text:
\section{Introduction}

One way to learn about a physical system is to study its reactions to external perturbations. The fluctuation-dissipation theorem of linear-response theory in statistical mechanics provides a classical example for this approach. Simon and Wolff \cite{artRSO1986SiWo}
adhered to the same idea when they derived powerful localisation and delocalisation  criteria in the theory of random Schr\"odinger operators \cite{artRSO1989Kir, carlac1990random, pastfig1992random, artRSO2008Kir, AizWarBook}.
In this paper, we analyse the response of continuum random Schr\"odinger operators to a perturbation in the region of complete localisation. The response can be used as a criterion for the existence of other spectral phases.

More precisely, we estimate the probabilistic expectation of the Schatten-von Neumann $p$-norm
of $\chi_{a}\big(f(H_G) - f(H_G^{\tau})\big) \chi_{b}$. Here, $H_G$ is the Dirichlet restriction of an alloy-type random Schr\"odinger operator in multi-dimensional Euclidean space $\R^{d}$ to an open region $G \subseteq \R^{d}$. The operator $H_G^{\tau}$ is a perturbation of $H_G$ by a bounded and compactly supported potential with coupling constant $\tau$, $\chi_{a}$ is the multiplication operator corresponding
to the indicator function of a unit cube centred about $a\in\R^{d}$ and $f$ is a function of bounded variation
on $\R$ 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 $H$. The easiest non-trivial example in this class of functions is the indicator function $f=\id_{(-\infty,E]}$, where $E$ is in the region of complete localisation. In Theorem~\ref{th:TrClBounds1} we find the bound
\begin{equation}
	\label{intro:eq1}
	\mathbb{E} \big[ \big\| \chi_a\big( f(H_G) - f(H^{\tau}_G) \big) \chi_b \big\|_p  \big]
	\leq C|\tau|^{s/2}			\NTV(f) \e^{-\mu(|a|+|b|)}
\end{equation}
with constants $C,\mu >0$ that depend on $s\in (0,1)$ and $p \in (0,\infty)$, but are uniform in the open region $G \subseteq \R^{d}$. Since the perturbation is compactly supported in space, the bound exhibits a separate exponential decay
in both $a$ and $b$. It is proportional to the total-variation norm $\NTV(f)$ of $f$ and vanishes algebraically in the
limit of small coupling constants. This local estimate is turned into the global estimate
\begin{align}
	\label{intro:eq2}
	\big(\mathbb{E} \left[ \| f(H_G) - f(H^{\tau}_G) \|_p^q  \right]\big)^{1/q} &\leq C_{\tau}\,  \NTV(f)
\end{align}
in Corollary \ref{cor:TrClBounds2}. In principle, this is done by summing over $a$ and $b$. Here, $q\in (0,\infty)$ is arbitrary, and the constant $C_{\tau}$ again vanishes in the limit as $\tau \downarrow 0$.
Analogous estimates hold for perturbations by boundary conditions and are stated in Theorems~\ref{th:TrClBounds3} and~\ref{Thm:prod}. In particular, they allow to control macroscopic limits of differences
$f(H_{L}) - f(H^{\tau}_L)$ in Schatten-von Neumann $p$-norms for finite-volume operators restricted to
boxes of size $L$.

There is a vast literature on Schatten-von Neumann-class properties of operator differences $f(H)- f(H^\tau)$, where $H$ and $H^\tau$ satisfy assumptions which are often motivated by scattering theory. For example, if the unperturbed operator is given by the Laplacian $-\Delta$ and
the potential $V$ decays sufficiently fast at infinity, the function $f$ needs to have certain differentiability properties
in order to ensure that $f(-\Delta)- f(-\Delta +V)$ is trace class; see e.g.\ \cite{yafaev1992mathscattering,MR3314510} and references therein. We emphasise that we are concerned with the opposite regime in this paper. In the region of complete localisation for random Schr\"odinger operators, even discontinuous functions $f$ lead to super-trace-class properties of the operator difference $f(H)- f(H^\tau)$.

Our proofs  of \eqref{intro:eq1} and the related results in Sect.\ \ref{sect:TraceClassEst} are structural in that they do not depend on the details of the model. They exploit general properties of Schr\"odinger operators such as the
deterministic a priori bounds \eqref{eq:aPrioriStat}
for bounded functions of $H^{(\tau)}$ and the Combes-Thomas estimate. In addition, localisation enters through
exponentially decaying fractional-moment bounds \eqref{eq:DefFMB} for the resolvents.

Before we turn to applications, we mention two auxiliary results in our proofs whose validity is not restricted to the region of complete localisation and which may be of independent interest in different contexts.
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-Sj\"ostrand formula in Lemma~\ref{lem:LocHelfSjo} which applies to compactly supported functions of bounded variation. In order to treat such general functions $f$ we have traded regularity of $f$ against boundedness of (spatially localised) resolvents. In the case $K= H^{(\tau)}$, the latter is provided
by the a priori bounds \eqref{eq:AppAPriori}.
The second auxiliary result is a scale-free unique continuation principle for averaged local traces of non-negative functions of \emph{infinite-volume} ergodic random Schr\"odinger operators in Lemma~\ref{lem:UCP}. It follows from a scale-free unique continuation principle for spectral projections of \emph{finite-volume} random Schr\"odinger operators \cite{MR3106507, Nakic:2015is} by a suitable limit procedure.
We are not aware that this consequence has been noticed elsewhere.

We apply the estimates \eqref{intro:eq1}, \eqref{intro:eq2} and their relatives in two different ways. Firstly, a particular choice for the function $f$ is the indicator function $f=\id_{(-\infty,E]}$ with $E$ lying inside the region of complete localisation. In this way we derive properties of the spectral shift function for random Schr\"odinger operators in the region of complete localisation.
In Theorem~\ref{th:SSFandTransOp} 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 also show H\"older continuity of the averaged spectral shift function and find a \emph{lower} bound in Theorem~\ref{lem:SSFcont} which establishes strict positivity under appropriate conditions. This bound relies on the latest quantitative unique continuation principle for spectral projections of random Schr\"odinger operators \cite{MR3106507, Nakic:2015is}.

The second application relates to the phenomenon known as Anderson orthogonality in the physics literature. Anderson orthogonality is a fundamental property of fermionic systems \cite{artAOC1967And, RevModPhys.62.929, mahan-book00}. Given two non-interacating systems of $N$ Fermions, which differ by a local perturbation, it is defined as the vanishing of the overlap
\begin{equation}
	S_{N,L} := \big|\big\<\boldsymbol{\Phi}_{N,L},\boldsymbol{\Psi}_{N,L}\big\>_{\H_{N,L}}\big|
\end{equation}
of their $N$-particle ground states $\boldsymbol{\Phi}_{N,L}$ and $\boldsymbol{\Psi}_{N,L}$
in the thermodynamic limit $L\to\infty$, $N\to\infty$ with a non-vanishing particle density  $N/L^{d} \to \rho(E)>0$, which we parametrise in terms of the Fermi energy $E$. In the above, $\<\,\pmb\cdot\,,\,\pmb\cdot\,\>_{\H_{N,L}}$ denotes the scalar product on the fermionic $N$-particle Hilbert space.
 Despite the attention Anderson orthogonality had and still has within the physics community, first mathematical results are only rather recent. There are by now several works \cite{artAOC2014KOS, artAOC2014GKM,
artAOC2015GKMO} which prove vanishing upper bounds on $S_{N,L}$ and, in more special one-dimensional situations, even exact asymptotics \cite{MR3376020,MR3405952}. The common conceptional point of these works is that if the Fermi energy belongs to the absolutely continuous spectrum of the operator $H$, then there is algebraic decay of $S_{N,L}$ in the thermodynamic limit.
In contrast, no attention has been paid to Anderson orthogonality for random Schr\"odinger operators with Fermi energy in the region of complete localisation, as it was established wisdom that the effect would not occur in this case \cite{PhysRevB.65.081106}. It was only the very recent studies \cite{Khemani-NaturePhys15, PhysRevB.92.220201}
which revealed interesting phenomena arising in this situation: \; (i)~\,~Anderson orthogonality does occur for random Schr\"odinger operators when the Fermi energy is in the region of complete localisation, but with a probability strictly between 0 and 1.  \; (ii)~\,~If $S_{N,L}$ vanishes in the thermodynamic limit, it vanishes even exponentially fast in the system size.
In this paper, we provide a mathematical understanding of most of these findings. For a Fermi energy $E$ in the region of complete localisation, we prove in Theorem~\ref{th:IntMainRes} that the finite-volume overlap $S_{N,L}$ converges in expectation and almost surely to a random limit $S(E)$, given by a Fredholm determinant related to the spectral shift operator $\id_{(-\infty,E]}(H) - \id_{(-\infty,E]}(H^{\tau})$. Anderson orthogonality occurs if and only if the spectral shift operator 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.

Finally, the results of this paper lead to new criteria which offer a possibility to detect spectral phases of random Schr\"odinger operators other than the region of complete localisation.  The first one is about the divergence of the averaged Schatten-$p$-norm of finite-volume spectral shift operators in the macroscopic limit. It is formulated in Corollary~\ref{phase-transition}. The second criterion, see Corollary~\ref{rm:notFMB},
concerns the almost-sure occurrence of Anderson orthogonality along a sequence of sufficiently fast growing length scales, no matter how small the coupling constant of the perturbation potential is.

The plan of this paper is as follows. In the next section we introduce the alloy-type Anderson model and
fix our notation. Section~\ref{sect:results} contains the results. In Section~\ref{sect:TraceClassEst}
we present bounds on disorder-averaged Schatten-von Neumann norms for differences of functions of
the unperturbed and perturbed random Schr\"odinger operator, which hold in the region of complete localisation.
We proceed by describing two applications of these bounds:
in Section~\ref{subsec:ssf} we are concerned with applications towards the spectral shift function.
In Section~\ref{subsec:aoc} we study the implications for Anderson orthogonality in the region of complete localisation.
All proofs are deferred to Sections~\ref{sec:4} to~\ref{sec:6}.
In Appendix~\ref{app:FMB} we verify the stability of fractional moment bounds under local perturbations.
Schatten von Neumann-class estimates for Schr\"odinger operators are reviewed in Appendix~\ref{app:Schatten}.
Finally, we collect some properties of the index of a pair of spectral projections in Appendix~\ref{App:SuppMater}.

\section{The model}

We consider a random Schr\"odinger operator
\begin{equation}
	\label{eq:TheOperator}
	\omega \mapsto H_{\omega} := H_0+V_\omega := H_0 +\lambda \sum_{k\in\Zd} \omega_k u_k
\end{equation}
acting on a dense domain in the Hilbert space $L^{2}(\Rn)$ for $d\in\N$ and $\lambda\geq 0$. 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.

\begin{MyDescription}
\item[(K)]{
	\label{assK}
	The non-random operator $H_0 := -\Delta+V_0$ consists of the non-negative Laplacian $-\Delta$ on
	$d$-dimensional Euclidean space $\Rn$ and of a deterministic, $\Zd$-periodic and bounded background potential
	$V_0 \in L^{\infty}(\R^{d})$.
	}
\item[(V1)]{
	\label{assV1}
	The family of canonically realised random coupling constants $\omega:=(\omega_k)_{k\in\Zd}
	\in \R^{\Z^{d}}$ is distributed independently and identically according to the Borel probability measure
	$\PP := \bigotimes_{\Z^{d}}P_{0}$ on $\R^{\Z^{d}}$. We write $\E$ for the corresponding expectation.
	The single-site distribution $P_0$ is absolutely continuous with respect to Lebesgue measure on $\R$
	and the corresponding Lebesgue density $\rho$ obeys $0 \le \rho\in L^\infty_{c}(\R)$ and
	$\supp(\rho) \subseteq [0,1]$.
  }
\item[(V2)]{
	\label{assV2}
	The single-site potentials $u_k(\,\cdot\,):=u(\,\cdot\,-k)$, $k\in\Zd$, are translates
	of a non-negative function $0 \le u\in L^\infty_{c}(\R^d)$. Moreover, there exists a constant
	$C_{u,-}$ such that the covering condition
	\begin{equation}
		\label{uk-bounds}
		0< C_{u,-} \leq \sum_{k\in\Zd} u_k
	\end{equation}
	holds.
	}
\end{MyDescription}

\noindent
The assumption $\supp(\rho) \subseteq [0,1]$ on the support of the single-site probability density is
made for convenience only and is not stronger than requiring $\supp(\rho)$ to be compact.
The regularity of $P_0$ as well as the covering condition \eqref{uk-bounds}
guarantee that the fractional-moment bounds \eqref{eq:DefFMB} and the a priori bounds
\eqref{eq:AppAPriori} below hold in fair generality for our model, see also Remarks~\ref{rem:FMBS} 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
\textit{almost surely} if an event occurs  $\mathbb{P}$-almost surely, and, given an $E$-dependent statement for
$E \in B \in \Borel(\R)$,  we write \textit{for a.e.\ $E\in B$}, if this statement holds for
Lebesgue-almost every $E \in B$.

The alloy-type random Schr\"odinger operator \eqref{eq:TheOperator} is $\Zd$-ergodic with respect to
lattice translations. It follows that there exists a closed set $\Sigma\subset\R$, the non-random spectrum
of $H$, such that $\Sigma=\sigma(H)$ holds almost surely \cite{pastfig1992random}. Here, $\sigma(H)$ is
the spectrum of $H$. Such a statement also holds individually for the components of the spectrum in
the Lebesgue decomposition.

Given an open subset $G\subseteq \Rn$, we write $H_G$ for the Dirichlet restriction of $H$ to $G$.
Whenever it is convenient, we think of a finite-volume operator
$H_{G}$ to act in $L^{2}(\R^{d})$ by being trivially extended on $L^{2}(\R^{d}\setminus G)$.
We define the random finite-volume eigenvalue counting function
\beq
	\label{eq:ev-count}
	\R \ni E \mapsto N_L(E):= \Tr \left(\id_{(-\infty,E]}(H_L)\right)
\eeq
for $L>0$, where $\id_{B}$ stands for the indicator function of a set $B\in\Borel(\R)$, $H_L:=H_{\Lambda_L}$
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 \ref{assV1} on the single-site distribution ensures a (standard) Wegner
estimate \cite{MR2362242}. The Wegner estimate and ergodicity imply that the limit
\beq
	\label{eq:DOSFinVol}
	\cN(E):= \lim_{L\to\infty} \frac 1 {|\Lambda_L|} N_L(E)
\eeq
exists for all $E\in\R$ almost surely. The limit function $\cN$ is called the
\emph{integrated density of states} of $H$.
Moreover, the Wegner estimate implies that it is an absolutely continuous
function under our assumptions. Its Lebesgue derivative $\cN'$ is called the \emph{density of states}.
We refer to, e.g., \cite{Kirsch:2007il,MR2378428} for an overview.

This paper is concerned with local perturbations of the random Schr\"odinger operator $H$.
For a bounded and compactly supported function $W \in L^{\infty}_c(\Rn)$, not necessarily of definite sign, we set
\beq
	\label{def:perturbedoperator}
	H^{\tau}:=H+\tau W,
\eeq
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 \emph{region of complete localisation} -- following the terminology of \cite{Germinet2006}. In the continuum setting, it has mostly been studied by the multi-scale analysis. We
refer to \cite{artRSO2001GeKl, MR2180453, MR2314108, zbMATH06126557} for newer developments in this direction and to the literature cited therein. In this paper, we characterise the region of complete localisation in terms of fractional-moment bounds
\cite{artRSO2006AizEtAl2}, see also \cite{MR2303305}. 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
$z\in \C\setminus \sigma(A)$. We write $Q_a:=a+[-1/2,1/2]^d$ for the (closed) unit cube centred at
$a\in\R^d$ and $\chi_a:=\id_{Q_a}$ for its indicator function.
We denote the supremum norm on $\R^d$ by $|x|:= \sup_{j=1,\ldots,d} |x_j|$, where
$x= (x_{1}, \ldots, x_{d}) \in\Rn$.
Finally, $\dist(U,V):=\inf \{  |x-y|:\ x\in U,\ y\in V\}$ stands for the distance of two subsets
$U,V\subset \R^d$ with respect to the supremum norm.

\begin{definition}[Fractional-moment bounds]
	\label{DefFMB}
	We write $E\in \FMB:=\FMB(H)$ if there exists a neighbourhood $U:=U_E$ of $E$ in $\R$ and an exponent
	$0<s<1$	such that the following holds: there exist constants $C,\mu>0$ such that for all open subsets
	$G\subseteq \Rn$ and all $a,b \in \Rn$ the bound
	\begin{equation}
		\label{eq:DefFMB}
		\sup_{\substack{E'\in U_E\\ \eta\neq 0}} \mathbb{E}\left[\|\chi_a R_{E'+\i\eta}(H_G) \chi_b\|^s \right]
		\leq C \e^{-\mu|a-b|}
	\end{equation}
	holds true. Here, $\|\boldsymbol\cdot\|$ stands for the operator norm.
\end{definition}

\begin{remarks}
\label{rem:FMBS}
\item
	Suppose that $I\subset\FMB$ for a compact set $I \subset \R$. Then there exist
	constants $C,\mu$ such that \eqref{eq:DefFMB} holds uniformly for every $E\in I$ with these constants.
\item
	\label{FMBalways}
	Suppose that \eqref{eq:DefFMB} holds for \emph{some} exponent
	$0<s<1$, then it holds for \emph{all} exponents $0<s<1$ with constants $C$ and $\mu$ depending on $s$.
	The original proof of this fact in \cite[Lemma B.2]{artRSO2001AizEtAl} for discrete models extends to
	the continuum setting, see Lemma~\ref{lem:PossFrac}.
\item
	\label{FMBalways2}
	We show in Lemma~\ref{Lem:PertPersPot} that the stability of the regime of complete localisation
	$\FMB(H)=\FMB(H^\tau)$ 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 \eqref{eq:DefFMB} for the unperturbed operator
	$H$ only and -- according to the previous remark -- with a fixed exponent $s$.
\item
	\label{rem:FMBholds}
	Bounds of the form \eqref{eq:DefFMB} have first been derived for the lattice Anderson model in
	\cite{artRSO1993AizMol}, see also \cite{artRSO1998AizGr,artRSO2001AizEtAl}, either for sufficiently strong
	disorder or in the Lifshitz-tail regime. They were generalized to continuum random Schr\"odinger operators
	in \cite{artRSO2006AizEtAl2}. The formulation there differs with respect to the distance function
	that is used in \eqref{eq:DefFMB}. We refer to \cite[(8) in App.\ A]{artRSO2006AizEtAl2} for an
	interpretation. Bounds as in \eqref{eq:DefFMB} have been derived in \cite{MR2303305} for the
	fluctuation-boundary regime by adapting the methods from \cite{artRSO2006AizEtAl2}.
		Fractional-moment bounds for the resolvent imply Anderson localisation in the strongest form, see e.g.\
	\cite{Germinet2006,AizWarBook}.
\item
	The fractional-moment bounds \eqref{eq:DefFMB} hold for energies below $\inf(\Sigma)$ as follows from a
	Combes-Thomas estimate \cite[Cor.\ 1]{MR1937430}, which extends to finite-volume operators.
\item
	\label{rem:aPrioriholds}
	The proof of \eqref{eq:DefFMB} in \cite{artRSO2006AizEtAl2} for alloy-type random Schr\"odinger 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 $I \subset \R$ there exists a finite constant $C_{s}>0$ such that
	\begin{equation}
		\label{eq:AppAPriori}
		\sup_{\tau\in[0,1]}\sup_{\substack{E\in I,\eta\neq 0\\G\subseteq\Rn,\, a,b\in\Rn}} \mathbb{E}
		\left[ \Vert \chi_{a}R_{E+\i\eta}(H^{\tau}_G)\chi_{b} \Vert^{s} \right] \leq C_{s}.
	\end{equation}
	Except for the supremum in $\tau$, this is the content of Lemma 3.3 and the subsequent Remark (4) in \cite{artRSO2006AizEtAl2}. 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	$\tau\in[0,1]$.
\end{remarks}

\section{Results}
\label{sect:results}

The results in Subsection \ref{sect:TraceClassEst} are at the heart of the proofs for the applications on the
spectral shift function and Anderson's orthogonality in the following two subsections.

