E\in\mathbb{R}$ and let $\Gamma\subset\mathbb{R}^{d}$ be a Borel set with $\operatorname{int}(\Gamma)\neq\varnothing$. Then there exists a finite constant $\gamma>0$ such that for every non-negative, measurable function $f:\mathbb{R}\rightarrow[0,\infty)$ with support $\operatorname{supp}(f)\subseteq(-\infty,E]$ the lower bound

[TABLE]

holds. If the operator $\mathds{1}_{\Gamma}f(H)$ is not trace class, we define its trace to be $+\infty$.

• Proof.

Even though a unique continuation principle for infinite-volume operators is known [TV16], it seems more convenient here to use the one for finite-volume operators cited above. By usual integration theory, the lemma follows if it is proven for indicator functions $f$ of arbitrary Borel sets $B\subseteq(-\infty,E]$. In turn, by the comparison theorem for measures, see e.g. [Els05, Thm. II.5.8], it is enough to prove it for semi-open intervals $I\subset(-\infty,E]$.

So let $I$ be such an interval. Since the probability for the endpoints of $I$ to coincide with an eigenvalue of $H$ is zero, we obtain from the definition and self-averaging of the integrated density of states

[TABLE]

The left-hand side of (5.13) is monotone in $\Gamma$ and invariant under $\mathbb{Z}^{d}$-translations of $\Gamma$. Therefore we assume without loss of generality that there exists an interior point of $\Gamma$ inside the unit cube $\Lambda_{1}$ and that $\Gamma\subseteq\Lambda_{1}$. Then, the scale-free unique continuation principle for spectral projections, see [NTTV15, Cor.] or [Kle13, Thm. 1.1], provides us with the deterministic estimate

[TABLE]

for $L\in\mathbb{N}$, where \Gamma_{L}:=\Lambda_{L}\cap\big{(}\bigcup_{k\in\mathbb{Z}^{d}}(k+\Gamma)\big{)} and the finite non-random constant $\gamma>0$ depends only on $d$, $\Gamma$, $E$ and $\mathcal{V}:=\|V_{0}\|_{\infty}+|\lambda|\,\|\sum_{k\in\mathbb{Z}^{d}}u_{k}\|_{\infty}<\infty$, see Assumptions 2 and 2. Next, we insert (5.15) into (5.14), exploit cyclicity of the trace and add and subtract the desired $L$-independent term. This yields

[TABLE]

It remains to show that the error term in the second line of (Proof.) vanishes as $L\to\infty$. This is a question of convergence of measures. Since a given real number is not an eigenvalue of $H$ with probability one, the limiting measure has no atoms, and the notion of vague convergence is sufficient. Thus, we consider the Laplace transforms, and the lemma follows if

[TABLE]

holds for every $t>0$ [Fel71, Thm. 2a in Sect. XIII.1]. Here, we reformulated the term with the $\mathbb{Z}^{d}$-ergodic infinite-volume Hamiltonian by using the the covariance relation for $H$ and the disjoint decomposition $\Gamma_{L}=\bigcup_{k\in\Lambda_{L}\cap\mathbb{Z}^{d}}(k+\Gamma)$, which holds for $L$ odd. We analyse the semigroups in the Feynman-Kac representation [Sim05a], see also [BHL00, Sect. 6], and obtain

[TABLE]

where $\mathbf{E}_{0,x}^{t,x}$ denotes the Brownian-bridge expectation over continuous closed paths starting at time [math] at $x$ and returning to $x$ at time $t$, and $\chi^{t}_{\Lambda_{L}}$ is the indicator function of the set of Brownian-bridge paths which stay inside $\Lambda_{L}$ for all times up to $t$. But the integral on the right-hand side of (5.18) vanishes as $L\to\infty$ according to [Kir89, p. 341]. ∎

We are now ready for the

• Proof of Theorem 3.17.