\subsection{Schatten-von Neumann-class estimates}
\label{sect:TraceClassEst}

First, we fix our notation. Given a compact operator $A$ on a separable Hilbert space, we order its
singular values $b_1(A)\geq b_2(A)\geq...$ in a non-increasing way. For $p>0$ we say that $A\in \S^p$,
the \emph{Schatten-von Neumann $p$-class}, if
\beq
	\label{def:SchattenClass}
	\|A\|_p := \l(\Tr (|A|^p)\r)^{1/p} = \Big(\sum_{n\in\N} b_n(A)^p\Big)^{1/p}<\infty.
\eeq
Definition \eqref{def:SchattenClass} induces a complete norm on the linear space $\mathcal{S}^p$ for $p\geq 1$,
while for $0<p<1$ it induces a complete quasi-norm for which the adapted triangle inequality
\beq\label{lem:conkav}
	\|A+B\|_p^p \leq \|A\|_p^p+\|B\|_p^p
\eeq
holds \cite[Thm. 2.8]{McCarthy}.
We state our results for functions $f$ on $\R$ which belong to the space of functions of bounded variation
\beq
\TV(\R):=\big\{ f:\R\to\R\ \text{measurable}: \ \NTV(f)<\infty\big\}.
\eeq
Here, the \emph{total variation} is given by $\NTV(f):= \sup_{(x_{p})_{p} \in \mathcal P}\sum_{p}
|f(x_{p+1}) - f(x_p)|$ with the supremum being taken over the set $\mathcal P$ of all finite partitions of $\R$.
We write $\TV_c(\R)$ for the subspace of all functions in $\TV(\R)$ with compact support.
Finally, given a bounded interval $I=[I_-,I_+]\subset\R$, we define the vector space of functions
\begin{equation}
	\mathcal{F}_I := \Big\{f\in \TV(\R):\  f\big|_{(-\infty,I_-]}\equiv \text{ const.} , \,
	f\big|_{[I_+,\infty)}\equiv 0 \, \Big\} \subset L^{\infty}(\R),
\end{equation}
which is a normed space with respect to the total variation.

\begin{theorem}
	\label{th:TrClBounds1}
	Fix $p>0$, $0<s<1$ and let $I \subset \FMB$ be a compact interval. Then there exist finite constants
	$C,\mu>0$ such that the following holds: for all $G\subseteq\Rn$ open, $a,b\in \R^d$, $\tau \in [0,1]$
	and $f \in \mathcal{F}_I$ we have
	\begin{equation}
		\label{eq:TrClBounds1Stat}
		\mathbb{E} \big[ \big\| \chi_a\big( f(H_G) - f(H^{\tau}_G) \big) \chi_b \big\|_p  \big]
		\leq C|\tau|^{s/2}			\NTV(f) \e^{-\mu(|a|+|b|)}.
	\end{equation}
\end{theorem}

\noindent
The method we use in the proof of Theorem \ref{th:TrClBounds1} can also be applied to treat
a perturbation by a boundary condition.

\begin{theorem}
	\label{th:TrClBounds3}
	Fix $p>0$ and let $I \subset \FMB$ be a compact interval. Then there exist finite constants $C,\mu>0$
	such that the following holds: for all open subsets $G\subset \wtilde{G}\subseteq \R^d$ with
	$\dist(\partial\wtilde{G},\partial G)\geq 1$, for all $a,b\in \R^d$ such that
	$Q_a\cap {G}\ne \emptyset$ or $Q_b\cap {G}\ne \emptyset$, for all $\tau\in[0,1]$ and
	all $f \in \mathcal{F}_I$ we have
	\begin{equation}
		\label{eq:TrClBounds3Stat}
		\mathbb{E} \big[\big\| \chi_a\big(f(H_{G}^{\tau}) - f(H^{\tau}_{\wtilde{G}})\big)\chi_b \big\|_p  \big]
		\leq C  \, \NTV(f)
		\e^{-\mu [\dist(a, \partial {G}) + \dist(b, \partial {G})]}.
	\end{equation}
\end{theorem}

\begin{remarks}
\item
	The separate exponential decay of \eqref{eq:TrClBounds1Stat} 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 \eqref{eq:TrClBounds3Stat}.
\item
	Choosing $\wtilde{G}=\R^{d}$, Theorem~\ref{th:TrClBounds3} controls the macroscopic limit of
	$f(H^\tau_{G})$.
\item
	From a structural point of view it is only the fractional-moment bound \eqref{eq:DefFMB}
	and the a priori bound \eqref{eq:AppAPriori} rather than the precise form of the Schr\"odinger
	operator which enter in our proofs of	Theorems \ref{th:TrClBounds1} and \ref{th:TrClBounds3}.
	Similar methods are used in \cite{DGHKM2016} to treat a change of boundary conditions from Dirichlet to Neumann.
\item
	Our proof of the above results relies on an application of the Helffer-Sj\"ostrand formula,
	Corollary \ref{cor:HSrandom} below, to express the operators $f(H_{G})$ and $f(H^\tau_{G})$
	in terms of the resolvents of $H_{G}$ and $H^\tau_{G}$. To handle functions $f\in\mathcal{F}_I$,
	which may have jump discontinuities,
	we elaborate on the assumptions of the Helffer-Sj\"ostrand formula
	by trading regularity of $f$ against regularity of the resolvent, see Lemma \ref{lem:LocHelfSjo}.
	In the case of random Schr\"odinger operators this regularity is provided by the a priori bound
 \eqref{eq:AppAPriori}.
\end{remarks}

\noindent
A consequence of Theorem~\ref{th:TrClBounds1} is

\begin{corollary}
	\label{cor:TrClBounds2}
	Fix $p,q>0$ and $\tau \in [0,1]$. Let $I \subset \FMB$ be a compact interval.
	Then there exists a finite constant $C_{\tau}>0$ such that for all $G\subseteq\Rn$ open and $f \in \mathcal{F}_I$
	\begin{align}
	\label{eq:TrClBounds2Stat}
	\big(\mathbb{E} \left[ \| f(H_G) - f(H^{\tau}_G) \|_p^q  \right]\big)^{1/q} &\leq C_{\tau}\,  \NTV(f).
	\end{align}
	The constant $C_{\tau}$ vanishes algebraically in $\tau$ as $\tau \downarrow 0$.
\end{corollary}

\begin{remark}
	\label{rem:T-transition}
	If $U$ is a bounded and sufficiently fast decaying potential,
	it is shown in \cite{MR3314510} that the operator difference
	$f(-\Delta)-f(-\Delta+U)\in \S^1$ for functions $f$ in an appropriate
	Besov space of weakly differentiable functions. In contrast, for $f=\id_{(-\infty,E]}$, $E\in\R$,
	the operator $\id_{(-\infty,E]}(-\Delta) - \id_{(-\infty,E]}(-\Delta+U)$
	exhibits bands of absolutely continuous spectrum whose widths is
	determined by the scattering phase shifts at energy $E$ \cite{Pushnitski:2008fa}.
	In contrast, Corollary~\ref{cor:TrClBounds2} shows that if $-\Delta$ is replaced by a random
	Schr\"odinger operator, $E\in\FMB$ and $U$ has compact support, then
	$\id_{(-\infty,E]}(H) - \id_{(-\infty,E]}(H^{\tau})$ belongs to any Schatten-von Neumann class.
	The next subsection deals with more applications towards the spectral shift.
\end{remark}

\noindent
Theorem \ref{th:TrClBounds1} and Theorem \ref{th:TrClBounds3} together imply

\begin{theorem}
	\label{Thm:prod}
	Fix $p,q>0$ and let $I \subset \FMB$ 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 $L>0$
	\begin{equation}
		\label{thm:prod:eq1}
		\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}
		\leq C\NTV(f) \e^{-\mu L}.
	\end{equation}
	Moreover, given a sequence $(L_n)_{n\in\N} \subset (0,\infty)$ of lengths with $L_{n}/\ln n \to \infty$
	as $n \to\infty$, then for every $\gamma\in (0,\mu)$ the convergence
	\begin{equation}
		\label{eq:ConvergencePi}
	  \lim_{n\to\infty} \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
	\end{equation}
	holds almost surely.
\end{theorem}

\begin{remark}
	\label{fast-convergence}
	If $(X_{n})_{n\in\N} \subset (0,\infty)$ is a sequence of random variables with summable expectation,
	$(\E [X_{n}])_{n\in\N} \in \ell^{1}(\N)$, then $\lim_{n\to\infty} X_{n} =0$ holds almost surely.
	This elementary fact shows that \eqref{eq:ConvergencePi} follows directly from \eqref{thm:prod:eq1}.
\end{remark}

\noindent
Corollary~\ref{cor:TrClBounds2} or Theorem~\ref{Thm:prod} offer a way to detect spectral phases
of random Schr\"odinger operators other than the region of complete localisation.

\begin{corollary}
	\label{phase-transition}
	Let $E \in \Sigma$ and $p>0$, then
	\begin{equation}
		\label{nonFMBcrit}
 		\sup_{L>0} \E \big[ \|T(E,{H}_{L}, {H}^\tau_{L})\|_{p} \big] = \infty \quad\; \Longrightarrow \quad\;
		E \notin \FMB.
	\end{equation}
\end{corollary}

\begin{remark}
	The criterion of Corollary~\ref{phase-transition} should be compared to the behaviour of the spectral shift operator
	in the (hypothetical) conducting phase, provided $W\ge 0$:
	the finite-volume spectral shift operator $T(E,{H}_{L}, {H}^\tau_{L})$ is finite rank for all
	$E\in\R$, hence even super trace class. Lemma \ref{th:SimonsTrace1} then implies the representation
	\begin{align}
		\label{eq:shift-product}
		\|T(E,{H}_{L}, {H}^\tau_{L})\|_{2n}^{2n} & =  \|\id_{(-\infty,E]}({H}_{L})
			\id_{(E,\infty)}(H^\tau_{L}) \id_{(-\infty,E]}({H}_{L})\|_{n}^{n} \notag\\
		& \quad\; + \|\id_{(E,\infty)}({H}_{L}) \id_{(-\infty,E]}({H}^\tau_{L}) \id_{(E,\infty)}({H}_{L})\|_{n}^{n}
	\end{align}
	for every $n\in \N$.
	Now, suppose we knew there exists a
	spectral  interval $J \subset\R$ such that $H$ has absolutely continuous spectrum in $J$ almost surely.
	It follows from \eqref{eq:shift-product} and \cite[Thm.\ 3.4]{artAOC2015GKMO} that, almost surely given any $p>0$,
	$\|T(E,{H}_L,{H}_L^\tau)\|_{p}$ 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 \eqref{gamma} below is strictly positive.
	This demonstrates that the criterion \eqref{nonFMBcrit} is potentially useful because its assumption holds in this case.
	The suppression of the logarithmic divergence in the region of complete localisation resembles the
	suppression of the logarithmic correction of the area law for the
	entanglement entropy observed in the discrete Anderson model in the region of complete localisation
	\cite{PhysRevLett.113.150404,ElgartPasturShcherbina2016}.
\end{remark}

\subsection{Application 1: The spectral shift function}\label{subsec:ssf}

In this subsection we apply the findings of the previous subsection to prove convergence of the finite-volume
spectral shift functions in the macroscopic limit for energies in the region of complete localisation.
Moreover, we show that the limit, which is the infinite-volume spectral shift function, coincides with the trace of
the spectral shift operator and also with the index of the corresponding pair of Fermi projections.
In addition, we establish H\"older continuity and positivity of the disorder-averaged spectral shift function
in the region of complete localisation.

First, we introduce the relevant quantities. Let $A$ and $B$ be self-adjoint operators in a Hilbert space.
For any given $E \in \R$, we call
\begin{equation}
	\label{def:T(E)}
 	 T(E,A,B):= \id_{(-\infty,E]}(A) - \id_{(-\infty,E]}(B)
\end{equation}
a \emph{spectral shift operator} (for $A$ and $B$). Being a difference of two projections, we recall that
\begin{equation}
	\label{eq:T<1}
 	\|T(E,A,B) \| \le 1.
\end{equation}
Now, assume further that $A$ and $B$ are bounded from below
and that $\e^{-A}- \e^{-B}\in \S^1$.
Then there exists a unique function $\xi := \xi(\,\pmb\cdot\,,A,B)\in L_{\text{loc}}^{1}(\mathbb{R})$, the
\emph{spectral shift function} (for $A$ and $B$), such that the equality
\begin{equation}
	\label{eq:Definition1SSF}
	\Tr \big(f(A)-f(B)\big) = - \int_{\mathbb{R}}  \d\lambda\, f'(\lambda)\,\xi(\lambda,A,B)
\end{equation}
holds for all test functions $f\in C^{\infty}(\mathbb{R})$ with $\lim_{\lambda\to\infty}f(\lambda)=0$
and $\supp (f')$ compact. We refer to, e.g., \cite{yafaev1992mathscattering} for details and further properties.
Finally, given two self-adjoint projections $P$ and $Q$ such that
$\pm1 \notin\sigma_{\text{ess}}(P-Q)$, the essential spectrum of $P-Q$,
we define their \emph{index} according to \cite[Prop.\ 3.1]{artIND1994AvSeSi} as
$\ind(P,Q) := \dim\ker (P-Q-1) - \dim\ker(P-Q+1)$. Further background on the index can be found in
\cite{artIND1994AvSeSi}.
In the particular case of spectral projections $P=\id_{(-\infty,E]}(A)$ and $Q=\id_{(-\infty,E]}(B)$, we write
\begin{align}
	\label{def:index}
	\theta(E,A,B) &:= 	\ind \big\{ \id_{(-\infty,E]}(A), \id_{(-\infty,E]}(B) \big\} \notag\\
	& \phantom{:}= \dim \ker\big( T(E,A,B)-\id\big) - \dim \ker\big( T(E,A,B)+\id\big).
\end{align}

Next, we turn to the cases of interest for us, namely $A=H_{(L)}$ and $B=H_{(L)}^{\tau}$ for $L>0$. We recall from
Corollary \ref{cor:TrClBounds2} that
\beq
	\label{eq:Ttraceclass}
	T(E,H_{(L)}, H^\tau_{(L)}) \in \S^{1} \qquad \text{almost surely for every $E\in\FMB$}
\eeq
and $L>0$. Hence, the index $\theta(E,H_{(L)},H^\tau_{(L)})$ is well defined almost surely for every $E\in\FMB$
and every $L>0$. On the other hand, we infer from \cite[Thm.~1]{MR2200269} that
$\e^{-H_{(L)}} - \e^{-H^\tau_{(L)}} \in\S^{1}$ holds almost surely for all $L>0$.
The spectral shift function $\xi(\,\pmb\cdot\,, H_{(L)}, H^\tau_{(L)})$ 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 spectral shift function,
the index and the trace of the spectral shift operator, are well-defined and
coincide, see e.g.\ Prop.\ 2.1 and its proof in \cite{art2009PushnIndex}.
We show in the next theorem that, in the case of the operators $H$ and $H^{\tau}$,
the equality of the three quantities extends to the region of complete localisation.

\begin{theorem}
	\label{th:SSFandTransOp}
	Let $\tau \in [0,1]$. Then,
	\begin{nummer}
	\item
		\label{th:SSFandTransOpi}
		the spectral shift function, the trace of the shift operator and the Fredholm index
	  coincide, i.e.
		\begin{equation}
			\label{th:SSFandTransOpstat1}
			\xi(E,H,H^\tau)  = 	\Tr\big( T(E,H,H^\tau)\big)
			=\theta(E,H,H^\tau) \qquad \text{ for a.e. } E\in \FMB
		\end{equation}
		holds almost surely.
	\item
		\label{th:SSFandTransOpii}
		given a compact interval $I \subset \FMB$, there exist constants $C,\mu >0$ such that the bound
		\begin{equation}
			\label{th:SSFandTransOpstat3}
			\mathbb{E}\left[\l| \xi(E,H_{L},H^\tau_{L})-\xi(E,H,H^\tau)\r| \right]
			\leq C \e^{-\mu L} \qquad \text{ for a.e. } E\in I
		\end{equation}
		holds for all $L>0$. Moreover, given a sequence of lengths with $L_{n}/\ln n \to \infty$
		as $n\to\infty$, the convergence
		\begin{equation}
			\label{th:SSFandTransOpstat2}
			\lim_{n\to\infty} \xi(E,H_{L_n},H^\tau_{L_n})=  \xi(E,H,H^\tau)\qquad \text{ for a.e. } E\in I
		\end{equation}
		holds almost surely.
	\end{nummer}
\end{theorem}

\begin{remarks}
\item
	\label{rem:right-equality-clear}
	It follows from \cite[Thm.\ 4.1]{artIND1994AvSeSi} that \eqref{eq:Ttraceclass} implies the equality
	\beq
		\label{eq:trShift-index}
		\Tr \big( T(E,H_{(L)}, H^\tau_{(L)}) \big) = \theta(E,H_{(L)},H^\tau_{(L)})
	\eeq
	almost surely for every $E\in\FMB$ and $L>0$. The news in Theorem~\ref{th:SSFandTransOpi}
	is the left equality. The right equality demonstrates that all quantities take on an integer value.
\item
 	In general, one cannot expect the spectral shift function 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 spectral shift function may
	be well defined. But in this case, the spectral shift function cannot be expected to be integer-valued,
	whereas the index is by definition.
\item For the very same reason, the convergence \eqref{th:SSFandTransOpstat2} cannot hold in general.
	As there is no essential spectrum, the finite-volume spectral shift function is
	given as a difference of eigenvalue-counting functions
	\begin{equation}
		\label{eq:xi-T}
 		\xi(E,H_{L},H_{L}^{\tau}) = \Tr \big(T(E, H_{L},H_{L}^{\tau}) \big)
		= \Tr \big(\id_{(-\infty,E]}(H_{L})\big) - 	\Tr \big(\id_{(-\infty,E]}(H_{L}^{\tau})\big)
	\end{equation}
	for all $E\in\R$ and are, thus, integer valued by definition. The infinite-volume quantity need not be.
  However, for a general Schr\"odinger operator $-\Delta+V_0$ and a perturbation by a
  bounded, compactly supported potential, vague convergence of the finite-volume spectral shift functions
  is known \cite{MR2596053}.
\item
	Theorem \ref{th:SSFandTransOpii} is formulated in terms of the spectral shift function.
	It holds for the index as well, even without having to exclude a Lebesgue null set of exceptional energies.
	Given a sequence of lengths with $L_{n}/\ln n \to \infty$ as $n\to\infty$, the analogue of
	\eqref{th:SSFandTransOpstat2} reads
	\begin{equation}
		\label{eq:SSFandIndex}
		\lim_{n\to\infty}\theta(E,H_{L_n},H^\tau_{L_n}) = \theta(E,H,H^\tau)
	\end{equation}
	almost surely for \emph{every} $E\in \FMB$. This follows directly from \eqref{eq:trShift-index} and an
	application of Theorem~\ref{Thm:prod} to the function $f=\id_{(-\infty, E]}$.
\end{remarks}

\noindent
The previous theorem immediately implies the boundedness of the
finite-volume spectral shift functions with respect to the volume.

\begin{corollary}
	\label{cor:bbdfiniteSSF}
	Let $\tau\in [0,1]$. Given a sequence of lengths with $L_{n}/\ln n \to \infty$ as $n\to\infty$, then
	\beq
		\label{bdd:finiteSSF}
		\sup_{n\in\N} |\xi(E,H_{L_n},H_{L_n}^{\tau})| <\infty \qquad \text{ for a.e. } E\in \FMB
	\eeq
	holds almost surely.
\end{corollary}

\begin{remarks}
\item
	Bounds like \eqref{bdd:finiteSSF} are unknown for general deterministic continuum Schr\"odinger operators
	in $d\ge 2$, and it is not even clear whether they are true. In the case of the pair $-\Delta$ and
	$-\Delta+W$ with $0\leq W\in L^\infty_c(\R^d)$, a partial converse is shown in
	\cite{artSSF1987Kir}: for any diverging sequence $(L_n)_{n\in\N}$ there exists a dense
	subset $I\subseteq (0,\infty)$ such that for all $E\in I$
	\beq
		\sup_{n\in\N} \xi(-\Delta_{L_{n}},-\Delta_{L_{n}}+W,E) =\infty.
	\eeq
	In contrast, $L^{p}$-bounds for finite- or infinite-volume spectral shift functions of deterministic
	continuum Schr\"odinger operators are known, see for example \cite{MR2200269, MR1824200, MR1945282}.
\item
	In the case of random Schr\"odinger operators and for a particular choice of the perturbation,
	\cite{MR2352262} show that the disorder-averaged finite- and infinite-volume spectral shift functions
	are locally bounded uniformly in the system size.
\end{remarks}

\noindent
The next lemma implies H\"older continuity of the averaged trace of the shift operator in the region of complete localisation.

\begin{lemma}
	\label{lem:HoelderCont}
	Let $I\subset\FMB$ 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 $E,E'\in I$
	\begin{equation}
 		\E \left[\big\| T(E,H_{(L)},H_{(L)}^{\tau})- T(E',H_{(L)},H_{(L)}^{\tau}) \big\|_{1} \right]
		\leq C |E-E'|^{\alpha}.
	\end{equation}
\end{lemma}

\begin{remark}
  Choosing representatives of the spectral shift functions as in Theorem \ref{th:SSFandTransOpi} and
  \eqref{eq:xi-T}, we infer from Lemma~\ref{lem:HoelderCont} that
  \begin{equation}
 		\E \left[\big\vert \xi(E,H_{(L)},H_{(L)}^{\tau})-\xi(E',H_{(L)},H_{(L)}^{\tau}) \big\vert \right]
		\leq C |E-E'|^{\alpha}
	\end{equation}
	for all $E,E'\in I$.
\end{remark}

\noindent
Lemma~\ref{lem:HoelderCont} will be used in the proof of the next theorem which establishes
pointwise positivity of the averaged spectral shift function
in the region of complete localisation for sufficiently positive perturbations. This will also be useful
in the following section on Anderson orthogonality.

\begin{theorem}
	\label{lem:SSFcont}
	Assume that the Lebesgue density $\rho$ of the single-site distribution, which is specified in \ref{assV1},
	satisfies
	\begin{equation}
		\rho_-:= \essinf_{x\in[0,1]} \rho(x) > 0.
	\end{equation}
	Let $\tau \in (0,1]$ and assume that the perturbation $W$ and the bump function $u_0$ satisfy
	\beq
		\label{bounds:u0}
		W \geq C u_0\geq c \id_{B_\delta(x)}
	\eeq
	with constants $c,C, \delta>0$ and some $x\in\Lambda_1$. Then
	\begin{equation}
		\E\l[\xi(E,H,H^{\tau})\r]>0 \qquad \text{ for a.e. } E\in \FMB\cap\, \{E'\in\R :\, \cN'(E')>0\},
	\end{equation}
	where $\cN'$ is the density of states of $H$.
\end{theorem}

\begin{remark}
	\label{rem:SSFcont}
	It is proven in \cite{DGHKM2016} that under the assumptions of Theorem \ref{lem:SSFcont} we have
	\begin{equation}
		\label{eq:dospos}
		\essinf_{E\in I} \cN'(E) > 0
	\end{equation}
	for every compact interval $I\subset \FMB\cap \Int\big(\sigma(H_0) + [0,C_{u,-}]\big)$,
	where $C_{u,-}$ is the constant from \eqref{uk-bounds} and $\text{int} (A)$ stands for the interior of a set
	$A\in\Borel(\R)$. This implies that
	\beq
		\E\l[\xi(E,H,H^{\tau})\r]>0 \qquad \text{ for a.e. } E\in \FMB\cap \Int\big(\sigma(H_0) + [0,C_{u,-}]\big).
	\eeq
\end{remark}

\subsection{Application 2: Anderson orthogonality}\label{subsec:aoc}

We apply the Schatten-von Neumann-class estimates from Section \ref{sect:TraceClassEst} 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
\beq
	\lambda_1^L\le \lambda_2^L\le\cdots\qquad\text{and}\qquad \mu_1^L\le \mu_2^L \le \cdots,
\eeq
respectively, ordered by magnitude and repeated according to their multiplicities. The corresponding
orthonormal bases of eigenfunctions are $(\varphi_k^L)_{k\in\N}$ and $(\psi_k^L)_{k\in\N}$,
respectively. For $N\in\N$ we consider two non-interacting $N$-Fermion systems
in the box $\Lambda_L$ with single-particle Schr\"odinger operators $H_L$ and $H^\tau_L$.
The corresponding $N$-particle operators are
\begin{equation}
  \boldsymbol{H}^{(\tau)}_{N,L} :=
  \sum_{j=1}^N \underbrace{\id\otimes\dotsm\otimes \id}_{(j-1) \text{times}} \otimes H_L^{(\tau)}
  \otimes \underbrace{\id\otimes\dotsm\otimes \id}_{(N-j) \text{times}}.
\end{equation}
They act on the totally antisymmetrised tensor product $\H_{N,L}:=\bigwedge_{j=1}^N L^2(\Lambda_L)$
with standard scalar product $\langle\,\pmb\cdot\,, \,\pmb\cdot\,\rangle_{\H_{N,L}}$.
The respective ground states of $\boldsymbol{H}_{N,L}$ and $\boldsymbol{H}^{\tau}_{N,L}$ are then
given by the totally antisymmetrised and normalised tensor products
\begin{equation}
  \boldsymbol{\Phi}_{N,L} := \varphi_1^L\wedge\dotsm\wedge\varphi_N^L
  \qquad\text{and}\qquad
  \boldsymbol{\Psi}_{N,L} := \psi_1^L\wedge\dotsm\wedge\psi_N^L.
\end{equation}
The \textit{modulus} of their scalar product is given by
\begin{equation}
	\label{intro:defS}
	S_{N,L} := \big|\big\<\boldsymbol{\Phi}_{N,L},\boldsymbol{\Psi}_{N,L}\big\>_{\H_{N,L}}\big|
	= \left|\det \begin{pmatrix}
          \<\varphi^L_1,\psi^L_1\>& \cdots & \<\varphi^L_1,\psi^L_N\>\\
          \vdots                  &        &  \vdots \\
          \<\varphi^L_N,\psi^L_1\>& \cdots &  \<\varphi^L_N,\psi^L_N\>
    \end{pmatrix} \right|,
\end{equation}
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 $K\in \S^1$
with eigenvalues $(b_n)_{n\in\N}$, listed according to their algebraic multiplicities,
the Fredholm determinant of $\id-K$ is defined by
\beq
	\det( \id- K) := \prod_{n\in\N} ( 1- b_n).
\eeq
It is well defined by the trace-class assumption on $K$ \cite[Sect.\ XIII.17]{reed1980methods4}.
The following representation of the overlap in terms of Fredholm determinants will be useful.

\begin{lemma}
	\label{Fredholm:Repr}
	Let $N\in\N$, $L>0$ and define the orthogonal projections
	\begin{equation}
		\label{eq:projections_till_N}
  	P:= P_{N,L} := \sum_{j=1}^N |\varphi_j^L\>\<\varphi_j^L|
	  \quad\text{and}\quad
  	Q:= Q_{N,L} := \sum_{k=1}^{N} \big|\psi_k^L\big\> \big\<\psi_k^L\big|,
	\end{equation}
	which we have written down in Dirac notation. Then
	\begin{align}
		\label{fredholm1}
		S_{N,L} &= \det\big( \id- (P - Q)^2\big)^{1/4}  \notag \\
    & =  \det\big( \id- P(\id - Q) P \big)^{1/2}
    =  \det\big( \id- (\id - P) Q (\id- P) \big)^{1/2}.
	\end{align}
\end{lemma}

For fixed $E\in\R$, which will be referred to as the \emph{Fermi energy} in this context,
we define the $L$-dependent particle number
\begin{equation}
	\label{eq:DefParticleNumberSeq}
	N_{L}(E) := \Tr \big(\id_{(-\infty,E]} (H_L)\big).
\end{equation}
The limit $L \to\infty$ is a realisation of the thermodynamic limit with particle density given by
the integrated density of states $\lim_{L\to\infty} N_{L}(E)/|\Lambda_L| = \cN(E)$ of the Hamiltonian $H$.
We define the \emph{finite-volume ground-state overlap}
\begin{equation}
\label{eq:DefOverlap}
	 S_L(E):= \left\{\begin{array}{c@{\quad\text{~if}\quad}l} S_{N_{L}(E),L}, & N_{L}(E) \in\N, \\[.5ex]
		 1, & N_{L}(E)=0.\end{array}\right.
\end{equation}
Following physics terminology, we say that \emph{Anderson orthogonality} occurs at an energy $E$, if $S_{L}(E)$ vanishes as $L\to\infty$.

Note that $P_{N_{L}(E),L} = \id_{(-\infty, E]}(H_{L})$. If $\xi(E,H_{L},H_{L}^{\tau})=0$ holds,
then we also have the identity $Q_{N_{L}(E),L} = \id_{(-\infty, E]}(H_{L}^{\tau})$. In this case we obtain
 \begin{equation}
 	\label{eq:Srep:xi0}
 	S_{L}(E) = \det\big(\id - T(E, H_{L},H_{L}^{\tau})^{2}\big)^{1/4}.
\end{equation}
This motivates to define the \emph{infinite-volume ground-state overlap} as
\begin{equation}
	\label{fredholm2}
	S(E):= \left\{\begin{array}{c@{\quad}l} \det \big( \id - T(E,H,H^\tau)^2 \big)^{{1/4}},
		&\text{~ if } \;T(E,H,H^\tau)\in \mathcal{S}^2, \\[.5ex]
		0,  &\text{~ otherwise.} \end{array} \right.
\end{equation}
We note that if $E\in \FMB$, then $T(E,H,H^\tau)\in\mathcal{S}^p$ almost surely for any $p>0$ by
Corollary \ref{cor:TrClBounds2}.

\begin{theorem}
	\begin{nummer}
	\item
		\label{th:IntMainResi}
		For all $E\in\FMB\cap \Int(\Sigma)$ the convergence
		\begin{equation}
			\label{eq:ResultNoAocStat1}
			\lim_{L\to\infty} \mathbb{E}\big[S_{L}(E)\big] = \mathbb{E}\big[S(E)\big]
		\end{equation}
		holds. If, in addition, $L_{n}/\ln n \to \infty$ as $n\to\infty$, then the pointwise convergence
		\begin{equation}
			\label{eq:ResultNoAocStatPointw}
			\lim_{n\to\infty} S_{L_n}(E) = S(E)
		\end{equation}
		holds almost surely.
	\item
		\label{th:IntMainResiprime}
		Let $E\in\FMB$. Then
		\begin{equation}
			\label{eq:ResultNoAocStat2}
			S(E)=0 \iff 1\in\sigma\big(T(E,H,H^\tau)^2\big)
		\end{equation}
		holds almost surely.
	\item
		\label{th:IntMainResii}
		Let  $I\subset \FMB$ be a compact interval.
		Then we have
		\beq\label{AbsAoc2}
			\lim_{\tau\downarrow 0} \inf_{E\in I}\,\E\big[S(E) \big] =  1.
		\eeq
	\item
		\label{th:IntMainResiii}
		Suppose that the assumptions of Theorem \ref{lem:SSFcont} are fulfilled. Then
		\begin{equation}\label{AbsAoc}
			\PP\big[S(E)= 0\big] > 0 \qquad\text{ for a.e. } E\in \FMB \cap \,\{ E'\in\R: \cN'(E') >0\},
		\end{equation}
		where $\cN'$ is the density of states of $H$.
	\end{nummer}
	\label{th:IntMainRes}
\end{theorem}

\begin{remarks}
\item
The set of energies for which \eqref{AbsAoc} holds is not empty for our model, see Remark \ref{rem:SSFcont}.
\item
	Theorem \ref{th:IntMainResii} 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. Indeed,
	if we suppose that the constant $C\equiv C_{\lambda}$ in Definition~\ref{DefFMB} vanishes in
	the strong-disorder limit,
	\begin{equation}
		\label{eq:const-large-lambda}
 		\lim_{\lambda\to\infty} C_{\lambda} =0,
	\end{equation}
	then the proof of Theorem~\ref{th:IntMainResii} shows that we also have
	\begin{equation}
 		\lim_{\lambda\to\infty} \inf_{E\in I} \E\big[S(E)\big] =1.
	\end{equation}
	Whether \eqref{eq:const-large-lambda} holds for alloy-type models in the continuum seems to be unknown
	\cite[App.\ A]{artRSO2006AizEtAl2}, but it does hold for the discrete Anderson model.
\item
	Theorem \ref{th:IntMainResiii} shows that Anderson orthogonality is not absent almost surely in the
	region of complete localisation, contrary to established wisdom
	in the physics literature \cite{PhysRevB.65.081106}.
	Only very recently physical reasoning and numerical evidence was put forward that Anderson orthogonality
	does occur in the region of complete localisation \cite{Khemani-NaturePhys15,PhysRevB.92.220201}.
\item
	For sign-definite perturbations $W$, the condition \eqref{eq:ResultNoAocStat2} for Anderson orthogonality to
	occur in the region of complete localisation is
	equivalent to a non-vanishing index,
	\begin{equation}
	\label{AOCequiIndex}
		S(E)=0 \iff \theta(E,H,H^\tau) \neq 0
	\end{equation}
	almost surely for every $E\in\FMB$.
	This follows from \eqref{eq:SimonsTrace2stat3} in Appendix \ref{App:SuppMater}.
\item
	Let $I \subset \FMB$ be compact. Then, Theorem \ref{th:IntMainRes} \itemref{th:IntMainResii} and
	\itemref{th:IntMainResiii} imply that there exists $\tau_I>0$ such that for all $0<\tau<\tau_I$
	and a.e.\ $E\in I \cap \,\{ E'\in\R: \cN'(E')\} >0$
	\beq
		0<\PP\big[S(E)=0\big]<1.
	\eeq
	Thus, $S(E)$ is not almost surely constant, and both presence and absence of Anderson orthogonality occur with positive probability for energies in the
	region of complete localisation.
\item
	\label{rm:expfastconv}
	Let $W$ be a sign-definite perturbation and let $E\in \FMB \cap \Int(\Sigma)$ such that
	$\theta(E,H,H^\tau) = 0$,
	that is, Anderson orthogonality
	cannot occur at this energy according to \eqref{AOCequiIndex}. In this case our proof of
	Theorem~\ref{th:IntMainResi} provides exponential speed of convergence of the overlap in
	\eqref{eq:ResultNoAocStatPointw}: we conclude from
	\eqref{eq:ProofIndexZero}  and \eqref{eq:ConvergencePi} 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
	\beq
		\label{expfastconv}
		\lim_{n\rightarrow\infty} \e^{cL_n}\big|S_{L_n}(E) - S(E)\big| = 0
	\eeq
	holds almost surely. According to the physical arguments in \cite{PhysRevB.92.220201},
	exponential speed of convergence occurs with positive probability for $E\in \FMB$ even if
	$\theta(E,H,H^\tau) \neq 0$.
	More precisely, it is argued there that $\E[\ln S_{L}(E)] \sim -L$ for large $L$.
  This phenomenon is dubbed
	\emph{statistical Anderson orthogonality} in \cite{PhysRevB.92.220201} to distinguish it from the
	usual algebraic decay for energies in the scattering regime, see also the next remark.
\end{remarks}

\noindent
Similarly to Corollary \ref{phase-transition}, Theorem~\ref{th:IntMainResi} and~\itemref{th:IntMainResii} offer a way to detect spectral phases
of random Schr\"odinger operators other than the region of complete localisation.

\begin{corollary}
	\label{rm:notFMB}
	Let $E \in \Int(\Sigma)$. Assume there exists a
	null sequence of coupling constants $(\tau_{k})_{k\in\N} \subset (0,1)$ and a sequence of
	lengths $(L_{n})_{n\in\N}$ with $L_{n}/\ln n \to \infty$ as $n\to\infty$ such that for every $k\in\N$
	\begin{equation}
 		\lim_{n \to \infty} S_{L_{n}}(E) =0 \qquad \text{almost surely},
	\end{equation}
	then $E \notin \FMB$.
\end{corollary}

\begin{remark}
	The criterion of Corollary \ref{rm:notFMB} should be compared to the behaviour of the overlap in the
	(hypothetical) conducting phase, provided $W\ge 0$:
	suppose we knew there exists a spectral interval $J \subset \Sigma$
	such that $H$ has absolutely continuous spectrum in $J$ almost surely. Let $(L_{n})_{n\in\N}$ be a
	sequence of lengths such that $L_{n} / \e^{n^{\alpha}} \to \infty$ as $n\to\infty$ for some $\alpha>1$.
	Then, Theorem 2.2 in \cite{artAOC2015GKMO}, see also \cite{artAOC2014GKM}, 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 $E\in J$
	\begin{equation}
		\label{AOC}
		S_{L_{n}}(E) \le  L_{n}^{-\gamma(E)/2 + o(L_{n}^{0})} \qquad\text{as $n\to\infty$}.
	\end{equation}
	The decay exponent
  \begin{equation}
  	\label{gamma}
		\gamma(E) := \frac{1}{2\pi^{2}} \Big\| \arcsin \Big|\frac{\id- \mathbb{S}_E}{2}\Big| \Big\|_2^2
	\end{equation}
	relates to the energy-dependent scattering matrix $\mathbb S_{E}$ and is strictly positive if the
	perturbation $W$ causes non-trivial scattering at energy $E$.
 	This demonstrates that the criterion in Corollary \ref{rm:notFMB} is potentially useful
	because its assumption holds in this case.
	We remark that the superexponential growth
	of the length scales $(L_{n})_{n\in\N}$ avoids the necessity of passing to a subsequence
	in \eqref{AOC} as is done in
	\cite[Theorem 2.2]{artAOC2015GKMO}, because it implies a sufficiently fast $L^{1}$-convergence in
	the proof of \cite[Lemma 3.3(i)]{artAOC2015GKMO} so that subsequences can be avoided in that lemma.
\end{remark}

\section{Proofs of the results from Section \ref{sect:TraceClassEst}}
\label{sec:4}

\subsection{Helffer-Sj\"ostrand formula}

Our analysis relies on an application of the \mbox{(Dynkin-)} Helffer-Sj\"ostrand formula which works for
compactly supported functions of bounded variation. Let $\mathcal{H}$ be a Hilbert space, $K$ a
self-adjoint operator on $\mathcal{H}$
and $f: \R \rightarrow \C$ a sufficiently regular function. According to the Helffer-Sj\"ostrand formula,
see e.g.\ \cite{MR1349825}, there exists a complex Borel measure $\zeta_{f}$ on $\R^{2}$ such that
\begin{equation}
	\label{eq:ClassHelfSjo}
	f(K) = \frac{1}{2\pi}\int_{\R^2} \d\zeta_f(x,y) \, R_{x+\i y}(K).
\end{equation}
The smoothness of $f$ determines how fast the measure $\zeta_f$ vanishes in the vicinity of the
horizontal axis $y=0$. This has to compensate the potential divergence of the resolvent as $|y| \to 0$.
For instance, if $f\in C_c^2(\R)$, then $\zeta_f$ can be chosen as $\d\zeta_{f}(x,y) = \d x \,\d y \,\tilde{f}(x,y)$ with
\begin{equation}
	\label{def:omega}
	\tilde{f}(x,y) := \left(\partial_x+\i\partial_y\right) \left((f(x)+\i yf'(x)) \Xi(x,y) \right).
\end{equation}
Here, $\Xi\in C_c^{\infty}(\R^2)$
is a cutoff function with $\Xi \equiv 1$ in a neighbourhood $N_f$ of $\supp (f)\times \{0\}\subset \R^2$.
In particular, the integral in \eqref{eq:ClassHelfSjo} is well defined in this case. The
Helffer-Sj\"ostrand formula \eqref{eq:ClassHelfSjo} implies
\begin{equation}
	\label{eq:ClassHelfSjo2}
	Af(K)B = \frac{1}{2\pi}\int_{\R^2} \d\zeta_f(x,y) \, A R_{x+\i y}(K) B
\end{equation}
for all bounded operators $A,B$ on $\mathcal{H}$. Now, it may be the case that the product $A R_{x+iy}(K) B$
has a less severe divergence as $|y|\to 0$ than the resolvent alone. In such cases, the right-hand side
of \eqref{eq:ClassHelfSjo2} typically is well defined for functions $f$ which are less regular than
what is needed for \eqref{eq:ClassHelfSjo}.
This is made precise in the next lemma.

For its formulation we consider functions $f\in \TV_c(\R)$. Given such $f$, we choose a cutoff function
$\Xi\in C_{c}^{\infty}(\R^{2})$ with $\Xi\equiv 1$ in a neighbourhood $N_f$ of $\supp(f)\times \{0\} \subset \R^2$.
We then define the complex Borel measure $\zeta_{f}$ on $\R^{2}$ by
\begin{equation}
	\label{eq:LocHelfSjoDef1}
	\d\zeta_f(x,y) := \d f(x)\, \d y\, \Xi(x,y) + \d x\,\d y\, f(x) \left(\partial_x+\i\partial_y\right) \Xi(x,y),
\end{equation}
where $\d f$ denotes Lebesgue-Stieltjes integration with respect to $f$. We write $|\zeta_{f}|$ for the
total variation measure of $\zeta_{f}$.

\begin{lemma}
	\label{lem:LocHelfSjo}
	Let $f\in \TV_c(\R)$ and $\zeta_{f}$ as in \eqref{eq:LocHelfSjoDef1}. Let $K$ be a self-adjoint operator and
	let $A$, $B$ be bounded operators on the Hilbert space $\mathcal{H}$. If
	\beq
		\label{eq:LocHelfSjoAssumpt}
		\int_{\R^2} \d|\zeta_{f}|(x,y)\,   \| A \, R_{x+\i y}(K) \, B \| <\infty,
	\eeq
	then
	\begin{equation}
		\label{eq:LocHelfSjoStat}
		A f(K) B = \frac{1}{2\pi} \int_{\R^{2}} \d\zeta_{f}(x,y) \, A \, R_{x+\i y}(K) \, B \,
	\end{equation}
	holds, where the right-hand side is a Bochner integral with respect to the operator norm.
\end{lemma}

\begin{remark}
	Lemma \ref{lem:LocHelfSjo} can be extended to appropriate Besov spaces, $B_{p,q}^s$, $1\leq p,q\leq \infty$
	and $0<s<1$, by using Dynkin's characterization of Besov spaces  in terms of quasi-analytic
	extensions \cite{zbMATH03808305}.
\end{remark}

Before we prove the lemma, we apply it to spatially localised functions of random Schr\"odinger operators which obey the a priori estimate \eqref{eq:AppAPriori}.