Throughout the proof we abbreviate the SSF by $\xi_{(L)}:=\xi(\,\boldsymbol{\cdot}\,,H_{(L)},H_{(L)}^{\tau})$ and choose representatives which coincide with the trace of the corresponding spectral shift operator on $\Sigma_{\mathrm{FMB}}$. Since $\xi_{(L)}$ depends monotonously on the perturbation and $W\geqslant Cu_{0}$, we assume without loss of generality that $W=Cu_{0}$.

By Lemma 3.15, the function $E\mapsto\mathbb{E}[\xi(E)]$ is Hölder continuous on compact intervals in $\Sigma_{\mathrm{FMB}}$ for any $\alpha\in(0,1)$. Let $E\in\Sigma_{\mathrm{FMB}}$ and $\varepsilon_{0}>0$ such that $I_{\varepsilon_{0}}:=[E-\varepsilon_{0},E+\varepsilon_{0}]\subset\Sigma_{\mathrm{FMB}}$. Consider any $\varepsilon\in(0,\varepsilon_{0}]$. Then

[TABLE]

where the Hölder constant depends on $\alpha$, $E$ and $\varepsilon_{0}$, but not on $\varepsilon$. We denote by $\mathbb{E}_{\neq 0}[\,\boldsymbol{\cdot}\,]$ the averaging with respect to all random variables but $\omega_{0}$ and infer from the Birman-Solomyak formula [BS75]

[TABLE]

We fix a parameter $s_{0}\in(0,\min\{1,C\}]$ to be determined later. Performing the change of variables $\omega_{0}\mapsto\omega_{0}+s$ and restricting first the $s$-integration to $[0,s_{0}]$ and then the $\omega_{0}$-integration to $[s_{0},1]$, we obtain the estimate

[TABLE]

Here, the last inequality follows from the right estimate in (3.27) together with the unique continuation principle stated in Lemma 5.2. Furthermore, we introduced $\rho_{+}:=\operatorname{ess\,sup}_{x\in[0,1]}\rho(x)<\infty$ and $J:=\int_{0}^{s_{0}}\mathrm{d}\omega_{0}\,\mathbb{E}_{\neq 0}\left[\operatorname{tr}\left(u_{0}\mathds{1}_{I_{\varepsilon}}(H)\right)\right].$

To estimate $J$ we first exchange the operator $H$ by its finite-volume restriction $H_{L}$. The error arising from this modification of the operator can be bounded by Theorem 3.3, which yields finite constants $C_{2},\mu>0$ such that $\mathbb{E}\left[\left|\operatorname{tr}\left(u_{0}\mathds{1}_{I_{\varepsilon}}(H)\right)-\operatorname{tr}\left(u_{0}\mathds{1}_{I_{\varepsilon}}(H_{L})\right)\right|\right]\leqslant C_{2}\operatorname{e}^{-\mu L}$ for every $L>0$. This estimate implies the bound

[TABLE]

for every $L>0$. Let $e_{1}:=(1,0,...,0)\in\mathbb{R}^{d}$ be the unit vector along the first coordinate axis. We define the subset of lattice points A_{L}:=\big{(}e_{1}+(3\mathbb{Z})^{d}\big{)}\cap\Lambda_{L} and introduce the random background operator $\widetilde{H}_{L}:=H_{0,L}+\lambda\sum_{k\in\mathbb{Z}^{d}\setminus A_{L}}\omega_{k}u_{k}$, where $H_{0,L}$ is the Dirichlet restriction of $H_{0}$ to $\Lambda_{L}$. However, for any fixed realisation of coupling constants $\{\omega_{k}\}_{k\in\mathbb{Z}^{d}\setminus A_{L}}$, we view $\widetilde{H}_{L}$ as a non-random operator with a non-periodic background potential that is bounded uniformly in $\{\omega_{k}\}_{k\in\mathbb{Z}^{d}\setminus A_{L}}$. Now, the Dirichlet restriction $H_{L}$ takes the form $H_{L}=\widetilde{H}_{L}+\lambda\sum_{k\in A_{L}}\omega_{k}u_{k}$ and, after scaling by a factor $1/3$ and (if required) introducing an energy shift, it constitutes a crooked Anderson Hamiltonian in the sense of [Kle13]. We apply the Wegner estimate [Kle13, Thm. 1.4] and obtain a finite constant $C_{3}>0$ such that