\begin{corollary}
	\label{cor:HSrandom}
 	Let $G\subseteq\R^d$ be open, $a,b\in G$ and let $H^{\tau}_G$ be defined as in \eqref{eq:TheOperator} and
	\eqref{def:perturbedoperator}. Then, for $f\in \TV_c(\R)$, the equality
	\begin{equation}
		\label{eq:LocHelfSjorandom}
		\chi_{a} f(H^{\tau}_{G}) \chi_{b} = \frac{1}{2\pi} \int_{\R^{2}} \d\zeta_{f}(x,y) \, \chi_{a} \,
		R_{x+\i y}(H^{\tau}_{G}) \, \chi_{b}
	\end{equation}
	holds almost surely.
\end{corollary}

\begin{proof}
	We verify that \eqref{eq:LocHelfSjoAssumpt} holds almost surely. There exists $\delta>0$ (independent of $f$)
	such that $\supp(\Xi) \subset \R \times [-\delta,\delta]$. For fixed $0<s<1$, we use the deterministic norm bound
	$\| R_z(H^{\tau}_G)\|^{1-s}\leq 1/ |\text{Im} z|^{1-s}$, $z\in\C\setminus\R$, and estimate
 	\begin{align}
		\label{eq:LocHelfSjo1}
		& \mathbb{E}\left[	\int_{\R^2} \d|\zeta_{f}|(x,y)\,  \| \chi_a\, R_{x+\i y}(H^{\tau}_G) \, \chi_b \| \right]
		\leq  C_s  \int_{\R^2} \frac{\d|\zeta_{f}|(x,y)}{|y|^{1-s}},
	\end{align}
	where
	\beq
		C_s:=
		\sup_{\substack{(E,\eta)\in \supp(\zeta_f)\\ \eta\neq 0}} \mathbb{E}
		\left[ \Vert \chi_{a}R_{E+\i\eta}(H^{\tau}_G)\chi_{b} \Vert^{s} \right] <\infty
	\eeq
   is finite by the a priori estimate \eqref{eq:AppAPriori}. The claim then follows from
	\begin{align}
		\label{eq:zeta-int}
 		\int_{\R^2} \frac{\d|\zeta_{f}|(x,y)}{|y|^{1-s}} \le \frac{2}{s}\, \delta^{s} \big(\NTV(f) \|\Xi\|_{\infty} +
		2 \|f\|_{1} \| |\nabla\Xi|\|_{\infty}\big).
		\\[-1ex]	\tag*{\qedhere}
	\end{align}
\end{proof}

\begin{proof}[Proof of Lemma \ref{lem:LocHelfSjo}]
	The first part of the proof closely follows the standard proof of the Helffer-Sj\"ostrand formula.
	We note that assumption \eqref{eq:LocHelfSjoAssumpt} implies that the right-hand side of
	\eqref{eq:LocHelfSjoStat} is well defined as a Bochner integral with respect to the operator norm.
	To show \eqref{eq:LocHelfSjoStat} we let $\varepsilon>0$ and introduce the closed horizontal strip
	$\mathcal{C}_{\varepsilon} := \R \times [-\varepsilon,\varepsilon]$ in $\R^{2}$.
	We split the integral according to
	\begin{align}
		\label{eq:LocHelfSjo2}
		\int_{\mathbb{R}^{2}}  \d\zeta_{f}(x,y) \,  A \, R_{x+\i y}(K) \, B
		&= \int_{\mathcal{C}_{\varepsilon}}	 \d\zeta_f(x,y) \, A \, R_{x+\i y}(K)  \,B  \notag\\
		& \quad	+\int_{\R^{2}\setminus \mathcal{C}_{\varepsilon}} \d\zeta_f(x,y) \, A\,R_{x+\i y}(K) \,B \notag\\
		&=: I_1^{\varepsilon} + I_2^{\varepsilon}.
	\end{align}
	Because of \eqref{eq:LocHelfSjoAssumpt}, dominated convergence yields
	$\lim_{\varepsilon\downarrow 0} \|I_{1}^{\varepsilon}\| =0$.
  As for the second integral, we note that $C \mapsto ACB$ is a norm-continuous linear map on the
  Banach space of bounded linear operators. The properties of the Bochner integral then imply
	\begin{equation}
		I_{2}^{\varepsilon} = A  \bigg( \int_{\R^{2}\setminus \mathcal{C}_{\varepsilon}} \d\zeta_f(x,y) \,
		R_{x+\i y}(K) \bigg) B.
	\end{equation}
 	Here, the right-hand side is well defined because $\supp(\zeta_{f})$ is compact and the
	norm of the resolvent is uniformly bounded on $\R^{2}\setminus \mathcal{C}_{\varepsilon}$.
	Now we choose $\varepsilon>0$ so small that $\Xi \equiv 1$ on $\supp(f) \times [-\varepsilon,\varepsilon]$.
	It follows from Fubini's theorem that $I_{2}^{\varepsilon} = 2\pi A f_{\varepsilon}(K)B$ with
	\begin{align}
		\label{eq:LocHelfSjo5}
 		f_{\varepsilon}(\lambda)
		&:= \frac{1}{2\pi} \int_{\R^{2}\setminus \mathcal{C}_{\varepsilon}}  \frac{\d\zeta_{f}(x,y)}{\lambda -x -\i y}
			=\frac{\i}{2\pi} \int_{\R} \d x\left(-\frac{f(x)\Xi(x,\varepsilon)}{\lambda-x-\i\varepsilon}
				+ \frac{f(x)\Xi(x,-\varepsilon)}{\lambda-x+\i\varepsilon}\right) \notag \\
		&\phantom{:}= \int_{\R} \d x \, f(x)\, \frac{\varepsilon/\pi}{(\lambda-x)^{2} + \varepsilon^{2}}
			= \int_{\R} \d x \, \frac{1/\pi}{x^{2} + 1} \, f(\lambda+ \varepsilon x)
	\end{align}
	for $\lambda\in\R$, where the first equality relies on integration by parts and holomorphy
	$(\partial_{x} + \i\partial_{y})(\lambda-x-\i y)^{-1} =0$ on $\R^{2}\setminus \mathcal{C}_{\varepsilon}$.
	The second equality follows from $\Xi(x,\varepsilon) = \Xi(x,-\varepsilon)=1$ for all $x \in\supp(f)$.

	The second part of the proof deals with the problem that discontinuity points of $f$ challenge the convergence of
	$I_{2}^{\varepsilon}$ as $\varepsilon\downarrow 0$. However, the regularity condition \eqref{eq:LocHelfSjoAssumpt}
	ensures that they form only a null set of the relevant spectral measures and, thus, weak convergence still holds.
	To see this, let
	$\varphi,\psi \in\mathcal{H}$ and define the complex spectral measure $\mu_{\varphi,\psi} := \langle \varphi, A \id_{\raisebox{-.4ex}{$\boldsymbol\cdot$}}(K) B\psi\rangle$ of $K$. The functional calculus, \eqref{eq:LocHelfSjo5} and dominated convergence imply
\begin{equation}
	\label{eq:f-eps-weak}
	 \lim_{\varepsilon\downarrow 0} \langle \varphi, A f_{\varepsilon}(K) B\psi\rangle =
	 \int_{\R}\d\mu_{\varphi,\psi}(\lambda) \, \lim_{\varepsilon\downarrow 0} \int_{\R} \d x\,
	 \frac{1/\pi}{x^{2} + 1} \, f(\lambda+ \varepsilon x).
\end{equation}
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 \eqref{eq:f-eps-weak} yields $\lim_{\varepsilon\downarrow 0} I_{2}^{\varepsilon} = 2\pi Af(K)B$ 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 $f\in \TV_{c}(\R)$, it has left and right limits at all points. Hence, the set
$\mathcal{U} := \{\lambda\in\R: f \text{ not continuous in } \lambda \}$ consists of jump discontinuities only. Moreover,
$\mathcal{U}$ is countable so that
\begin{equation}
\label{eq:U-count}
 \mu_{\varphi,\psi}(\mathcal{U}) = \sum_{\lambda \in \mathcal{U}} \mu_{\varphi,\psi}(\{\lambda\}).
\end{equation}
We fix an arbitrary $\lambda\in\mathcal{U}$ and set $\delta f_{\lambda} := \lim_{\varepsilon\downarrow 0} [f(\lambda+\varepsilon) - f(\lambda -\varepsilon)] \neq0$. We choose $y_{0} >0$ small enough such that $\Xi(\lambda,y)=1$ whenever $|y| \le y_{0}$.
Assumption \eqref{eq:LocHelfSjoAssumpt} implies
\begin{align}
	\label{eq:ass-cons}
	\infty & >  \int_{\R^2} \d|\zeta_{f}|(x,y)\,   \| A \, R_{x+\i y}(K) \, B \|
		\ge |\delta f_{\lambda}|  \int_{-y_{0}}^{y_{0}} \d y\, \| A \, R_{\lambda+\i y}(K) \, B \| \notag\\
	& \ge  |\delta f_{\lambda}|  \int_{-y_{0}}^{y_{0}} \frac{\d y}{|y|}\, |h(y)|,
\end{align}
 where $h(y) := \int_{\R}\d\mu_{\varphi,\psi}(\lambda') \, y/(\lambda' - \lambda -\i y)$. Dominated convergence implies
that $\R \ni y \mapsto h(y)$ is continuous and that $\lim_{y\to 0} h(y) = \i\,\mu_{\varphi,\psi}(\{\lambda\})$.
Now, we assume that $\mu_{\varphi,\psi}(\{\lambda\}) \neq 0$. Then there exists $0<y_{1} \le y_{0}$ such that
$|h(y)| \ge  |\mu_{\varphi,\psi}(\{\lambda\})|/2 $ whenever $|y| \le y_{1}$, and we conclude that \eqref{eq:ass-cons}
yields a contradiction. Therefore we must have $\mu_{\varphi,\psi}(\{\lambda\})=0$, and \eqref{eq:U-count} implies the desired continuity of $f$.
\end{proof}

\subsection{Proof of Theorem \ref{th:TrClBounds1} and Corollary \ref{cor:TrClBounds2}}
\label{pf:th:TrClBounds1}

We introduce a lower bound of the spectra of all considered random Schr\"odinger operators,
\begin{align}
	\label{eq:DefE0}
	E_0 &:= \min \big\{\inf_{x\in\R^d}  \{V_0(x)\}, \inf_{x\in \R^d} \{V_0(x)  + W(x)\}\big\} \notag\\
	& \phantom:\leq \min\big\{ \min\{\sigma(H_G)\}, \min\{\sigma(H^\tau_G)\}\big\}.
\end{align}
Here, the inequality holds almost surely for all $\tau \in [0,1]$ and all $G\subseteq \R^d$ open.

We begin with an operator-norm version of Theorem \ref{th:TrClBounds1}. Given a function $f\in \TV(\R)$,
we use the notation $\supp(f')$ for the support of the complex measure defined by Lebesgue-Stieltjes
integration with respect to $f$.

\begin{theorem}
	\label{th:TrClBounds1:OpNorm}
	Fix $0<s<1$ and a compact set $S \subset\FMB$. Then there exist finite constants $C,\mu>0$ such that
	for all functions $f\in\BV_c(\R)$ with $\supp(f')\subseteq S$, for all $\tau\in[0,1]$, all open $G\subseteq\Rn$
	and all $a,b\in \R^d$ we have
	\begin{equation}
		\label{eq:OpBounds1Stat2}
		\mathbb{E} \left[ \left\| \chi_a\big( f(H_G) - f(H^\tau_G) \big) \chi_b \right\|  \right]
		\leq C|\tau|^{s/2}\,\big( \|f\|_1+\NTV(f) \big) \e^{-\mu\left(|a|+|b|\right)}.
	\end{equation}
\end{theorem}

\begin{proof}
	We fix $f\in\BV_c(\R)$ such that $\supp(f')\subseteq S$. Let $S_{-} := \inf S$, $S_{+} := \sup S$ and
	observe $\supp(f) \subseteq [S_{-},S_{+}]$. We choose a cutoff function $\Xi=\Xi_{f}$ subject to
	\begin{MyDescription}
	\item[(P1)]{
		\label{K1}
		$\Xi\in C_c^{\infty}(\R^2)$ with $0\leq \Xi \leq 1$ and $\|\partial_{x}\Xi\|_{\infty},\|\partial_{y}
		\Xi\|_{\infty} \leq 3$,}
	\item[(P2)]{
		\label{K2}
		$\supp(\Xi)\subseteq [S_- -1,S_+ +1] \times [-1,1]$,}
	\item[(P3)]{
		\label{K3}
		$\Xi\equiv 1$ on $[S_- -1/2,S_+ + 1/2] \times [-1/2,1/2]$.}
	\end{MyDescription}
	Now fix $G\subseteq\Rn$ open and $a,b\in \R^d$. Let $\zeta_h$ be the complex Borel measure defined
	in \eqref{eq:LocHelfSjoDef1}. We apply Corollary~\ref{cor:HSrandom} to the operators $f(H_G)$ and $f(H^\tau_G)$.
	This gives the estimate
	\begin{align}
		\label{combes-thomas:eq10}
		\mathbb{E} &\left[ \big\| \chi_a\big( f(H_G) - f(H^\tau_G) \big) \chi_b \big\|  \right] \notag\\
		& \leq \frac 1 {2\pi} \E \l[ \int_{\mathbb{R}^2} \d|\zeta_{f}|(x,y) \;
			\big\|\chi_a \big(R_{x+\i y}(H_G)-R_{x+\i y}(H^\tau_G)\big)\chi_b \big\| \r] \notag\\
		&	\leq \frac {2^{1-s/2}} {2\pi} \E \l[ \int_{\mathbb{R}^2} \frac{\d|\zeta_{f}|(x,y)}{|y|^{1-s/2}}  \,
			\big\|\chi_a \big(R_{x+\i y}(H_G)-R_{x+\i y}(H^\tau_G)\big)\chi_b \big\|^{s/2} \r],
	\end{align}
	where, in the last step, we chose some $0<s<1$ and used the norm bound
	$\| R_z(H_{G}^{(\tau)})\|^{1-s/2}\leq 1/|\text{Im}z|^{1-s/2}$
	for the resolvent of a self-adjoint operator.
	We introduce the subset of lattice points
	\begin{equation}
		\label{def:GammaW}
 		\Gamma_{W} := \big\{n\in\Z^{d}: \dist\big(n,\supp(W)\big)\leq 1/2)\big\}
	\end{equation}
	needed as centres to cover
	$\supp(W)$ by closed unit cubes and deduce from the resolvent equation and the Cauchy-Schwarz inequality
	\begin{align}
		\label{combes-thomas:eq2}
		\E &\bigg[ \int_{\mathbb{R}^2} \frac{\d|\zeta_f|(x,y)}{|y|^{1-s/2}}\, \big\|\chi_a
			\big(R_{x+\i y}(H_G)-R_{x+\i y}(H^\tau_G)\big)\chi_b \big\|^{s/2} \bigg] \notag \\
 		& \leq  \|\tau W\|_\infty^{s/2} \sum_{c\in\Gamma_{W}} \int_{\mathbb{R}^2} \frac{\d|\zeta_f|(x,y)}{|y|^{1-s/2}} \notag\\
		& \hspace{3.5cm} \times \, \E \l[\|\chi_a R_{x+\i y}(H_G)\chi_c\|^{s/2}\|\chi_c R_{x+\i y}(H^\tau_G)\chi_b
			\|^{s/2} \r] \notag \\
		& \leq \|\tau W\|_\infty^{s/2} \sum_{c\in {\Gamma_{W}}}  \int_{\mathbb{R}^2} \frac{\d|\zeta_f|(x,y)}{|y|^{1-s/2}} \;
		  \notag \\
		& \hspace{3.5cm} \times \, \mathbb{E} \big[ \|\chi_a R_{x+\i y}(H_G)\chi_c \|^{s}\big]^{1/2} \mathbb{E}
			\big[ \|\chi_c R_{x+\i y}(H^\tau_G) \chi_b\|^{s}\big]^{1/2}.
	\end{align}
	We infer from \eqref{eq:LocHelfSjoDef1}, \ref{K2} and \ref{K3} that
	\begin{equation}
		\label{def:suppomega}
		\supp ( \zeta_f ) \subseteq \Big(\supp(f')\times [-1,1]\Big)\cup \Big([S_- ,S_+ ] \times \big([-1,-1/2]
		\cup [1/2, 1]\big)\Big)
		=: Z_1\cup Z_2.
	\end{equation}
	On the set $Z_2$ we estimate the right-hand side of \eqref{combes-thomas:eq2} by the Combes-Thomas estimate
	stated in \cite[Cor.\ 1]{MR1937430} for deterministic Schr\"odinger operators. Even though this result is
	formulated for Schr\"odinger operators on $L^2(\R^d)$, the argument extends to Schr\"odinger operators on
	$L^2(G)$ for arbitrary $G\subseteq \R^d$ open -- see also  \cite{Shen2014}. Thus, there exist constants
	$C_{0},\mu_{0}>0$, which are independent of $G\subseteq\R^d$, $\tau\in [0,1]$ and $\omega\in \Omega$, such that
	\begin{equation}
		\label{CombesThomas:cont}
		\|\chi_a R_z(H^{\tau}_G)\chi_b \| \leq C_{0} \e^{-\mu_{0} |z|^{-1}|a-b|}
	\end{equation}
	for all $z\in\C$ subject to $\dist(z,\sigma(H_G^{\tau})) \geq 1/2$. For $(x,y)\in Z_2$ we have
	\beq\label{eq:distspec}
 		|x+ \i y| \leq  {|x| + |y|} \leq  \max\{|S_-|, |S_+|\} + 1 =: C_{S}
	\eeq
	and therefore we obtain finite constants $C_1>0$ and $\mu_{1} := 2s\mu_{0}/C_{S}>0$ such that
	\beq
		\label{Combeseq1}
		\sup_{(x,y)\in Z_2} \Big\{ \mathbb{E} \big[ \|\chi_a R_{x+ \i y}(H_G)\chi_c \|^{s}\big] \,
		\mathbb{E} \big[ \|\chi_c R_{x+\i y}(H^\tau_G) \chi_b\|^{s}\big] \Big\}
		\leq C_1 \e^{-\mu_{1}(|a-c| + |c-b| )}.
	\eeq

	On the set $Z_1$ we use the fractional-moment bounds \eqref{eq:DefFMB} for $H$ and $H^\tau$, which can be applied
	because of $\supp( f') \subseteq \FMB$ and Remark~\ref{FMBalways2}.
	Hence, there exist finite constants $C_{2}>0$ and $\mu_{2}>0$ such that
	\beq
		\label{Combeseq2}
		\sup_{(x,y)\in Z_1} \Big\{ \mathbb{E} \big[ \|\chi_a R_{x+\i y}(H_G)\chi_c \|^{s}\big] \,
		\mathbb{E} \big[ \|\chi_c R_{x+\i y}(H^\tau_G) \chi_b\|^{s}\big] \Big\}
		\leq C_{2} \e^{-\mu_{2}(|a-c| + |c-b| )}.
	\eeq

	Collecting the estimates in \eqref{combes-thomas:eq10}, \eqref{combes-thomas:eq2}, \eqref{Combeseq1} and
	\eqref{Combeseq2}, 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
	\begin{align}
		\label{combes-thomas:eq3}
		\mathbb{E} \left[ \big\| \chi_a\big( f(H_G) - f(H^\tau_G) \big) \chi_b \big\|  \right]
		& \leq C_{3} \|\tau W \|_\infty^{s/2} \sum_{c\in {\Gamma_{W}}} \e^{-\mu(|a-c| + |c-b|)}
			\int_{\R^2} \frac{\d|\zeta_f|(x,y)}{|y|^{1-s/2}}  \notag\\
		& \leq  C |\tau|^{s/2} \big(\|f\|_1+\NTV(f)\big) \e^{-\mu\left(|a|+|b|\right)}.
	\end{align}
	To obtain the last inequality, we applied \eqref{eq:zeta-int}  and the estimate
	\begin{align}
		\label{eq:expproduct}
		\sum_{c\in {\Gamma_{W}}} \e^{-\mu(|a-c| + |c-b|)} \le \e^{-\mu(|a| + |b|)} \sum_{c\in {\Gamma_{W}}}
		\e^{2\mu|c|}.
		\\[-2ex]	\tag*{\qedhere}
	\end{align}
\end{proof}

\begin{proof}[proof of Theorem \ref{th:TrClBounds1}]
	We set $I_{-}:= \min I$, $I_{+} := \max I$ and assume without restriction that $I_{+} \ge E_{0}$,
	the lower bound \eqref{eq:DefE0} for the spectrum of $H_{G}^{\tau}$ (otherwise the statement is trivial).
	We fix $G\subseteq\Rn$ open and $a,b\in \R^d$.
	Since $\|A\|_{p} \le \|A\|_{1}$ for any Schatten norm with $p \ge 1$, we restrict ourselves to $0<p \le 1$.

 	Let $f\in \mathcal F_I$. As we want to apply Theorem \ref{th:TrClBounds1:OpNorm},
	which requires functions of compact support, we introduce $h:= f \id_{[E_{0}-1, \infty)}$.
	Then we have  $h\in \text{BV}_c(\R)$, $\NTV(h)\leq 2 \NTV(f)$,
	\beq
		\label{pf:Thm3.1equality}
 		f(H^{\tau}_G) =  h(H^{\tau}_G)
	\eeq
	for every $\tau\in[0,1]$ and $\supp(h') \subseteq I\cup \{E_0-1\} \subset\FMB$. The last inclusion holds
	by the definition of $E_{0}$ and the Combes-Thomas estimate \cite[Cor.\ 1]{MR1937430},
	which extends to finite-volume operators.

	 Because of \eqref{pf:Thm3.1equality} and Lemma \ref{lem:schatten-hoelder}
	we obtain for any $0<r<p$ that
	\begin{multline}
		\label{trclbound1:eq1}
		\big\| \chi_a\left( f(H_G) - f(H^\tau_G) \right) \chi_b \big\|_p
		= \big\| \chi_a\left( h(H_G) - h(H^\tau_G) \right) \chi_b \big\|_p\\
		\leq  \big\|\chi_a\left( h(H_G) - h(H^\tau_G) \right) \chi_b \big\|^{r/p}
		\big\|\chi_a\left( h(H_G) - h(H^\tau_G) \right) \chi_b \big\|_{p-r}^{1- r/p}.
	\end{multline}
	The adapted triangle inequality \eqref{lem:conkav} and the deterministic a priori estimate from Lemma \ref{lem:aPriori}
	imply
	\begin{align}
		\label{trclbound1:eq2}
		\big\|\chi_a\left( h(H_G) - h(H^\tau_G) \right) \chi_b \big\|_{p'}^{p'} \leq
		\|\chi_a h(H_G) \chi_b\|_{p'}^{p'} + \|\chi_a h(H^\tau_G)  \chi_b \|_{p'}^{p'}
		\leq C_{p'}  \|h\|^{p'}_\infty \notag\\
	\end{align}
	for every $0< p' \le 1$, where $C_{p'}$ depends also on $I_{+}$, but is uniform in the disorder,
	$G$ and $\tau \in [0,1]$, and independent of $h$.
	We apply \eqref{trclbound1:eq2} with $p'=p-r$ to estimate the expectation of \eqref{trclbound1:eq1} by
	\begin{align}
		\label{TrCl-done}
		\mathbb{E} \Big[ \big\| \chi_a\left( f(H_G) - f(H^\tau_G) \right) \chi_b \big\|_p \Big]
 		\le C_{p-r}^{1/p} \, \| h \|_{\infty}^{1-s_1}  \,
		\mathbb{E} \Big[ \big\| \chi_a \big( h(H_G) - h(H^\tau_G) \big) \chi_b \big\| \Big]^{s_1}. \notag\\
	\end{align}
	Here, we introduced $s_1:=r/p < 1$ and also used Jensen's inequality.

	Now, we choose $0<s_2<1$ and apply Theorem \ref{th:TrClBounds1:OpNorm} with $S= I \cup \{E_{0} -1\}$
	to the expectation on the right-hand side of \eqref{TrCl-done}. This yields finite constants $C_{1}, \mu_{1} >0$
	(which depend on $s_{2}$ and $I$) such that
	\begin{equation}
		\label{pf:Thm3.1equality3}
		\mathbb{E} \Big[ \big\| \chi_a \big( h(H_G) - h(H^\tau_G) \big) \chi_b \big\| \Big] \leq C_{1}| \tau|^{s_2/2}\,
		(I_+- E_0 + 2) \NTV(f)\, \e^{-{\mu_{1}}\left(|a|+|b|\right)},
	\end{equation}
	where we used
	\begin{equation}
 	\label{eq:1normTV}
		\|h\|_1+\NTV(h) \leq  2(I_+ - E_0+ 2) \NTV(f).
	\end{equation}
	Inserting \eqref{pf:Thm3.1equality3} into \eqref{TrCl-done} and observing
	\begin{equation}
		\label{eq:inftynormTV}
 		\|h\|_{\infty} \leq \NTV(h) \leq 2 \NTV(f),
	\end{equation}
	we obtain
	\begin{align}
		\label{pf:Thm3.1equality4}
		\E \l[ \big\| \chi_a\left( f(H_G) - f(H^\tau_G) \right) \chi_b \big\|_p\r]
		\leq C |\tau|^{s_1s_2/2} \NTV(f) \e^{-s_1\mu_{1}\left(|a|+|b|\right)}
	\end{align}
	with a suitable finite constant $C>0$. Since $s_{1},s_{2} \in (0,1)$ are both arbitrary,
	the claim follows with $s:= s_{1}s_{2}$.
\end{proof}

\begin{proof}[Proof of Corollary \ref{cor:TrClBounds2}]
	The (quasi-) norm estimates
	\begin{equation}
		\mathbb{E}\left[ |X|^{q_1} \right]^{1/q_1} \leq \mathbb{E}\left[ |X|^{q_2} \right]^{1/q_2}
		\quad \text{ and } \quad \|A\|_{p_1}\leq \|A\|_{p_2}
	\end{equation}
	for random variables $X$ and Schatten-class operators $A$ hold for all $0< p_2\leq p_1 < \infty$ and all
	$0< q_1\leq q_2 < \infty$.
	Thus, we assume without loss of generality that $p\leq 1$, $q\geq 1$ such that $k:= q/p\in\N$.

	For $f\in\mathcal F_I$ and $G\subseteq \R^d$ open we abbreviate
	\beq
		T_f:= f(H_{G})- f(H^\tau_{G}).
	\eeq
	Because of the adapted triangle inequality \eqref{lem:conkav} for $p\le 1$ we estimate
	\begin{equation}
		\label{pf:CorTrBndNew1}
		\|T_f\|_p^p \leq \sum_{a,b\in\Z^d}\|\chi_a T_f\chi_b \|_p^p.
	\end{equation}
	A $k$-fold application of \eqref{pf:CorTrBndNew1} yields
	\begin{equation}
		\label{pf:CorTrBndNew2}
		\mathbb{E}\left[ \|T_f\|_p^{q} \right] = \mathbb{E}\big[ \|T_f\|_p^{pk} \big]
		\leq \sum_{\substack{a_1,...,a_k\in\Z^d\\b_1,...,b_k\in\Z^d}} \mathbb{E}\Big[
		\prod_{l=1}^k \|\chi_{a_l}T_f\chi_{b_l}\|_p^p \Big].
	\end{equation}
	Next, we apply H\"older's inequality to the expectation in \eqref{pf:CorTrBndNew2} and obtain
	\begin{equation}
		\label{pf:CorTrBndNew3}
		\mathbb{E}\left[ \|T_f\|_p^{q} \right] \leq
		\sum_{\substack{a_1,...,a_k\in\Z^d\\b_1,...,b_k\in\Z^d}}
		\prod_{l=1}^k  \mathbb{E}\big[\|\chi_{a_l}T_f\chi_{b_l}\|_p^{pk} \big]^{1/k}.
	\end{equation}
	Now, we choose $0<s<1$. Theorem \ref{th:TrClBounds1} implies the existence of finite constants $C,\mu>0$
	(depending only on $p, I, s$) such that
	\begin{equation}
		\mathbb{E}\big[\|\chi_{a_l}T_f\chi_{b_l}\|_p \big] \leq C \tau^{s/2} \NTV(f) \e^{-\mu(|a_l|+|b_l|)}
	\end{equation}
	for all $a_{l},b_{l}\in \Z^{d}$.
	The deterministic a priori estimate from Lemma \ref{lem:aPriori} provides the existence of a finite
	constant $C_{1}$ (depending only on $p, I, s$, but not on $\omega$) such that
	\begin{equation}
		\|\chi_{a_l}T_f\chi_{b_l}\|_p^{pk-1}  \leq C_{1} \NTV(f)^{pk-1}.
	\end{equation}
	We thus obtain
	\begin{equation}
		\label{pf:CorTrBndNew4}
		\mathbb{E}\big[\|\chi_{a_l}T_f\chi_{b_l}\|_p^{pk} \big]
		\leq C C_{1} \tau^{s/2} \NTV(f)^{pk}  \e^{- \mu (|a_l|+|b_l|)}
	\end{equation}
	for all $a_{l},b_{l}\in \Z^{d}$.
	Estimating the right-hand side of \eqref{pf:CorTrBndNew3} by \eqref{pf:CorTrBndNew4} and using $q=pk$, we arrive at
	\begin{equation}
		\mathbb{E}\left[ \|T_f\|_p^{q} \right] \leq C_{2} \tau^{s/2} \NTV(f)^q ,
	\end{equation}
	where the finite constant $C_{2}$ depends on $p, I, s$.
	In particular, the constant $C_{\tau}$ in the statement vanishes algebraically as $\tau\downarrow 0$.
\end{proof}

\subsection{Proof of Theorem \ref{th:TrClBounds3} and Theorem \ref{Thm:prod}}

\begin{proof}[Proof of Theorem \ref{th:TrClBounds3}]
	We will follow the strategy in the proofs of Theorems \ref{th:TrClBounds1:OpNorm} and \ref{th:TrClBounds1}
	and use the notation introduced there. As it is done there, we assume without restriction that
	$I_{+} \ge E_{0}$ and that $0<p\leq 1$. Moreover, we restrict ourselves to the case $b\in\R^d$
	with $Q_b\cap G\neq\emptyset$.

	Let $a \in\R^{d}$, $\tau\in [0,1]$, $f\in \mathcal F_I$  and define again the
	truncation $h:= f \id_{[E_{0}-1, \infty)}$.
	We write $\zeta_h$ for the complex measure defined as in \eqref{eq:LocHelfSjoDef1} with a cutoff
	function $\Xi=\Xi_h$ that satisfies \ref{K1} -- \ref{K3}, where $f$ is replaced by $h$.
	Proceeding along the lines of \eqref{TrCl-done} and
	\eqref{combes-thomas:eq10}, we obtain for any $s,s' \in (0,1)$
	\begin{align}
		\label{th:trbounds3:eq1}
		\mathbb{E} \Big[\big\| \chi_a & \big(f(H^\tau_G) - f(H^\tau_{\wtilde{G}})\big)\chi_b \big\|_p  \Big]
			= \mathbb{E} \left[\big\| \chi_a\big(h(H^\tau_G) - h(H^\tau_{\wtilde G})\big)\chi_b \big\|_p  \right] \notag\\
		& \leq C_{1} \, \|h\|_{\infty}^{1-s'}\, \mathbb{E} \left[\big\| \chi_a\big(h(H^\tau_G) -
			h(H^\tau_{\wtilde{G}})\big)\chi_b \big\|  \right]^{s'}\notag\\
		& \leq C_{2}\, \, \|h\|_{\infty}^{1-s'} \, \E \l[  \int_{\mathbb{R}^2} \frac{\d|\zeta_h|(x,y)}{|y|^{1-s/2}}\,
			\big\|\chi_a \big(R_{x+\i y}(H^\tau_G)-R_{x+\i y}(H^\tau_{\wtilde{G}})\big)\chi_b \big\|^{s/2} \r]^{s'} \notag\\
	\end{align}
	with finite constants  $C_{1}=C_{1,p,s',I_{+}}>0$ and $C_{2}=C_{2,p,s,s',I_{+}}>0$.

	\emph{Case 1.} We assume $\dist(a,\partial G)>1$ and
	$\dist(b,\partial G)>1$. Hence, we have $Q_{b} \subset G$ in this case.
	We apply the geometric
	resolvent inequality, see e.g.\ \cite[Lemma 2.5.2]{MR1935594}, to the operator norm in the last line
	of \eqref{th:trbounds3:eq1}. Even though it is only stated for boxes there, the key estimate,
	\cite[Lemma 2.5.3]{MR1935594}, covers our setup.
	Hereby we use the assumption $\dist(\partial G, \partial \wtilde{G}) >1$.
	We obtain in analogy to \cite[Eq.\ (4.9)]{DGHKM2016}
	\begin{equation}
		\big\|\chi_a \big(R_{z}(H^\tau_{G})-R_{z}(H^\tau_{\wtilde{G}})\big)\chi_b \big\|
		\leq C_{3} \sum_{c\in \delta G^{\#}} \|\chi_a R_{z}(H^\tau_{\wtilde{G}}) \chi_c \|
		\|\id_{\Lambda_2(c)} R_{z}(H^\tau_{G}) \chi_b\|,
	\end{equation}
	where $\Lambda_2(c):= c+\Lambda_2$ and
	\begin{equation}
		\label{eq:Gplusraute}
 		\delta G^{\#}:=\big\{n\in\Z^d:\, \dist(n,{\partial G})\leq 1\big\}
	\end{equation}
	denotes the set of lattice points needed as centres to cover a strip of width $1$ around
	the boundary $\partial G$ of $G \subset \R^{d}$ by unit cubes.
	The constant $C_3$ is uniform in $z\in \C$ on each compact subset of $\C$. It is also uniform
	in $a\in\R^d$  and $b\in G$ with $\dist(b,\partial G)> 1$.
	Inserting this into \eqref{th:trbounds3:eq1} and using the Cauchy-Schwarz inequality, we get
	\begin{align}
		\label{th:trbounds3:eq2}
		\E \bigg[  \int_{\mathbb{R}^2} \frac{\d|\zeta_h|(x,y)}{|y|^{1-s/2}} \, &
		\big\|\chi_a \big(R_{x+\i y} (H^\tau_G)-R_{x+\i y}(H^\tau_{\wtilde{G}})\big)\chi_b \big\|^{s/2} \bigg]\notag\\
		& \leq C_{4} \sum_{c\in \delta G^{\#}} \int_{\mathbb{R}^2} \frac{\d|\zeta_h|(x,y)}{|y|^{1-s/2}} \,
			\E \Big[ \big\|\chi_a R_{x+\i y}(H^\tau_{\wtilde{G}})\chi_c\big\|^s\Big]^{1/2} \notag\\
		&	\hspace{2.6cm} \times \E\Big[ \big\| \id_{\Lambda_2(c)} R_{x+\i y}(H^\tau_{G})\chi_b \big\|^s\Big]^{1/2}
	\end{align}
	with a constant $C_{4}= C_{4,s,I_+}$.
	We decompose the support of $\zeta_{h}$ as in \eqref{def:suppomega}
	and treat the product of the expectations on $Z_{1}$ with the fractional-moment estimate as in \eqref{Combeseq2}
	and on $Z_{2}$ with the Combes-Thomas estimate as in and \eqref{Combeseq1} (with $H_{G}$ replaced by
	$H^\tau_{\wtilde{G}}$ in both estimates).
	The remaining integral is estimated as in \eqref{eq:zeta-int}. We then arrive at
	\begin{align}
		\label{th:trbounds3:eq3.1}
		\E \bigg[  \int_{\mathbb{R}^2}  \frac{\d|\zeta_h|(x,y)}{|y|^{1-s/2}}   \,
			\big\|\chi_a \big(R_{x+\i y} (H^\tau_G) & - R_{x+\i y}(H^\tau_{\wtilde{G}})\big)\chi_b \big\|^{s/2} \bigg]
			\notag\\
 		& \le C_{5} \sum_{c\in \delta G^{\#}} \e^{-\mu(|a-c| + |c-b| )} \big(\|h\|_{1} + \NTV(h) \big) \notag\\
		& \le C_{6} \NTV(f) \e^{-(\mu/2) [\dist(a, \partial {G}) + \dist(b, \partial {G})]}
	\end{align}
	with finite constants $C_{5}, C_{6}, \mu >0$, all depending only on $s$ and $I$.
	In the last step we used \eqref{eq:1normTV}, the estimate
	\begin{equation}
		\label{eq:partialGsum}
 		\sum_{c\in \delta G^{\#}} \e^{-\mu(|a-c| + |c-b| )} \le
		\e^{-(\mu/2) [\dist(a, \partial {G}) + \dist(b, \partial {G})-2]}
		\sum_{c\in \Z^{d}} \e^{-(\mu/2) (|a-c| + |c-b| )}
	\end{equation}
	and that the sum on the right-hand side of \eqref{eq:partialGsum} is
	bounded from above uniformly in $a, b \in\R^{d}$ by, e.g., the Cauchy-Schwarz inequality.
	Now, the claim follows upon inserting \eqref{th:trbounds3:eq3.1} into \eqref{th:trbounds3:eq1} and observing
	\eqref{eq:inftynormTV}. This finishes Case 1.

	\emph{Case 2.} We assume $\dist(a,\partial G) \leq  1$ or $\dist(b,\partial G) \leq  1$.
	Hence, we have
	\begin{equation}
		\label{eq:ab-dist}
 		|a-b| \ge \max\big\{\dist(a,\partial G), \dist(b,\partial G)\big\} -1
	\end{equation}
	in this case.	We estimate the operator norm on the right-hand side of \eqref{th:trbounds3:eq1}
	by the triangle inequality and each of the resulting two terms by the fractional-moment estimate
	\eqref{eq:DefFMB}, see also Remark \ref{FMBalways2}.
	The remaining integral is again estimated by \eqref{eq:zeta-int} and \eqref{eq:1normTV}, and we obtain
	\begin{equation}
		\label{th:trbounds3:eq3.2}
		\E \bigg[  \int_{\mathbb{R}^2}  \frac{\d|\zeta_h|(x,y)}{|y|^{1-s/2}}   \,
		\big\|\chi_a \big(R_{x+\i y} (H^\tau_G) - R_{x+\i y}(H^\tau_{\wtilde{G}})\big)\chi_b \big\|^{s/2} \bigg]
 		\le C_{7}  \e^{-\mu|a-b|} \NTV(f)
	\end{equation}
	with finite constants $C_{7}, \mu >0$ depending only on $s$ and $I$.
	Now, the claim follows upon inserting \eqref{th:trbounds3:eq3.2} into \eqref{th:trbounds3:eq1} and observing
	\eqref{eq:inftynormTV} and \eqref{eq:ab-dist}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{Thm:prod}]
	As in the proof of Corollary~\ref{cor:TrClBounds2} we assume without loss of generality that
	$p\le 1$, $q \ge 1$ and we define and $k:= q/p \in\N$. Let $f\in\mathcal F_I$.
	The argument leading to \eqref{pf:CorTrBndNew3} in the proof of Corollary~\ref{cor:TrClBounds2}
	implies in the present context the estimate
	\begin{equation}
		\mathbb{E}\left[ \|T_{f,L}\|_p^{q} \right] \leq
		\sum_{\substack{a_1,...,a_k\in\Z^d\\b_1,...,b_k\in\Z^d}}
		\prod_{l=1}^k  \mathbb{E}\big[\|\chi_{a_l}T_{f,L}\chi_{b_l}\|_p^{pk} \big]^{1/k},
	\end{equation}
	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
	\begin{equation}
 		\label{eq:decomp1}
		T_{f,L}^{(1)} := f(H_{L}) - f(H) \quad \text{and} \quad T_{f,L}^{(2)} := f(H_{L}^{\tau}) - f(H^{\tau}),
	\end{equation}
	and the second is with the terms
	\begin{equation}
 		\label{eq:decomp2}
		T_{f,L}^{(1)} := f(H_{L}) - f(H_{L}^{\tau}) \quad \text{and} \quad T_{f,L}^{(2)} := f(H) - f(H^{\tau}).
	\end{equation}
	In any case, the adapted triangle inequality \eqref{lem:conkav} and Minkowski's inequality on
	$L^{k}(\Omega,\PP)$ imply
	\begin{equation}
 		\mathbb{E}\big[\|\chi_{a_l}T_{f,L}\chi_{b_l}\|_p^{pk} \big]^{1/k} \le \sum_{j=1}^{2}
		\mathbb{E}\big[\|\chi_{a_l}T_{f,L}^{(j)}\chi_{b_l}\|_p^{pk} \big]^{1/k}.
	\end{equation}
	Moreover, for either choice of the decomposition, the deterministic a priori bound from
	Lemma~\ref{lem:aPriori} implies $\|\chi_{a_l}T_{f,L}^{(j)}\chi_{b_l}\|_{p}^{pk-1} \le C_{1} \NTV(f)^{pk-1}$
	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 \eqref{eq:DefE0} for the spectra. We conclude that
	\begin{equation}
		\label{eq:TeTraceConvergence1}
 		\mathbb{E}\left[ \|T_{f,L}\|_p^{q} \right]
		\leq C_{1} \NTV(f)^{pk-1}
		\sum_{\substack{a_1,...,a_k\in\Z^d\\b_1,...,b_k\in\Z^d}}
		\prod_{l=1}^k  \Big(\sum_{j=1}^{2}\mathbb{E}\big[\|\chi_{a_l}T_{f,L}^{(j)}\chi_{b_l}\|_p \big]^{1/k}\Big).
	\end{equation}
	Next we split the summation over each pair $(a_{l},b_{l}) \in\Z^{d} \times \Z^{d}$ into two parts:
	the box $\Lambda^2_{L/2}:=(\Lambda_{L/2}\times\Lambda_{L/2})\cap (\Z^{{d}} \times \Z^{d}) $ and its complement
	$\Lambda^{2,c}_{L/2}:= (\ Z^{{d}} \times \Z^{d})\setminus \Lambda^{2}_{L/2}$.
	For $(a_{l},b_{l}) \in\Lambda^2_{L/2}$ we compare infinite- and finite-volume operators, that is,
	we choose the decomposition \eqref{eq:decomp1}. Theorem \ref{th:TrClBounds3} then provides
	finite constants $C_2 = C_{2,p,I}>0$ and $\mu_2=\mu_{2,p,I}>0$ such that
	\begin{align}
		\label{eq:TeTraceConvergence2}
  	\sum_{(a,b)\in \Lambda^2_{L/2}} \sum_{j=1}^{2} &
  		\mathbb{E}\big[\|\chi_{a_l}T_{f,L}^{(j)}\chi_{b_l}\|_p \big]^{1/k}	\notag\\
 		& \leq C_{2} \NTV(f)^{1/k} \sum_{(a,b)\in \Lambda^2_{L/2}}
			\e^{-\mu_2[(\dist(a,\partial\Lambda_{L})+\dist(b,\partial\Lambda_{L})]/k} \notag\\
		& \leq C_{2}\NTV(f)^{1/k}\, (L+1)^{2d}  \e^{-\mu_{2} L/(2k)}.
	\end{align}

	For $(a_{l},b_{l}) \in\Lambda^{2,c}_{L/2}$ we compare unperturbed and perturbed operators, that is,
	we choose the decomposition \eqref{eq:decomp2}. We fix $0<s<1$. Theorem~\ref{th:TrClBounds1} then
	provides finite constants $C_3 = C_{3,p,s,I}>0$ and $\mu_3 = \mu_{3,p,s,I}>0$ such that
	\begin{align}
		\label{eq:TeTraceConvergence3}
  	\sum_{(a,b)\in \Lambda^{2,c}_{L/2}}\sum_{j=1}^{2}
  	\mathbb{E}\big[\|\chi_{a_l}T_{f,L}^{(j)}\chi_{b_l}\|_p \big]^{1/k}
 		& \leq C_{3} \NTV(f)^{1/k}\, \sum_{(a,b)\in \Lambda^{2,c}_{L/2}} \e^{-\mu_3(|a|+|b|)/k} 	\notag\\
		&	\leq 2C_{3} \NTV(f)^{1/k}\, \sum_{\substack{a\in\mathbb{Z}^d\\ b\in \Z^{d} \setminus\Lambda_{L/2}}}
			\e^{-\mu_3(|a|+|b|)/k}\notag\\
		& \leq C_{3}' \NTV(f)^{1/k} (L+1)^{d-1} \e^{-\mu_3 L/(4k)},
	\end{align}
	where $C_{3}' = C'_{3,p,s,I}>0$ is another finite constant. We conclude from \eqref{eq:TeTraceConvergence1},
	\eqref{eq:TeTraceConvergence2} and \eqref{eq:TeTraceConvergence3} that
	\begin{equation}
		\label{eq:TeTraceConvergence4}
 		\mathbb{E}\left[ \|T_{f,L}\|_p^{q} \right] \leq  C \NTV(f)^{pk} \e^{-\mu L}
	\end{equation}
	with finite constants $C =C_{p,q,s,I}>0$ and  $\mu = \mu_{p,q,s,I}>0$. This proves \eqref{thm:prod:eq1}.