[TABLE]

where $\mathbb{E}_{A_{L}}[\,\boldsymbol{\cdot}\,]$ denotes the average over the couplings $\{\omega_{k}\}_{k\in A_{L}}$. The estimate (Proof of Theorem 3.17.) holds for all $\varepsilon\in(0,\varepsilon_{1}]$ and all length-scales $L\geqslant L_{1}$ such that $L/3\in\mathbb{N}$ is odd, and $\varepsilon_{1}$ and $L_{1}$ depend only on model parameters. We insert (Proof of Theorem 3.17.) into (5.22) and obtain

[TABLE]

Combining (5.19), (Proof of Theorem 3.17.) and (5.24), we conclude

[TABLE]

for all $\varepsilon\in(0,\min\{\varepsilon_{0},\varepsilon_{1}\}]$, all $s_{0}\in(0,\min\{1,C\}]$ and all $L\geqslant L_{1}$ such that $L/3\in\mathbb{N}$ is odd.

Now, suppose that $E$ is a Lebesgue point of the integrated density of states $\mathcal{N}$, which is the case Lebesgue-almost everywhere. Then we have $\mathcal{N}(I_{\varepsilon})/2\varepsilon\to\mathcal{N}^{\prime}(E)$ as $\varepsilon\downarrow 0$. Suppose also that $\mathcal{N}^{\prime}(E)>0$. Then, there exists $\varepsilon_{2}\in(0,\min\{\varepsilon_{0},\varepsilon_{1}\}]$ such that $\mathcal{N}(I_{\varepsilon})/2\varepsilon\geqslant\mathcal{N}^{\prime}(E)/2$ for every $\varepsilon\in(0,\varepsilon_{2}]$. Finally, we choose $L\geqslant L_{1}$ with $L/3\in\mathbb{N}$ odd so large that $\varepsilon:=L^{-4d/\alpha}\leqslant\varepsilon_{2}$ and $s_{0}:=L^{-2d}\leqslant\min\{1,C\}$. In this case (5.25) yields

[TABLE]

The right-hand side of (5.26) is strictly positive by possibly enlarging $L$ even further. ∎

6. Proofs of the results from Section 3.3

A crucial observation to prove Theorem 3.19 is the following Fredholm determinant representation of the ground-state overlap.

Lemma 6.1**.**

Let $N\in\mathbb{N}$, $L>0$ and define the orthogonal projections

[TABLE]

which we have written down in Dirac notation. Then

[TABLE]

• Proof.

Consider the $N\times N$-matrix M:=\big{(}\langle\varphi^{L}_{j},\psi^{L}_{k}\rangle\big{)}_{1\leqslant j,k\leqslant N}. Then, the matrix entries of $MM^{*}$, respectively $M^{*}M$, read $(MM^{*})_{jl}=\langle\varphi_{j}^{L},PQP\varphi_{l}^{L}\rangle$, respectively $(M^{*}M)_{jl}=\langle\psi_{j}^{L},QPQ\psi_{l}^{L}\rangle$, for $1\leqslant j,l\leqslant N$. Thus, $S_{N,L}$ can be written as

[TABLE]

where $Q^{c}:=\mathds{1}-Q$. Likewise, we have

[TABLE]

Here, the last equality follows from the fact that the non-zero singular values of $QP^{c}$ coincide with the non-zero singular values of its adjoint $P^{c}Q$. The determinants in (6.3) and (6.4) are well-defined Fredholm determinants because $PQ^{c}P$ and $QP^{c}Q$ are of finite rank. Multiplying the expressions in (6.3) and (6.4) yields

[TABLE]

Now, the lemma follows from the identity

[TABLE]

∎

The remainder of this section is devoted to the proof of Theorem 3.19. For $L>0$ and $E\in\mathbb{R}$ we abbreviate the index and the shift operator of $H_{(L)}$ and $H_{(L)}^{\tau}$ by $\theta_{(L)}(E):=\theta(E,H_{(L)},H^{\tau}_{(L)})$ and $T_{(L)}(E):=T(E,H_{(L)},H^{\tau}_{(L)})$, respectively.