	The almost-sure convergence \eqref{eq:ConvergencePi} for a super-logarithmically growing
	sequence of lengths follows from \eqref{thm:prod:eq1} with Remark~\ref{fast-convergence}.
\end{proof}

\section{Proofs of the results from Section \ref{subsec:ssf}}
\label{sec:5}

\subsection{Proof of Theorem \ref{th:SSFandTransOp}}

\begin{proof}[Proof of Theorem \ref{th:SSFandTransOp}]
	We fix $\tau \in [0,1]$.
  Let $I\subset \FMB$ be a compact interval and $E\in I$. Theorem \ref{th:SSFandTransOpii} follows from
  Theorem \ref{th:SSFandTransOpi}  and  Theorem \ref{Thm:prod} applied to the Fermi function $f=\id_{(-\infty,E]}$.

	It remains to prove the left equality in Theorem \ref{th:SSFandTransOpi}, because the right equality follows
	already from Remark \ref{rem:right-equality-clear}.
	Let $I \subset \FMB$ be again a compact interval. Then, Corollary~\ref{cor:TrClBounds2} and Fubini's Theorem imply
	\begin{equation}
		\mathbb{E} \Big[\int_{I}\d E\, \big\|T(E,H, H^\tau)\big\|_{1} \Big] <\infty.
	\end{equation}
	Hence, we have $\|T(\,\pmb\cdot\,, H, H^\tau)\|_{1} \in L^{1}(I)$ almost surely. Thus, the left inequality
	in Theorem \ref{th:SSFandTransOpi} follows from Lemma~\ref{lem:SSFprep2ndproof} below, and the proof is
	complete.
\end{proof}

It remains to prove the following deterministic lemma.

\begin{lemma}
 	\label{lem:SSFprep2ndproof}
 	Let $A$ and $B$ be two self-adjoint operators in a Hilbert space $\mathcal H$ which are bounded from below.
	We assume that
 	$\e^{-A}- \e^{-B}\in \mathcal{S}^{1}$ and that, for some open interval $I\subset \R$, the mapping
	\begin{equation}
 		\label{eq:SSFprepAssumpt}
 		I \ni E\mapsto \|T(E,A,B)\|_{1}
 	\end{equation}
 	is an $L^{1}(I)$-function. Then the spectral shift function and the trace of the shift operator coincide, i.e.\
 	\begin{equation}
	 	\label{eq:SSFprep2ndproofstat}
 		\xi(E,A,B)=\Tr\big( T(E,A,B) \big)\qquad \text{ for a.e. } E\in I.
 	\end{equation}
\end{lemma}

\begin{remark}
	The assumption $\e^{-A}- \e^{-B}\in \mathcal{S}^{1}$ in the lemma is only needed for the spectral
	shift function to be well defined according to \eqref{eq:Definition1SSF} and could be relaxed.
\end{remark}

\begin{proof}[Proof of Lemma~\ref{lem:SSFprep2ndproof}]
	We show that the function $E \mapsto \Tr\big(T(E,A,B)\big)$  satisfies \eqref{eq:Definition1SSF} for
  every $f\in C^{\infty}(\mathbb{R})$ with $\supp(f')\subseteq I$ and $\lim_{\lambda\to\infty}f(\lambda)=0$.

  Let $f\in C^\infty(\R)$ be such a function. Assumption \eqref{eq:SSFprepAssumpt} implies that
	\beq
	  \R \ni E\mapsto |f'(E) |\|T(E,A,B) \|_1 \in L^1(\R).
	\eeq
	Hence, $\int_\R \d E\, f'(E) \, T(E,A,B)$ is a trace-norm convergent Bochner integral.
	Moreover, the identity
	\beq
		\label{eq:bochner}
		f(B)- f(A)=\int_\R \d E\, f'(E) \,T(E,A,B)
	\eeq
	holds, as we argue below. Since the mapping $\S^1\ni B\mapsto \Tr (B)$ is a bounded linear
	functional on $\S^1$, it interchanges with the Bochner integral, and \eqref{eq:bochner} implies
	the assertion
	\beq
		\Tr \big( f(B)- f(A)\big) = \int_\R \d E\,f'(E) \Tr\big(  T(E,A,B) \big).
	\eeq
	It remains to prove \eqref{eq:bochner}. We fix $\phi \in \H$ and compute, using Fubini's theorem,
	\begin{equation}
 		\<\phi , f(A) \phi\> = - \int_{\mathbb{R}}\d\mu_{\phi,\phi}^{A}(\lambda)\int_{\lambda}^{\infty}\d E\,
			f^{\prime}(E) = -\int_{\mathbb{R}}\d E\, f^{\prime}(E) \< \phi,\id_{(-\infty,E]}(A)\phi \>,
	\end{equation}
	where we introduced the spectral measure $\mu^{A}_{\phi,\phi}:=\< \phi, \id_{\,\pmb\cdot\,}(A)\phi\>$.
	The analogous computation for $B$ gives
	\begin{align}
		\<\phi ,\big( f(B) - f(A) \big) \phi\>  &= \int_\R \d E\,  f'(E) \< \phi,  T(E,A,B) \phi \> \notag \\
		&= \< \phi, \bigg(\int_\R \d E\, f'(E) \, T(E,A,B) \bigg) \phi \>,
	\end{align}
	where the last equality follows again from the continuity of the functional $\S^1\ni B\mapsto \langle\varphi,
	B \varphi\rangle$, which therefore interchanges with the Bochner integral. The operator equality 		\eqref{eq:bochner} follows by polarisation.
\end{proof}

\subsection{Proof of Lemma \ref{lem:HoelderCont} and Theorem \ref{lem:SSFcont} }

\begin{proof}[Proof of Lemma \ref{lem:HoelderCont}]
	We fix $E,E'\in I$ arbitrary with $E<E'$ and define $J:=[E,E']$.
	The estimate \eqref{TrCl-done} with $f=h=\id_{J}$ implies
	\begin{align}
		\label{eq:PfLemSSFcont1}
	  \E \big[ \big\|  T(E,H_{(L)},H_{(L)}^{\tau})
	  & - T(E',H_{(L)},H_{(L)}^{\tau}) \big\|_{1}\big]\nonumber\\
		& \leq \E \left[ \| \id_J(H_{(L)})-\id_J(H^{\tau}_{(L)}) \|_1 \right] \nonumber\\
		& \leq C_{\theta} \sum_{a,b\in\Z^{d}} \E \big[ \big\| \chi_a \big(
			\id_J(H_{(L)}) - \id_J(H^{\tau}_{(L)}) \big)\chi_b \big\| \big]^{\theta},
	\end{align}
	where $\theta \in (0,1)$
	and the constant $C_{\theta}$ is independent of $L$ and $\tau \in [0,1]$.
	We will apply the Helffer-Sj\"ostrand formula to the operator $\id_{J}(H_{(L)}^{(\tau)})$
	and need an appropriate cutoff
	function for this purpose. Let $g\in C_c^\infty(\R)$ be a smooth indicator function such that
	$g(x):=1$ for $x\in [-1,1]$, $g(x):=0$ for $x\in \R \backslash [-2,2]$, $\|g\|_\infty\leq 1$ and
	$\|g'\|_\infty\leq 2$. We define the centre $J_{c}:= (E+E')/2$ of the interval $J$ and the
	cutoff function $\Xi_{J}\in C^\infty_c(\R^2)$ by
	\beq
		\label{cutoff-specified}
		\Xi_{J}(x,y)= g\Big( \frac{x-J_{c}}{|J|}\Big)g\Big( \frac{y}{|J|}\Big), \qquad (x,y) \in \R^{2}.
	\eeq
	Let $\zeta_{\id_J}$ be the complex measure defined in \eqref{eq:LocHelfSjoDef1} with the cutoff
	function \eqref{cutoff-specified}.
	We note that
	\begin{equation}
		\label{eq:supp-zeta-J}
 		\supp (\zeta_{\id_{J}}) \subseteq J \times  [-2|J|,2|J|].
	\end{equation}
	We now apply Lemma \ref{lem:LocHelfSjo} to  the difference in \eqref{eq:PfLemSSFcont1} and obtain
	for all $a,b\in\Z^{d}$
	\begin{equation}
		\label{eq:PfLemSSFcont1b}
		\E \left[ \big\|\chi_a \big(\id_J(H_{(L)})-\id_J(H^{\tau}_{(L)})\big)\chi_b \big\| \right]
	  \leq \frac 1 {2\pi} \int_{\mathbb{R}^2} \d|\zeta_{\id_{J}}|(x,y) \,
	  \E\left[\|\chi_a (R_z-R^{\tau}_z)\chi_b \|\right],
	\end{equation}
	where we abbreviated $z: =x+\i y$, $R_z:=R_z(H_{(L)})$ 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
	\begin{align}
		\label{eq:PfLemSSFcont2c}
		\E\big[\|\chi_a (R_z &- R^{\tau}_z) \chi_b \|\big]  \notag\\
		& = \E\left[\|\chi_a (R_z-R^{\tau}_z)\chi_b \|\right]^{1/p} \,
			\E\left[\|\chi_a (R_z-R^{\tau}_z)\chi_b \|\right]^{1/q} \nonumber\\
 		& \leq \frac{2^{\delta/p}(\tau\|W\|_{\infty})^{1/q}}{|y|^{\delta/p+ (1+\delta)/q}} \,
			\E\big[\|\chi_a (R_z-R^{\tau}_z)\chi_b \|^{1-\delta}\big]^{1/p}	\nonumber\\
		& \quad \times\sum_{c\in\Gamma_W} \E\left[\|\chi_a R_z\chi_c\|^{(1-\delta)/2}
			\|\chi_c R^{\tau}_z \chi_b \|^{(1-\delta)/2}\right]^{1/q} \nonumber\\
 		& \leq \frac{2^{\delta/p}(\tau\|W\|_{\infty})^{1/q}}{|y|^{\delta+ 1/q}} \,
			\left(\E\big[\|\chi_a R_z\chi_b \|^{1-\delta}\big]  +
			\E\big[\|\chi_a R^{\tau}_z\chi_b \|^{1-\delta}\big]\right)^{1/p} \nonumber\\
		& \quad \times\sum_{c\in\Gamma_W} \left(\E\big[\|\chi_a R_z\chi_c\|^{1-\delta} \big] \;
			\E\big[\|\chi_c R^{\tau}_z \chi_b \|^{1-\delta}\big]\right)^{1/(2q)},
	\end{align}
  where $\Gamma_{W}$ was defined in \eqref{def:GammaW}. For $(x,y) \in \supp (\zeta_{\id_{J}})$
  we have $x \in J \subseteq I \subset \FMB$ so that each expectation can be bounded with the
  localisation estimate from
  Lemma~\ref{Lem:PertPersPot}, which is uniform in $\tau\in [0,1]$, $L>0$ and $J\subseteq I$.
	Treating the resulting $c$-sum of exponentials as in \eqref{eq:expproduct} and throwing away the
	decay in $|a-b|$, we arrive at
	\beq
		\label{eq:PfLemSSFcont3a}
		\E\left[\|\chi_a (R_z-R^{\tau}_z)\chi_b \|\right] \leq C_{1} \frac{\e^{-\mu_{1}(|a|+|b|)}}{|y|^{\delta+ 1/q}}
	\eeq
	with finite constants $C_{1},\mu_{1}>0$ that depend only on $p$, $\delta$ and $I$.
	This estimate needs to be inserted into \eqref{eq:PfLemSSFcont1b},
	so the following integral is relevant
	\begin{equation}
		\label{eq:PfLemSSFcont3b}
		\int_{\R^2} \frac{\d|\zeta_{\id_{J}}|(x,y)}{|y|^{\delta+ 1/q}} \leq C_{2} |J|^{p^{-1}-\delta}.
	\end{equation}
	To obtain the bound, we used $\delta <1/p$ and applied \eqref{eq:zeta-int} with the
	particular choice \eqref{cutoff-specified} for the cutoff function $\Xi_{J}$. The constant
	$C_{2}>0$ depends only on $p$, $\delta$ and $I$.
	Collecting the estimates \eqref{eq:PfLemSSFcont1}, \eqref{eq:PfLemSSFcont1b}, \eqref{eq:PfLemSSFcont3a}
	and \eqref{eq:PfLemSSFcont3b}, we conclude that
	\beq
		\label{eq:PfLemSSFcont4}
		\E \left[ \big\| T(E,H_{(L)},H_{(L)}^{\tau}) - T(E',H_{(L)},H_{(L)}^{\tau}) \big\|_{1}\right]
		\le  C_{3} |J|^{\theta (p^{-1} -\delta)}
	\eeq
	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 $(0,1)$.
\end{proof}

Our proof of Theorem \ref{lem:SSFcont} uses the scale-free unique continuation principle
of  \cite{MR3106507, Nakic:2015is} applied to averaged
local traces of non-negative functions of \emph{infinite-volume} ergodic random Schr\"odinger operators.

\begin{lemma}
 	\label{lem:UCP}
	Let $E\in\R$ and let $\Gamma \subset\R^{d}$ be a Borel set with $\Int (\Gamma) \neq\emptyset$.
	Then there exists a finite constant $\gamma>0$ such that for every non-negative, measurable function
	$f: \R \rightarrow [0, \infty)$ with support $\supp(f) \subseteq (-\infty,E]$ the lower bound
	\begin{equation}
		\label{eq:UCPconsequence}
 		\E\big[ \Tr \big(\id_{\Gamma} f(H)\big)\big] \ge \gamma \int_{\R}\d E'\, \mathcal{N}'(E') f(E')
	\end{equation}
	holds. If the operator
	$\id_{\Gamma} f(H)$ is not trace class, we define its trace to be $+\infty$.
\end{lemma}

\begin{proof}
	Even though a unique continuation principle for infinite-volume operators is known \cite{MR3519210},
	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.\ \cite[Thm.\ II.5.8]{elstrodt2011measure},
	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
	\begin{equation}
		\label{eq:IDOSlimit}
 		\int_{I} \d E' \mathcal{N}'(E') = \lim_{L\to\infty} \frac{1}{L^{d}} \, \E\big[ \Tr \big(\id_{I}(H_{L})\big)\big].
	\end{equation}
	The left-hand side of \eqref{eq:UCPconsequence} is monotone in $\Gamma$ and invariant under
	$\Z^{d}$-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 \cite[Cor.]{Nakic:2015is} or \cite[Thm.\ 1.1]{MR3106507}, provides us with
	the deterministic estimate
	\begin{equation}
		\label{eq:UCP}
	 	\id_{I}(H_{L}) \le \frac{1}{\gamma}  \id_{I}(H_{L}) \, \id_{\Gamma_{L}} \id_{I}(H_{L})
	\end{equation}
	for $L\in\N$, where $\Gamma_{L}:= \Lambda_{L} \cap \big(\bigcup_{k \in\Z^{d}} (k + \Gamma)\big)$
	and the finite non-random constant $\gamma > 0$ depends only on $d$,
	$\Gamma$, $E$ and
	\begin{equation}
 		\mathcal{V} := \|V_{0}\|_{\infty} +  |\lambda| \, \Big\|\sum_{k\in\Z^{d}} u_{k}\Big\|_{\infty} < \infty,
	\end{equation}
	see Assumptions \ref{assK} and \ref{assV2}.
	Next, we insert \eqref{eq:UCP} into \eqref{eq:IDOSlimit}, exploit cyclicity of the trace and add and subtract
	the desired $L$-independent term.
	This yields
	\begin{align}
		\label{eq:UCPapplied}
 			\int_{I} \d E' \mathcal{N}'(E') & \le  \frac{1}{\gamma} \,  \E\big[ \Tr \big(\id_{\Gamma} \id_{I}(H)		\big)\big] \notag\\
		  & \quad + \frac{1}{\gamma} \,\liminf_{L\to\infty}
				\left(  \frac{1}{L^{d}}  \E \big[\Tr\big( \id_{\Gamma_{L}} \id_{I}(H_{L}) \big) \big] -
				\E \big[\Tr\big(\id_{\Gamma}\id_{I}(H) \big) \big] \right).
	\end{align}
	It remains to show that the error term in the second line of \eqref{eq:UCPapplied}
	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
	\begin{equation}
		\label{laplace-vanish}
 		\lim_{\substack{L\to\infty\\[.5ex] L \text{~odd}}} \frac{1}{L^{d}} \,\E \big[ \Tr \big( \id_{\Gamma_{L}}
		(\e^{-t H_{L}} - \e^{-t H})\big) \big]   =0
	\end{equation}
	holds for every $t>0$ \cite[Thm.\ 2a in Sect.\ XIII.1]{MR0270403}.
	Here, we reformulated the term with the $\Z^{d}$-ergodic infinite-volume Hamiltonian
	by using the the covariance relation for $H$ and the disjoint decomposition
	$\Gamma_{L} =\bigcup_{k\in \Lambda_{L} \cap \Z^{d}} (k+ \Gamma)$, which holds for $L$ odd.
	We analyse the semigroups in
	the Feynman-Kac representation \cite{MR2105995}, see also \cite[Sect.\ 6]{MR1756112},
	and obtain
	\begin{align}
		\label{BB-rep}
		\bigg|\frac{1}{L^{d}} & \,\E \big[ \Tr \big( \id_{\Gamma_{L}}
		(\e^{-t H_{L}} - \e^{-t H})\big) \big] \bigg|\notag\\
 		& \le \frac{\e^{t\mathcal{V}}}{(2\pi t)^{d/2}} \int_{\Gamma_{L}} \frac{\d x}{L^{d}} \;
			\mathbf{E}_{0,x}^{t,x}( 1- \chi^{t}_{\Lambda_{L}} )
			\le \frac{\e^{t\mathcal{V}}}{(2\pi t)^{d/2}} \int_{\Lambda_{L}} \frac{\d x}{L^{d}} \;
			\mathbf{E}_{0,x}^{t,x}( 1- \chi^{t}_{\Lambda_{L}} ),
	\end{align}
 	where
	$\mathbf{E}_{0,x}^{t,x}$ denotes the Brownian-bridge expectation over continuous closed paths starting
	at time $0$ 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 \eqref{BB-rep} vanishes as $L \to\infty$ according to
	\cite[p.\ 341]{artRSO1989Kir}.
\end{proof}

\noindent
We are now ready for the

\begin{proof}[Proof of Theorem \ref{lem:SSFcont}]
	Throughout the proof we abbreviate the spectral shift functions by
	$\xi_{(L)} := \xi(\,\pmb\cdot\,,H_{(L)},H_{(L)}^{\tau})$ and choose representatives which coincide with
	the trace of the corresponding spectral shift operator on $\FMB$. Since $\xi_{(L)}$ depends monotonously on the
	perturbation and $W\geq C u_0$, we assume without loss of generality that $W=C u_0$.