Note that $P_{N_{L}(E),L}=\mathds{1}_{(-\infty,E]}(H_{L})$. If $\theta_{(L)}(E)=0$ holds, then we also have the identity $Q_{N_{L}(E),L}=\mathds{1}_{(-\infty,E]}(H_{L}^{\tau})$. In this case we obtain

[TABLE]

• Proof of Theorem 3.19.

We address the different parts of the theorem in the order 2, 3, 4, 1.

Part 2.  Let $E\in\Sigma_{\mathrm{FMB}}$. Then Corollary 3.5 implies that $T(E)\in\mathcal{S}^{2}$ almost surely, and we have S(E)=\det\big{(}\mathds{1}-T(E)^{2}\big{)}^{1/4} almost surely. Clearly, if $1$ is an eigenvalue of $T(E)^{2}$, then $S(E)=0$. Conversely, suppose that $1$ is not an eigenvalue of $T(E)^{2}$. Observing (3.13), we denote by $1>\|T(E)\|^{2}=b_{1}\geqslant b_{2}\geqslant\ldots\geqslant 0$ the non-increasingly ordered sequence of eigenvalues of $T(E)^{2}$. Then we obtain

[TABLE]

Part 3.  We fix a compact interval $I\subset\Sigma_{\mathrm{FMB}}$, $E\in I$, a constant $c>0$ and define the event $\Omega_{0}:=\{\|T(E)\|_{1}<1\}$. Markov’s inequality implies

[TABLE]

where we used (Proof of Theorem 3.19.) to obtain the last inequality. Since $\|T(E)\|\leqslant\|T(E)\|_{1}<1$ on $\Omega_{0}$ and, thus, $\|T(E)\|_{2}^{2}\leqslant\|T(E)\|\|T(E)\|_{1}<\|T(E)\|_{1}$, we conclude that the fraction inside the exponent in the last line of (Proof of Theorem 3.19.) is bounded from above by $\|T(E)\|_{1}/(1-\|T(E)\|_{1})$. Therefore we get

[TABLE]

We apply Markov’s inequality again and obtain

[TABLE]

so that

[TABLE]

Corollary 3.5 gives \sup_{E\in I}\mathbb{E}\big{[}\|T(E)\|_{1}\big{]}\to 0 as $\tau\to 0$. Hence, the claim follows by letting $c\downarrow 0$.

Part 4.  We restrict ourselves to $W\geqslant 0$, the proof for $W\leqslant 0$ follows along the same lines. Since $\theta(E,H,H^{\tau})=\xi(E,H,H^{\tau})$ for a.e. $E\in\Sigma_{\mathrm{FMB}}$ almost surely by Theorem 1, the equivalence (3.41) implies

[TABLE]

for a.e. $E\in\Sigma_{\mathrm{FMB}}$. Hence, the claim follows from Theorem 3.17.

Part 1.  We fix $E\in\Sigma_{\mathrm{FMB}}\cap\operatorname{int}(\Sigma)$. The almost-sure statement (3.37) implies (3.36) by dominated convergence and a subsubsequence argument applied to the $\mathbb{R}$-valued sequence $\left(\mathbb{E}\left[S_{L}(E)\right]\right)_{L>0}$. Thus it remains to prove (3.37).

Let $(L_{n})_{n\in\mathbb{N}}$ be a sequence with $L_{n}/\ln n\to\infty$ as $n\to\infty$. Since the quantities $\theta_{(L_{n})}(E)$ are all integer-valued, (3.22) implies that there is a random variable $n_{0}:=n_{0}(E):\Omega\rightarrow\mathbb{N}$ such that almost surely

[TABLE]

Case 1: $\theta(E)=0$.  Let $n\geqslant n_{0}$. Because of (6.14), (3.20) and (3.21) for $L=L_{n}$ we have the representation (6.7) for the finite-volume overlap in this case. The continuity of the Fredholm determinant with respect to the trace norm [Sim05b, Thm. 3.4] implies

[TABLE]

Theorem 3.7 now yields the almost-sure convergence

[TABLE]

This and (Proof of Theorem 3.19.) imply the desired convergence (3.37) in the case $\theta(E)=0$.

Case 2: $\theta(E)\neq 0$.  We assume without loss of generality that $\theta(E)>0$. The other case follows along the same lines. Because of the equivalence (3.41) we have $S(E)=0$ in this case, which, in turn, is equivalent 1\in\sigma\big{(}T(E)^{2}\big{)} by (3.38). We abbreviate $P_{n}:=P_{N_{L_{n}}(E),L_{n}}$ and $Q_{n}:=Q_{N_{L_{n}}(E),L_{n}}$. Let us assume the strong convergence

[TABLE]

for the time being. Since 1\in\sigma\big{(}T(E)^{2}\big{)}, it implies almost surely the existence of a sequence of eigenvalues $(\alpha_{n})_{n\in\mathbb{N}}$ with \alpha_{n}\in\sigma\big{(}(P_{n}-Q_{n})^{2}\big{)} and $\alpha_{n}\to 1$ as $n\to\infty$ [RS80, Thm. VIII.24(a)]. Moreover, $0\leqslant(P_{n}-Q_{n})^{2}\leqslant 1$, and we conclude (note that $N_{L_{n}}(E)\neq 0$ for $n$ large enough because $E\in\operatorname{int}(\Sigma)$, whence the first line in (3.34) applies)

[TABLE]

as $n\to\infty$ almost surely. This is the assertion in the case $\theta(E)\neq 0$.

Thus, it remains to prove (6.17). For this it suffices to show strong convergence $P_{n}-Q_{n}\to T(E)$ as $n\to\infty$. Let $\eta\in L^{2}(\mathbb{R}^{d})$. We add and subtract the term $T_{L_{n}}(E)$ and estimate

[TABLE]

where we used that $P_{n}=\mathds{1}_{(-\infty,E]}(H_{L_{n}})$ by definition of the particle number (3.33). The second term on the right-hand side of (6.19) converges to [math] as $n\to\infty$ almost surely by Theorem 3.7. As to the first term on the right-hand side we recall from (6.14), (3.20) and $\operatorname{tr}P_{n}=\operatorname{tr}Q_{n}$ that \theta(E)=\theta_{L_{n}}(E)=\operatorname{tr}\big{(}T_{L_{n}}(E)\big{)}=\operatorname{tr}\big{(}Q_{n}-\mathds{1}_{(-\infty,E]}(H_{L_{n}}^{\tau})\big{)} almost surely for $n\geqslant n_{0}$. Since we assumed that $\theta(E)>0$ and $Q_{n}$ is the orthogonal projection on the eigenspaces of all eigenvalues up to the $N_{L_{n}}(E)$th eigenvalue of $H^{\tau}_{L_{n}}$, we obtain that $\mu_{N_{L_{n}}(E)-\theta(E)}\leqslant E\leqslant\mu_{N_{L_{n}}(E)-\theta(E)+1}\leqslant\mu_{N_{L_{n}}(E)}$ and that