	By Lemma \ref{lem:HoelderCont}, the function $E \mapsto \E[\xi(E)]$ is H\"older continuous on compact intervals in
	$\FMB$ for any $\alpha\in (0,1)$. Let $E\in \FMB$ and $\varepsilon_{0} >0$ such that
	$I_{\varepsilon_{0}} := [E-\varepsilon_{0}, E+ \varepsilon_{0}] \subset \FMB$. Consider any
	$\varepsilon \in (0,\varepsilon_{0}]$. Then
	\begin{equation}
		\label{eq:PfPosSSFcont1}
		\E \left[ \xi(E) \right] \geq \frac{1}{2\eps} \int_{I_{\varepsilon}}\d E'\, \E \left[ \xi(E') \right]
		- C_{1} \eps^{\alpha},
	\end{equation}
	where the H\"older constant depends on $\alpha$, $E$ and $\varepsilon_{0}$, but not on $\varepsilon$.
	We denote by $\E_{\neq 0}[\,\pmb\cdot\,]$ the averaging with respect to all random variables but $\omega_0$
	and infer from the Birman-Solomyak formula \cite{MR0315482}
	\begin{align}
		\label{eq:PfPosSSFcont2}
 		\int_{I_{\varepsilon}} \d E'\,\E \left[ \xi(E') \right]  &= \int_{0}^1 \d s\,
		\E \left[ \Tr\left( Cu_0 \,\id_{I_\eps}( H+s C u_0 )\right) \right]\nonumber  \\
		& = \int_{0}^{C} \d s \int_{0}^1 \d\omega_0\, \rho(\omega_0) \, \E_{\neq 0}
			\big[ \Tr\big( u_0 \id_{I_\eps}(H+su_0) \big) \big].
	\end{align}
	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
	\begin{align}
		\label{eq:PfPosSSFcont2b}
		\int_{I_{\varepsilon}} \d E'\,\E \left[ \xi(E') \right] & \geq
			s_0 \rho_- \int_{s_0}^{1} \d\omega_0\, \E_{\neq 0} \big[ \Tr\big( u_0 \id_{I_\eps}({H})\big) \big] \nonumber\\
		& \geq s_0 \rho_- \left( \frac{1}{\rho_+} \E \left[ \Tr\left( u_0 \id_{I_\eps}( H) \right) \right]
			- \int_{0}^{s_0} \d\omega_0\, \E_{\neq 0} \left[ \Tr\left( u_0 \id_{I_\eps}( H) \right) \right] \right)\nonumber\\
		& \ge  s_0 \rho_- \left(\frac{\gamma \mathcal{N}(I_{\varepsilon})}{\rho_+} - J\right).
	\end{align}
	Here, the last inequality follows from the right estimate in \eqref{bounds:u0} together with the unique
	continuation principle stated in Lemma~\ref{lem:UCP}. Furthermore, we introduced
	$\rho_+ := \esssup_{x\in[0,1]} \rho(x)<\infty$ and
	\begin{equation}
		J := \int_{0}^{s_0} \d\omega_0\, \E_{\neq 0} \left[ \Tr\left( u_0 \id_{I_\eps}( H) \right) \right].
	\end{equation}
	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 \ref{th:TrClBounds3},
	which yields finite constants $C_2,\mu>0$ such that
	\begin{equation}
		\label{eq:PfPosSSFcont3}
 		\E \left[\left\vert  \Tr\left( u_0 \id_{I_\eps}( H) \right) -
		\Tr\left( u_0 \id_{I_\eps}( H_L \right)\right\vert \right]
		\leq   C_2  \e^{-\mu L}
	\end{equation}
	for every $L>0$. The estimate \eqref{eq:PfPosSSFcont3} implies the bound
	\begin{equation}
		\label{eq:PfPosSSFcont3a}
		J \leq s_0 \sup_{\omega_0\in [0,s_0]}  \E_{\neq 0}\left[ \Tr\left( u_0 \id_{I_\eps}(H_L) \right) \right]
		+ \frac{C_{2}}{\rho_{-}} \, \e^{-\mu L}
	\end{equation}
	for every $L>0$. Let $e_1:=(1,0,...,0) \in\R^{d}$ be the unit vector along the first coordinate axis.
	We define the subset of lattice points $A_L:= \big( e_{1}+ (3\Z)^d \big) \cap \Lambda_L$ and introduce the
	random background operator
	\begin{equation}
		\wtilde{H}_L :=  H_{0,L} +  \lambda \sum_{k\in\Z^d\setminus A_L} \omega_k u_k,
	\end{equation}
	where $H_{0,L}$ is the Dirichlet restriction of $H_{0}$ to $\Lambda_{L}$.
	However, for any fixed realisation of coupling constants $\{\omega_k\}_{k\in \Z^{d}\setminus A_{L}}$,
	we view $\wtilde{H}_L$
	as a non-random operator with a non-periodic background potential that is bounded uniformly in
	$\{\omega_k\}_{k\in \Z^{d}\setminus A_{L}}$. Now, the Dirichlet restriction $H_{L}$ takes the form
	\begin{equation}
		H_L=\wtilde{H}_L + \lambda \sum_{k\in A_L} \omega_{k} u_{k}
	\end{equation}
	and, after scaling by a factor $1/3$ and (if required) introducing a energy shift, it
	constitutes a \emph{crooked Anderson Hamiltonian} in the sense of \cite{MR3106507}.
	We apply the Wegner estimate \cite[Thm.\ 1.4]{MR3106507} and obtain a finite constant $C_3 >0$ such that
	\begin{align}
		\label{eq:PfPosSSFcont8}
		\sup_{\omega_k\in [0,1], \, k\in \Z^d\setminus A_L} \E_{A_L}\left[  \Tr\left( u_0 \id_{I_\eps}( H_L \right) \right]
		& \leq \|u_{0}\|_{\infty} \sup_{\omega_k\in [0,1],\, k\in \Z^d\setminus A_L}
			\E_{A_L}\left[  \Tr\left( \id_{I_\eps}( H_L \right) \right]	\nonumber\\
		& \leq C_3 2\eps  L^d,
	\end{align}
	where $\E_{A_L}[ \,\pmb\cdot\, ]$ denotes the average over the couplings $\{\omega_k\}_{k\in A_L}$. The estimate
	\eqref{eq:PfPosSSFcont8} holds for all $\varepsilon \in (0,\varepsilon_{1}] $ and all length-scales $L  \ge L_{1}$
	such that $L/3 \in \N$ is odd, and $\varepsilon_{1}$ and $L_{1}$ depend only on model parameters. We insert
	\eqref{eq:PfPosSSFcont8} into \eqref{eq:PfPosSSFcont3a} and obtain
	\begin{equation}
		\label{eq:PfPosSSFcont9}
		J \leq C_3  2\eps s_0 L^d + \frac{C_2}{\rho_-} \e^{-\mu L}.
	\end{equation}
	Combining \eqref{eq:PfPosSSFcont1}, \eqref{eq:PfPosSSFcont2b} and \eqref{eq:PfPosSSFcont9}, we conclude
	\begin{align}
		\label{eq:PfPosSSFcont10}
		\E \left[ \xi(E) \right] & \geq  \rho_- s_{0} \left( \frac{\gamma}{\rho_{+}}
		\frac{\mathcal{N}(I_{\varepsilon})}{2 \varepsilon}  -  C_3  s_{0} L^d
		 - \frac{C_2}{\rho_-}  \, \frac{\e^{-\mu L}}{2\varepsilon}  \right)  - C_1 \varepsilon^{\alpha}
	\end{align}
	for all $\varepsilon \in (0, \min\{\varepsilon_{0},\varepsilon_{1}\}]$,
	all $s_{0} \in (0,\min\{ 1,C\}]$
	and all $L \ge L_{1}$ such that $L/3 \in \N$ 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
	$\mathcal{N}(I_{\eps})/2\eps \to \mathcal{N}'(E)$ as $\varepsilon\downarrow 0$.
	Suppose also that $\mathcal{N}'(E)>0$. Then, there exists
	$\varepsilon_{2} \in (0, \min\{\varepsilon_{0},\varepsilon_{1}\}]$ such that
	$\mathcal{N}(I_{\eps})/2\eps \ge \mathcal{N}'(E)/2$ for every $\varepsilon \in (0,\varepsilon_{2}]$.
	Finally, we choose $L \ge L_{1}$ with $L/3 \in \N$ odd so large that $\eps := L^{-4d/\alpha} \le \varepsilon_{2}$
	and $s_0:=L^{-2d} \le \min\{1,C\}$. In this case \eqref{eq:PfPosSSFcont10} yields
	\begin{align}
		\label{ssf-lb-final}
		\E \left[ \xi(E) \right] & \geq \frac{\gamma \rho_{-}}{2\rho_{+}} \, L^{-2d}
		\Big( 	\mathcal{N}'(E)  -  O(L^{-d})  \Big).
	\end{align}
	The right-hand side of \eqref{ssf-lb-final} is strictly positive by possibly enlarging $L$ even further.
\end{proof}

\section{Proofs of the results from Section \ref{subsec:aoc}}
\label{sec:6}

\begin{proof}[Proof of Lemma \ref{Fredholm:Repr}]
  Consider the $N\times N$-matrix $M:= \big(\<\varphi^L_j, \psi^L_k\>\big)_{1\le j,k\le N}$.
  Then, the matrix entries of $MM^*$ and $M^*M$ read
  \beq
  \begin{aligned}
  	(MM^*)_{jl} &= \sum_{k=1}^N \< \varphi_j^L, \psi_k^L\>\< \psi_k^L,\varphi_l^L\>
  		= \< \varphi_j^L,P Q P\varphi_l^L\>, \\
  	(M^*M)_{jl} &= \sum_{k=1}^N \< \psi_j^L, \varphi_k^L\>\< \varphi_k^L,\psi_l^L\>
			= \< \psi_j^L, Q P Q\psi_l^L\>
  \end{aligned}
  \eeq
  for $1\le j,l \le N$.
	Thus, $S_{N,L}$ can be written as
  \begin{equation}
    \label{Fredholm:eq4}
    S_{N,L}^2 = \det ( M M^*)
	      = \det \big( P Q P \big|_{\ran P}\big)
	      = \det \big( \id - P (\id- Q) P\big) ,
  \end{equation}
  and likewise as
  \begin{align}
    \label{Fredholm:eq5}
    S_{N,L}^2 = \det (M^*M)
	        =  \det \big( Q P Q  \big|_{\ran Q}\big)
	      & =  \det\big(\id - Q(\id - P) Q \big) \notag\\
	      & =  \det\big(\id - (\id - P) Q (\id - P) \big).
  \end{align}
  Here, the last equality follows from the fact that the non-zero singular values of $Q(\id - P)$
  coincide with the non-zero singular values of its adjoint $(\id - P)Q$.
	The determinants in \eqref{Fredholm:eq4} and \eqref{Fredholm:eq5} are well-defined Fredholm
	determinants because $P (\id- Q) P$ and $Q(\id - P) Q$ are of finite rank. Multiplying the expressions in \eqref{Fredholm:eq4} and \eqref{Fredholm:eq5} yields
  \begin{align}
    \label{Fredholm:eq1}
    S_{N,L}^4 &= \det \big( \id - P (\id- Q) P\big)  \det\big(\id - (\id - P) Q (\id - P) \big)
    	\notag \\
	   &= \det \big( \id - P (\id- Q) P -  (\id - P) Q (\id - P) \big).
  \end{align}
  Now, Lemma \ref{th:SimonsTrace1} completes the proof of the lemma.
\end{proof}

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

\begin{proof}[Proof of Theorem \ref{th:IntMainRes}]
	We address the different parts of the theorem in the order \itemref{th:IntMainResiprime},
	\itemref{th:IntMainResii}, \itemref{th:IntMainResiii}, \itemref{th:IntMainResi}.

	\textsc{Part \itemref{th:IntMainResiprime}.} \quad
	Let $E\in \FMB$. Then Corollary \ref{cor:TrClBounds2} implies that $T(E)\in\mathcal{S}^2$ almost surely,
	and we have $S(E)= \det\big(\id- T(E)^2\big)^{1/4}$ 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 \eqref{eq:T<1}, we denote by
	$1 > \|T(E)\|^{2} = b_{1} \ge b_{2} \ge \ldots \ge 0$
	the non-increasingly ordered sequence of eigenvalues of $T(E)^{2}$. Then we obtain
	\begin{align}
		\label{eq:S-expand}
 		S(E)^{4} &= \exp\bigg\{ \sum_{n\in\N} \ln (1-b_{n})\bigg\}
			= \exp\bigg\{ -\sum_{n\in\N} \sum_{k\in\N} \frac{b_{n}^{k}}{k}\bigg\} \notag\\
		& \ge \exp\bigg\{ - \sum_{n\in\N} b_{n}\sum_{k\in\N} b_{1}^{k-1}\bigg\}
			= \exp\bigg\{ - \frac{\|T(E)\|_{2}^{2}}{1- \|T(E)\|^{2}}\bigg\}
	 		>0.
	\end{align}

	%
	%
	\textsc{Part \itemref{th:IntMainResii}.} \quad
	%
	We fix a compact interval $I \subset\FMB$, $E\in I$, a constant $c>0$ and define the event
	$\Omega_{0} := \{ \|T(E)\|_{1} <1\}$. Markov's inequality implies
	\begin{align}
		\label{eq:FinalAbsOfAndOrth11}
		\mathbb{E}\big[ S(E) \big]
		& \geq \e^{-c/4}\ \mathbb{P}\left[ S(E)^4\geq \e^{-c} \right]
			\ge  \e^{-c/4}\ \mathbb{P}\big[ \Omega_{0} \cap \{ S(E)^4\geq \e^{-c}\}  \big] \notag\\
		& \ge \e^{-c/4}\ \mathbb{P}\bigg[ \Omega_{0} \cap \bigg\{ \exp \Big(- \frac{\|T(E)\|_{2}^{2}}{1- \|T(E)\|^{2}}
			\Big) \geq \e^{-c} \bigg\} \bigg],
	\end{align}
	where we used \eqref{eq:S-expand} to obtain the last inequality.
	Since $\|T(E)\| \le \|T(E)\|_{1}< 1$ on $\Omega_{0}$ and, thus,
	$\|T(E)\|_{2}^{2} \le \|T(E)\| \|T(E)\|_{1} < \|T(E)\|_{1}$, we conclude that the fraction inside
	the exponent in the last line of \eqref{eq:FinalAbsOfAndOrth11} is bounded from above by
	$\|T(E)\|_{1}/(1- \|T(E)\|_{1})$. Therefore we get
	\beq
		\mathbb{E}\big[ S(E) \big] \ge  \e^{-c/4}\ \mathbb{P}\Big[  \|T(E)\|_1 \leq \frac{c}{c+1} \Big].
	\eeq
	We apply Markov's inequality again and obtain
	\begin{equation}
		\label{eq:FinalAbsOfAndOrth13}
		\mathbb{P}\Big[ \|T(E)\|_1 \leq \frac{c}{c+1} \Big] = 1-\mathbb{P}\Big[  \|T(E)\|_1  > \frac{c}{c+1} \Big]
		\geq 1- \frac{c+1}{c} \,\mathbb{E}\big[ \|T(E)\|_1 \big]
	\end{equation}
	so that
	\begin{equation}
		\label{eq:FinalAbsAndOrth14}
		\mathbb{E}\big[ S(E) \big] \geq \e^{-c/4} \Big( 1-  \frac{c+1}{c} \,\mathbb{E}\big[ \|T(E)\|_1 \big] \Big).
	\end{equation}
	Corollary \ref{cor:TrClBounds2} gives $\sup_{E \in I} \E\big[\|T(E)\|_1\big]\to 0$ as $\tau\to 0$.
	Hence, the claim follows by letting $c\downarrow 0$.

	%
	%
	\textsc{Part \itemref{th:IntMainResiii}.} \quad
	%
	We restrict ourselves to $ W\geq 0$, the proof for $ W\leq 0$ follows along the same lines.
	Since $\theta(E,H,H^\tau) = \xi(E,H,H^\tau)$ for a.e.\ $E\in\FMB$ almost surely by
	Theorem \ref{th:SSFandTransOpi}, the equivalence \eqref{AOCequiIndex} implies
	\beq\label{eq:equivprob}
		\PP\big[S(E)=0\big] = \PP \big[ \theta(E,H,H^\tau) >0\big] = \PP\big[ \xi(E,H,H^\tau) >0 \big]
	\eeq
	for a.e.\ $E\in\FMB$. Hence, the claim follows from Theorem \ref{lem:SSFcont}.

	%
	%
	\textsc{Part \itemref{th:IntMainResi}.} \quad
	%
	We fix $E \in\FMB \cap \Int(\Sigma)$.
	The almost-sure statement \eqref{eq:ResultNoAocStatPointw} implies \eqref{eq:ResultNoAocStat1}
	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 \eqref{eq:ResultNoAocStatPointw}.

	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, \eqref{eq:SSFandIndex} implies
	that there is a random variable $n_{0}:= n_{0}(E):\Omega \rightarrow\N$ such that almost surely
	\begin{equation}
		\label{theta-const}
  	\theta(E)=\theta_{L_n}(E) \qquad \text{for all } n\geq n_0.
	\end{equation}

 	\emph{Case 1: $\theta(E)=0$.} \quad
	Let $n \ge n_{0}$. Because of \eqref{theta-const}, \eqref{eq:trShift-index} and
	\eqref{eq:xi-T} for $L=L_{n}$ we have the representation \eqref{eq:Srep:xi0} for the finite-volume overlap
	in this case. The continuity of the Fredholm determinant with respect to the trace norm
	\cite[Thm.\ 3.4]{simon2005trace} implies
	\begin{align}
		\label{eq:ProofIndexZero}
		\big|S_{L_{n}}(E)^{4} - S(E)^{4}\big| & =
		\big|\det\big( \id - T_{L_n}(E)^2\big) - \det\big(\id - T(E)^2\big)\big| \notag \\
		& \leq 2 \| T_{L_n}(E) - T(E) \|_1\exp\big( \|T_{L_n}(E)\|_2^2+ \|T(E)\|_2^2 +1\big).
	\end{align}
	Theorem \ref{Thm:prod} now yields the almost-sure convergence
	\begin{equation}
		\lim_{n\to\infty} \| T_{L_n}(E) - T(E) \|_1\exp\big( \|T_{L_n}(E)\|_2^2+ \|T(E)\|_2^2 +1\big) =  0.
	\end{equation}
	This and \eqref{eq:ProofIndexZero} imply the desired convergence \eqref{eq:ResultNoAocStatPointw}
	in the case $\theta(E)=0$.

	\emph{Case 2: $\theta(E) \neq 0$.} \quad
	We assume without loss of generality that $\theta(E)>0$. The other case follows along the same lines.
	Because of the equivalence \eqref{AOCequiIndex} we have $S(E)=0$ in this case, which, in turn, is equivalent
	$1\in \sigma\big(T(E)^2\big)$ by \eqref{eq:ResultNoAocStat2}.
	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
	\beq
		\label{eq:strongconv}
		\mathrm{s}\,\text{-}\!\!\lim_{n\rightarrow\infty} ( P_n -  Q_n)^2 =  T(E)^2
		\qquad\text{almost surely}
	\eeq
	for the time being. Since $1\in \sigma\big(T(E)^2\big)$, it implies almost surely the existence of a sequence of
	eigenvalues $(\alpha_n)_{n\in\N}$ with $\alpha_n\in \sigma \big( (P_n -Q_{n})^{2}\big)$ and
	$\alpha_n\to 1 $ as $n\to\infty$ \cite[Thm.\ VIII.24(a)]{reed1980methods1}. Moreover,
	$0 \leq (P_n -Q_{n})^2 \leq 1$, and we conclude
	(note that $N_{L_{n}}(E) \neq 0$ for $n$ large enough because $E \in \Int(\Sigma)$,
	whence the first line in \eqref{eq:DefOverlap} applies)
	\beq
  	S_{L_n}(E)^{4} = \det\big( \id - (P_n -  Q_n)^2\big) \leq 1- \alpha_n \rightarrow 0
	\eeq
	as $n \to\infty$ almost surely. This is the assertion in the case $\theta(E) \neq 0$.

	Thus, it remains to prove \eqref{eq:strongconv}. For this it suffices to show strong convergence
	$P_n - Q_{n} \to T(E)$ as
	$n\to\infty$. Let $\eta\in L^2(\R^d)$. We add and subtract the term $T_{L_n}(E)$ and estimate
	\begin{equation}
		\label{eq:pfAOC1}
		\big\| \big( P_n -   Q_n - T(E) \big) \eta \big\|
		\leq \big\| \big(\id_{(-\infty,E]}(H^\tau_{L_n}) - Q_n\big)\eta\big\|
		+	\big\|\big( T_{L_n}(E) - T(E)\big)\eta \big\|,
	\end{equation}
  where we used that $ P_{n} = \id_{(-\infty,E]}(H_{L_n})$ by definition of the particle number
  \eqref{eq:DefParticleNumberSeq}.
	The second term on the right-hand side of \eqref{eq:pfAOC1} converges to $0$ as $n\to\infty$ almost surely
	by Theorem~\ref{Thm:prod}.
  As to the first term on the right-hand side we recall from \eqref{theta-const},
	\eqref{eq:trShift-index} and $\Tr P_{n} = \Tr Q_{n}$ that
	\begin{equation}
 		\theta(E) = \theta_{L_{n}}(E) = \Tr\big( T_{L_{n}}(E) \big)
		= \Tr\big( Q_n - \id_{(-\infty,E]}(H_{L_{n}}^{\tau}) \big)
	\end{equation}
	almost surely for $n\ge n_{0}$. 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
	\beq\label{eq:pfAOC1b}
	\mu_{N_{L_n}(E)-\theta(E)}\leq E\leq \mu_{N_{L_n}(E)-\theta(E)+1} \leq\mu_{N_{L_n}(E)}
	\eeq
	and that
  \beq\label{eq:pfAOC1c}
  	Q_n - \id_{(-\infty,E]}(H^\tau_{L_n}) =
  		\displaystyle\sum_{k= 0}^{\theta(E)-1} |\psi^{L_{n}}_{N_{L_n}(E)-k}\> \<\psi^{L_{n}}_{N_{L_n}(E)-k}|
   \eeq
  hold almost surely for $n \ge n_{0}$.
  Since $E \in\Int(\Sigma) $ and $\Sigma = \sigma_{\text{ess}}(H^{\tau})$, we obtain for any $\eps>0$ by Fatou's lemma and strong resolvent convergence of $H_{L_{n}}^{\tau}$ to $H^{\tau}$ that
  	\begin{equation}
		 \lim_{n\to\infty} \Tr \big(\id_{[E,E+\eps)}(H_{L_{n}}^{\tau}) \big)
		\ge \Tr\big(\id_{[E,E+\eps)}(H^{\tau}) \big)= \infty
	\end{equation}
	almost surely.
 Since $\theta(E)$ is a finite number,  this implies that for any $\eps>0$ there is an $n_1>0$ such that
 \beq
 \displaystyle\sum_{k= 0}^{\theta(E)-1} |\psi^{L_{n}}_{N_{L_n}(E)-k}\> \<\psi^{L_{n}}_{N_{L_n}(E)-k}|\leq
 \id_{[E,E+\eps)}(H^{\tau})
 \eeq
 for all $n\geq n_1$ almost surely.
 This yields for any fixed $\varepsilon>0$ the bound
  \beq
  	\label{eq:pfaoc2}
  	\limsup_{n\to\infty} \big\| \big( Q_{n} - \id_{(-\infty,E]}(H^\tau_{L_n}) \big)\eta\big\|
  	\leq \limsup_{n\to\infty} \big\| \id_{[E,E+\eps)} (H^\tau_{L_n})\eta \big\|
  \eeq
  almost surely.
	Finally, we fix a null sequence $(\varepsilon_{l})_{l\in\mathbb{N}} \subset (0,\infty)$.
	Lemma \ref{lm:multiEigenvalues} implies that $E,E+\varepsilon_{l}$, $l\in\mathbb{N}$, are no eigenvalues of
	$H^\tau$ almost surely.  Therefore we have strong convergence $\id_{[E,E+\varepsilon_l)}(H^\tau_L)
	\to \id_{[E,E+\varepsilon_l)}(H^\tau)$ as $L\to\infty$ almost surely \cite[Thm.\ VIII.24(b)]{reed1980methods1}.
	This and \eqref{eq:pfaoc2} lead to
	\begin{align}
  	\label{eq:pfaoc4}
		\limsup_{n\to\infty} \big\| \big( Q_{n} - \id_{(-\infty,E]}(H^\tau_{L_n}) \big)\eta\big\|
  	& \leq  \limsup_{l\to\infty} \big\| \id_{[E,E+\eps_l)} (H^\tau)\eta \big\| \notag\\
  	& = \big\| \id_{\{E\}} (H^\tau)\eta \big\|=0
  \end{align}
	almost surely, where the last equality follows again from $E$ being almost surely not an eigenvalue of $H^\tau$.
	This proves \eqref{eq:strongconv}.
\end{proof}

\begin{appendix}

\section{Stability of fractional-moment bounds}
\label{app:FMB}

Here, we prove that the fractional-moment bounds given in Definition \ref{DefFMB} are stable under
local perturbations. Among others, we show that $\FMB(H)=\FMB(H +W)$ for per\-turbations
$W\in L_c^\infty(\R^d)$.