[TABLE]

hold almost surely for $n\geqslant n_{0}$. Since $E\in\operatorname{int}(\Sigma)$ and $\Sigma=\sigma_{\text{ess}}(H^{\tau})$, we obtain for any $\varepsilon>0$ by Fatou’s lemma and strong resolvent convergence of $H_{L_{n}}^{\tau}$ to $H^{\tau}$ that \lim_{n\to\infty}\operatorname{tr}\big{(}\mathds{1}_{[E,E+\varepsilon)}(H_{L_{n}}^{\tau})\big{)}\geqslant\operatorname{tr}\big{(}\mathds{1}_{[E,E+\varepsilon)}(H^{\tau})\big{)}=\infty almost surely. Since $\theta(E)$ is a finite number, this implies that for any $\varepsilon>0$ there is an $n_{1}>0$ such that

[TABLE]

for all $n\geqslant n_{1}$ almost surely. This yields for any fixed $\varepsilon>0$ the bound

[TABLE]

almost surely. Finally, we fix a null sequence $(\varepsilon_{l})_{l\in\mathbb{N}}\subset(0,\infty)$. Lemma A.2 implies that $E,E+\varepsilon_{l}$, $l\in\mathbb{N}$, are no eigenvalues of $H^{\tau}$ almost surely. Therefore we have strong convergence $\mathds{1}_{[E,E+\varepsilon_{l})}(H^{\tau}_{L})\to\mathds{1}_{[E,E+\varepsilon_{l})}(H^{\tau})$ as $L\to\infty$ almost surely [RS80, Thm. VIII.24(b)]. This and (6.22) lead to

[TABLE]

almost surely, where the last equality follows again from $E$ being almost surely not an eigenvalue of $H^{\tau}$. This proves (6.17). ∎

Appendix A Stability of fractional-moment bounds

Here, we prove that the fractional-moment bounds given in Definition 2.1 are stable under local perturbations. Among others, we show that $\Sigma_{\mathrm{FMB}}(H)=\Sigma_{\mathrm{FMB}}(H+W)$ for perturbations $W\in L_{c}^{\infty}(\mathbb{R}^{d})$.

Lemma A.1**.**

Let $I\subset\Sigma_{\mathrm{FMB}}$ be a compact interval. Then for any fixed $0<s<1$ there exist constants $C,\mu>0$ such that for all $G\subseteq\mathbb{R}^{d}$ open and $a,b\in\mathbb{R}^{d}$

[TABLE]

holds. In particular, $\Sigma_{\mathrm{FMB}}(H)=\Sigma_{\mathrm{FMB}}(H^{\tau})$ for every $\tau\in[0,1]$.

• Proof.

Let $z:=E+i\eta$ with $E\in I$ and $\eta\neq 0$. For the moment, we also fix $0<s<1/2$. The resolvent equation $R_{z}(H_{G}^{\tau})=R_{z}(H_{G})-\tau R_{z}(H_{G})WR_{z}(H_{G}^{\tau})$ yields the upper bound

[TABLE]

While the first term $I_{1}$ is independent of $\tau$ and can directly be estimated by the fractional-moment bound (2.6) for the unperturbed operator $H_{G}$, we estimate $I_{2}$ as

[TABLE]

where $\Gamma_{W}$ is defined in (4.14). Now, we recall Remark 1 and estimate the first expectation with the fractional-moment bound for the unperturbed operator $H_{G}$ and the exponent $2s<1$. And we estimate the second expectation with the a priori estimate (2.7) which is uniform in $\tau\in[0,1]$. This yields

[TABLE]

with finite constants $C_{1},C_{2}>0$ that are independent of $G\subseteq\mathbb{R}^{d}$ open, $a\in\mathbb{R}^{d}$ and $\tau\in[0,1]$. For the second inequality we used that $\Gamma_{W}$ is finite due to the compact support of $W$. Since $R_{z}(H_{G})WR_{z}(H_{G}^{\tau})=R_{z}(H_{G}^{\tau})WR_{z}(H_{G})$, we obtain along the same lines that $I_{2}\leqslant C_{2}\operatorname{e}^{-\mu|b|}$. Multiplying this inequality with (A.4), we infer

[TABLE]

Together with the bound for $I_{1}$, which decays exponentially in $|a-b|$, we obtain the assertion for $s<1/2$. For general $0<s<1$, the result then follows from Remark 1 where no additional $\tau$-dependence is generated. ∎

We finish this appendix with another simple stability result. It shows that the probability for a given energy to be an eigenvalue is zero. This property does not rely on ergodicity, it follows already from the existence of the a priori bounds (2.7).

Lemma A.2**.**

Let $E\in\mathbb{R}$ and $\tau\in[0,1]$ be given. Then $E$ is not an eigenvalue of $H^{\tau}$ almost surely.

• Proof.

Since, for $x\in\mathbb{R}$, $-i\varepsilon(x-E-i\varepsilon)^{-1}\to\mathds{1}_{\{E\}}(x)$ as $\varepsilon\to 0$, the spectral theorem implies the strong convergence $-i\varepsilon R_{E+i\varepsilon}(H^{\tau})\to\mathds{1}_{\{E\}}(H^{\tau})$ as $\varepsilon\to 0$. Using this, we obtain for some $0<s<1$ and any $a,b\in\mathbb{Z}^{d}$ that

[TABLE]

where the last equality follows from the a priori bound (2.7). Since $\mathbb{Z}^{d}$ is countable, $\mathds{1}_{\{E\}}(H^{\tau})=0$ almost surely. ∎

Appendix B Deterministic a priori estimate for Schrödinger operators

This appendix contains a local Schatten-class bound for functions of (non-random) Schrödinger operators. Estimates of this type appeared in [Sim82, Sect. B.9], [AENSS06, App. A] and [BNSS06]. We include the proofs for completeness, following closely [BNSS06].

We consider the non-random Schrödinger operator

1. $H:=-\Delta+U$ with a bounded potential $U\in L^{\infty}(\mathbb{R}^{d})$,

acting in $L^{2}(\mathbb{R}^{d})$. To formulate the deterministic a priori bound we introduce $E_{0}:=\inf_{x\in\mathbb{R}^{d}}U(x)$ so that $H_{G}\geqslant E_{0}$ for all $G\subseteq\mathbb{R}^{d}$ open.

Lemma B.1**.**

Assume (D). Let $p>0$ and $m\in\mathbb{N}$ with $m>d/(2p)$. Then there exists a finite constant $C$ such that for all open $G\subseteq\mathbb{R}^{d}$, all $a,b\in\mathbb{R}^{d}$ and all $g\in L^{\infty}(\mathbb{R})$ we have the estimate

[TABLE]

The constant $C$ in (B.1) depends on the potential $U$, but only through $\inf\sigma(H)-E_{0}$.

• Proof.

We use the abbreviation $H:=H_{G}$. Without loss of generality we assume $E_{0}\leqslant\sup\operatorname{supp}(g)<\infty$ (because otherwise the statement is trivial) and $0<p\leqslant 1$ (because $\|\boldsymbol{\cdot}\|_{p}\leqslant\|\boldsymbol{\cdot}\|_{1}$ for $p\geqslant 1$). Let $m\in\mathbb{N}$ such that $m>d/(2p)$. We insert the $m$-th power of the resolvent and apply Hölder’s inequality

[TABLE]

Next we set $a_{0}:=a$ and estimate with the adapted triangle inequality (3.1)

[TABLE]

Hölder’s inequality for Schatten classes and the Combes-Thomas estimate for Schatten norms in the form of [DGHKM17, Lemma A.2] (which equally holds for $H_{G}$) with $pm>d/2$ and $E=E_{0}-1$ imply

[TABLE]

with finite constants $\mu_{1}=\mu_{1,p,m}>0$ and $C_{1}=C_{1,p,m}>0$. The latter also depends on the potential $U$, but only through $\inf\sigma(H)-E_{0}$. Inserting (Proof.) into (Proof.) and repeatedly using

[TABLE]

where $C_{2}=C_{2,p,m}>0$ is a finite constant, we obtain the lemma from (Proof.). ∎