\begin{lemma}
	\label{Lem:PertPersPot}
	Let  $I\subset\FMB$ be a compact interval. Then for any fixed $0<s<1$ there exist constants $C,\mu>0$
	such that for all $G\subseteq\Rn$ open and $a,b \in \Rn$
	\begin{equation}
		\label{eq:PertPersPotStat}
		\sup_{\tau\in [0,1]}\sup_{E\in I,\eta\neq 0} \mathbb{E}\left[\|\chi_a R_{E+i\eta}(H^\tau_G)
		\chi_b\|^s \right] \leq C \e^{-\mu|a-b|}
	\end{equation}
	holds. In particular, $\FMB(H)=\FMB(H^\tau)$ for every $\tau \in [0,1]$.
\end{lemma}

In order to prove Lemma \ref{Lem:PertPersPot}, we need to know that if fractional-moment estimates hold
for one exponent $0 < s <1$, then they hold for every exponent $0 < s <1$. This was proven in
\cite[Lemma B.2]{artRSO2001AizEtAl} in the discrete case. This proof directly extends to the continuum.

\begin{lemma}
	\label{lem:PossFrac}
	If, for some $0<s_{0}<1$, there exist constants $C_{s_{0}},\mu_{s_{0}}>0$ such that for
	all $G\subseteq\Rn$ open and $a,b\in\R^{d}$ the bound
	\begin{equation}
		\label{FMBassump}
		\sup_{E\in I,\eta\neq 0}\mathbb{E} \left[ \Vert \chi_{a}R_{E+i\eta}(H_G)\chi_{b} \Vert^{s_{0}} \right]
		\leq C_{s_{0}} \e^{-\mu_{s_{0}}\vert a-b\vert}
	\end{equation}
	holds, then for every $0<s<1$ there exist constants $C_s,\mu_s>0$ such that for all $G\subseteq\Rn$ open
	and $a,b\in\R^{d}$ the bound
	\begin{equation}
		\label{s0boost}
		\sup_{E\in I,\eta\neq 0} \mathbb{E} \left[ \Vert \chi_{a}R_{E+i\eta}(H_G)\chi_{b} \Vert^{s} \right]
		\leq C_{s} \e^{-\mu_{s}\vert a-b\vert}
	\end{equation}
	holds.
\end{lemma}

\begin{proof}
	For every $0<s\leq s_{0}$, the claim \eqref{s0boost} follows from \eqref{FMBassump} by H\"older's inequality.
	For $s_0 < s<1$ we have $\frac{1-s_0}{s-s_0}>1$. We fix $q\in(1,\frac{1-s_0}{s-s_0})$ and let
	$p>1$ be its H\"older conjugate. Using H\"older's inequality, we estimate for all $z\in\C\setminus\R$,
	$a,b\in\R^{d}$ and all $G \subseteq\R^{d}$ open
	\begin{align}
		\label{appendix:ss0eq1}
		\mathbb{E} \left[ \Vert \chi_{a}R_{z}(H_G)\chi_{b} \Vert^{s} \right]
		& = \mathbb{E} \left[ \Vert \chi_{a}R_{z}(H_G)\chi_{b} \Vert^{s_0/p}
			\Vert \chi_{a}R_{z}(H_G)\chi_{b} \Vert^{s-s_0/p}\right] \notag\\
	 	& \leq \mathbb{E} \big[ \Vert \chi_{a}R_{z}(H_G)\chi_{b} \Vert^{s_{0}} \big]^{1/p} \,
			\mathbb{E} \big[ \Vert \chi_{a}R_{z}(H_G)\chi_{b} \Vert^{qs-s_0 q/p} \big]^{1/q}.
	\end{align}
	The choice of $q$ ensures that $qs-s_0 q/p = q(s-s_0) +s_0 <1$, and we use the a priori estimate
	\eqref{eq:AppAPriori} to bound the second expectation in the last line of \eqref{appendix:ss0eq1}.
	Finally, assumption \eqref{FMBassump} yields the claim.
\end{proof}

\begin{remark}
	\label{PossFracUni}
 	The proof of Lemma \ref{lem:PossFrac} shows that if one replaces $H_{G}$ by $H_{G}^{\tau}$ and assumes that
 	\eqref{FMBassump} holds even uniformly in $\tau\in[0,1]$, then the assertion also holds uniformly in
 	$\tau\in[0,1]$.
\end{remark}

\begin{proof}[Proof of Lemma \ref{Lem:PertPersPot}]
	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
	\begin{equation}
		R_{z}(H_{G}^{\tau}) = R_{z}(H_{G}) -\tau R_{z}(H_{G}) W R_{z}(H_{G}^{\tau})
	\end{equation}
	yields the upper bound
	\begin{align}
		\mathbb{E} \left[ \Vert\chi_{a} R_{z}(H_{G}^{\tau})\chi_{b}\Vert^s \right]
		& \le \mathbb{E} \left[ \Vert\chi_{a}R_{z}(H_{G})\chi_{b}\Vert^s \right]
			+ \tau^{s} \mathbb{E} \left[\Vert\chi_{a} R_{z}(H_{G}) W R_{z}(H_{G}^{\tau})\chi_{b}\Vert^s\right] \notag\\
		& =: I_{1}+ \tau^{s} I_{2}.
	\end{align}
	While the first term $I_{1}$ is independent of $\tau$ and can directly be estimated by
	the fractional-moment bound \eqref{eq:DefFMB}
	for the unperturbed operator $H_{G}$, we estimate $I_2$ as
	\begin{align}
		I_{2} & \leq \|W\|_\infty^{s} \sum_{c\in\Gamma_W} \mathbb{E} \left[ \Vert\chi_{a}R_{z}(H_{G})\chi_{c}\Vert^s
			\Vert\chi_{c} R_{z}(H_{G}^{\tau}) \chi_{b}\Vert^s \right] \notag\\
		& \leq  \|W\|_\infty^{s} \sum_{c\in\Gamma_W}
			\mathbb{E} \left[ \Vert\chi_{a} R_{z}(H_{G}) \chi_{c}\Vert^{2s}\right]^{1/2}
			\mathbb{E} \left[\Vert\chi_{c} R_{z}(H_{G}^{\tau}) \chi_{b}\Vert^{2s} \right]^{1/2} ,
	\end{align}
	where $\Gamma_{W}$ is defined in \eqref{def:GammaW}. Now, we estimate the first expectation
	with the fractional-moment bound from Lemma \ref{lem:PossFrac} for the unperturbed operator
	$H_{G}$ and the exponent $2s <1$. And we estimate the second expectation with the a priori estimate
	\eqref{eq:AppAPriori} which is uniform in $\tau \in [0,1]$. This yields
	\begin{equation}
		\label{AppLm1eq1}
		I_{2} \leq C_1 \sum_{c\in\Gamma_W}  \e^{-\mu_1|c-a|} \leq C_2 \e^{-\mu |a|}
	\end{equation}
	with finite constants $C_{1}, C_{2}>0$ that are independent of $G \subseteq\R^{d}$ open, $a \in\R^{d}$
	and $\tau \in [0,1]$. For the second inequality we used that $\Gamma_{W}$ is finite due to the
	compact support of $W$. Since $R_{z}(H_{G}) W R_{z}(H_{G}^{\tau}) = R_{z}(H_{G}^{\tau}) W R_{z}(H_{G})$,
	we obtain along the same lines that $I_2\leq C_2 \e^{-\mu |b|}$. Multiplying this inequality with
	\eqref{AppLm1eq1}, we infer
	\beq
		\label{AppLm1eq2}
		I_2\leq C_2 \e^{-(\mu /2)(|a|+|b|)} \le C_{2} \e^{-(\mu /2)(|a - b|)}.
	\eeq
	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 Lemma \ref{lem:PossFrac} together with Remark
	\ref{PossFracUni}.
\end{proof}

We finish this appendix with another simple stability result. Eigenvalues of ergodic random Schr\"odinger operators move
wildly as a function of the random coupling constants so 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
\eqref{eq:AppAPriori} and thus extends to more general situations.

\begin{lemma}\label{lm:multiEigenvalues}
	Let $E\in\R$ and $\tau\in [0,1]$ be given. Then $E$ is not an eigenvalue of $H^\tau$ almost surely.
\end{lemma}

\begin{proof}
Since, for $x\in\R$, $-i\eps (x-E-i\eps)^{-1}\to \id_{\{E\}}(x)$ as $\eps\to 0$, the spectral theorem implies  the strong convergence  $-i\eps R_{E+i\eps}(H^{\tau}) \to \id_{\{E\}}(H^{\tau})$ as $\eps\to 0$. Using this, we obtain for some $0<s<1$ and any
$a, b\in\Z^d$ that
\begin{equation}
\mathbb{E} \big[ \|\chi_a\id_{\{E\}}(H^{\tau}) \chi_b \|^s \big] \leq \liminf_{\eps \to 0} \eps^s\,  \mathbb{E} \big[ \|\chi_a R_{E+i\eps}(H^{\tau})\chi_b \|^{s} \big] = 0,
\end{equation}
where the last equality follows from the a priori bound \eqref{eq:AppAPriori}.
Since $\Z^d$ is countable, $\id_{\{E\}}(H^{\tau})= 0$ almost surely.
\end{proof}

\section{Deterministic a priori estimate for Schr\"odinger operators}
\label{app:Schatten}

This appendix contains a local Schatten-von Neumann-class bound for functions of (non-random)
Schr\"odinger operators. We state it in Lemma~\ref{lem:aPriori} and apply it in the proofs of
Theorem \ref{th:TrClBounds1} and Theorem \ref{th:TrClBounds3}.
Estimates of this type appeared in \cite[Sect.\ B.9]{artSEM1982Sim}, \cite[App.\ A]{artRSO2006AizEtAl2} and \cite{MR2303305}.
We include the proofs for completeness, following closely \cite{MR2303305}.

We frequently use the following inequalities for Schatten norms.

\begin{lemma}
	\label{lem:schatten-hoelder}
	Let $p>0$. Then, for $A\in \S^\infty$ and $B\in \S^p$ we have
	\beq
		\label{singular:eq1}
		\| AB \|_p^{p}\leq \| A \|^p \|B\|_p^p.
	\eeq
	If $0<\varepsilon< p$ is such that $A\in\S^{p-\eps}$, then the inequality
	\beq
		\label{singular:eq2}
		\|A\|_p^p\leq \|A\|^\varepsilon \|A\|_{p-\varepsilon}^{p-\varepsilon}
	\eeq
	holds.
\end{lemma}

\begin{proof}
	An application of the min-max principle implies
	\beq
		\label{ineq:singular}
		\mu_n(AB)\leq \|A\| \mu_n(B)
	\eeq
	for the singular values $\mu_n$, $n\in\N$, see e.g. \cite[Thm.\ 1.6]{simon2005trace}. Hence,
	\beq
		\|AB\|_p^p =\sum_{n\in\N} |\mu_n(AB)|^p \leq \|A\|^p \|B\|_p^p
	\eeq
	holds. Moreover, we have
	\beq
		\|A\|_p^p = \sum_{n\in\N} \mu_n(A)^p = \sum_{n\in\N} \mu_n(|A|)^p
	  = \sum_{n\in\N} \mu_n(|A|^{\varepsilon/p} |A|^{1-\varepsilon/p})^p.
	\eeq
	Together with \eqref{ineq:singular}, this gives \eqref{singular:eq2}.
\end{proof}

In the remaining part of this appendix we consider the non-random Schr\"odinger operator
\begin{MyDescription}
\item[(D)] {\quad $H:=-\Delta+U$ with a bounded potential $U\in L^{\infty}(\R^{d})$,} \label{AssD}
\end{MyDescription}
acting in $L^{2}(\R^{d})$.
To formulate the deterministic a priori bound we introduce $E_{0} := \inf_{x\in\Rn} U(x)$
so that $H_G\geq E_0$ for all $G\subseteq \R^d$ open.

\begin{lemma}
	\label{lem:aPriori}
	Assume {\upshape (D)}. Let $p>0$ and $m\in\N$ with $m>d/(2p)$. Then there exists a finite constant $C$
	such that for all open $G\subseteq \R^d$, all $a,b\in\mathbb{R}^{d}$ and all $g\in L^\infty(\R)$
	we have the estimate
	\begin{equation}
		\label{eq:aPrioriStat}
		\|\chi_a g(H_{G})\chi_b\|_p \leq C \|g\|_\infty \big( \max\{0, \sup\supp(g) -E_{0}+1\}  \big)^m   .
	\end{equation}
\end{lemma}

\noindent
The constant $C$ in \eqref{eq:aPrioriStat} depends on the potential $U$, but only through
$\inf\sigma(H) -E_{0}$.
The proof of Lemma~\ref{lem:aPriori} is based, up to some iteration procedure, on the following
Schatten-class Combes-Thomas estimate.

\begin{lemma}
	\label{lem:SchattenCombesThomas}
	Assume {\upshape (D)}.
	Let $p>d/2$, $E \in (-\infty,E_0)$ and $\tilde E:= E_0-E>0$. Then there exist finite constants
	$C_{p,\tilde E},\mu_{p,\tilde E}>0$ such that for all open $G\subseteq \R^d$ and all $a,b\in\mathbb{R}^{d}$
	\begin{equation}
	\label{eq:SchattenCTStat}
		\|\chi_a (H_{G}-E)^{-1}\chi_b \|_p \leq  C_{p,\tilde E} \, \e^{-\mu_{p,\tilde E}|a-b|}.
	\end{equation}
\end{lemma}

\noindent
Whereas the constant  $\mu_{p,\tilde{E}}$ in $\eqref{eq:SchattenCTStat}$ is independent of the potential $U$,
the constant $C_{p,\tilde{E}}$ depends on $U$, but only through $\inf\sigma(H) -E_{0}$.

\begin{proof}[Proof of Lemma \ref{lem:SchattenCombesThomas}]
	Let $E\in (-\infty,E_0)$ and $G\subseteq\R^d$ open. The form inequality
	\begin{equation}
		H_G \geq -\Delta + E_0
	\end{equation}
	and Eq.\ (2.21) in \cite[Sec.\ VI.2]{MR1335452} imply the bound
	\begin{equation}
		\label{eq:aPriNB3}
		\big\Vert (-\Delta + E_0 -E )^{1/2}(H_G- E)^{-1/2}\big\Vert \leq 1
	\end{equation}
	for $E < E_0$. We set $\tilde E:= E_0-E $.
	Using H\"older's inequality, we estimate for fixed $a\in \R^d$ and $p' \ge 1$ (to be determined later)
	\begin{align}
		\label{eq:aPriNB4}
		\|\chi_a(H_G-E)^{-1/2}\|_{p'}
		& = \big\|\chi_a(-\Delta+ \tilde E)^{-1/2}(-\Delta + \tilde E)^{1/2}(H_G-E)^{-1/2} \big\|_{p'}  \notag \\
		& \leq  \big\|\chi_a (-\Delta + \tilde E)^{-1/2}\big\|_{p'}=: C_{p', \tilde E},
	\end{align}
	where $C_{p',\tilde{E}}<\infty$ is finite for $p'>d$ \cite[Thm.\ B.9.3]{artSEM1982Sim} and independent of $a$.
	Now let $p>d/2$ and fix $\theta\in(0,p- d/2)$, whence $p':=2(p-\theta)>d$.
	We obtain with Lemma \ref{lem:schatten-hoelder}
	\begin{equation}
		\label{eq:SchattenCT4}
		\big\|\chi_a(H_{G}-E)^{-1}\chi_b\big\|_p^{p}
		\leq \big\|\chi_a(H_{G}-E)^{-1}\chi_b \big\|_{p-\theta}^{p-\theta}
			\big\|\chi_a(H_{G}-E)^{-1}\chi_b\big\|^{\theta}.
	\end{equation}
	H\"older's inequality and the bound \eqref{eq:aPriNB4} therefore imply
	\begin{align}
		\label{eq:SchattenCT40}
		\big\|\chi_a(H_{G}-E)^{-1}\chi_b\big\|_p^{p} & \leq \| \chi_a(H_{G}-E)^{-1/2}\|_{2(p-\theta)}^{p-\theta}
			\|(H_{G}-E)^{-1/2}\chi_b\|_{2(p-\theta)}^{p-\theta} \notag\\
		& \qquad\times\big\|\chi_a(H_{G}-E)^{-1}\chi_b\big\|^{\theta} \notag\\
		& \le C^{2(p-\theta)}_{2(p-\theta), \tilde E}\, \big\|\chi_a(H_{G}-E)^{-1}\chi_b\big\|^{\theta}.
	\end{align}
	Now, the finite-volume operator can be replaced by the infinite-volume operator
	\begin{equation}
		\label{eq:res-fin-inf}
		\big\|\chi_a(H_{G}-E)^{-1}\chi_b\big\| \le \big\|\chi_a(H-E)^{-1}\chi_b\big\|.
	\end{equation}
	This follows from the Feynman-Kac representation for Dirichlet restrictions, see e.g.\ \cite[Sect.\ 6]{MR1756112},
	which implies the estimate
	\begin{equation}
 		0 \le (\chi_{a}(H_{G}-E)^{-1}\chi_b)(x,y) \le (\chi_{a}(H-E)^{-1}\chi_b)(x,y)
	\end{equation}
	for the kernels and all $(x,y) \in\R^{d}\times\R^{d}$. Moreover, we use
	$\|A\psi\| \le \| B |\psi|\| \le \|B\| \|\psi\|$ for integral operators $A$ and $B$ with kernels
	$0\le A(x,y) \le B(x,y)$, proving \eqref{eq:res-fin-inf}.
	Finally, the Combes-Thomas estimate \cite[Eq.\ (19)]{MR1937430} gives
	\begin{equation}
		\label{eq:op-CT}
		\big\|\chi_a(H-E)^{-1}\chi_b\big\| \leq  C_{\text{CT}}\, \e^{-\mu_{\text{CT}}|a-b|}
	\end{equation}
	with finite constants $\mu_{\text{CT}} >0$, which depends only on $\tilde{E}$, and $C_{\text{CT}}$, which depends only on
	$\tilde{E}$ and on $\inf\sigma(H) -E_{0}$. The claim follows from
	\eqref{eq:SchattenCT4}, \eqref{eq:SchattenCT40}, \eqref{eq:res-fin-inf} and \eqref{eq:op-CT}.
\end{proof}

\begin{proof}[Proof of Lemma \ref{lem:aPriori}]
	We use the abbreviation $H:=H_G$. Without loss of generality we assume $E_{0} \le \sup\supp(g) < \infty$
	(because otherwise the statement is trivial) and $0<p \le 1$ (because $\|\pmb\cdot\|_{p} \le \|\pmb\cdot\|_{1}$
	for $p \ge 1$). Let $m\in\N$ such that $m> d/(2p)$. We insert the $m$-th power of the resolvent
	and apply H\"older's inequality
	\begin{align}
		\label{eq:aPriori1}
		\big\|\chi_a g(H)\chi_b\big\|_p^{p} &= \| \chi_a(H-E_0+ 1)^{-m}(H-E_0+1)^m g(H)\chi_b\|_p^p \notag \\
		& \leq \| \chi_a(H-E_0+ 1)^{-m}\|_p^p \, \| (H-E_0+1)^m g(H)\chi_b\|^p \notag \\
		& \leq (\sup\supp(g)-E_0+1)^{mp} \, \|g\|_{\infty}^p \, \| \chi_a(H-E_0+ 1)^{-m}\|_p^p.
	\end{align}
	Next we set $a_0:=a$ and estimate with the adapted triangle inequality \eqref{lem:conkav}
	\begin{align}
		\label{eq:aPriori2}
	  \|\chi_a(H  -E_0+1)^{-m}\|_p^p
		& =  \Big\| \chi_{a_0}(H-E_0+1)^{-1}\Big(\sum_{a_1\in\Zd}\chi_{a_1} \Big)(H-E_0+1)^{-1}  \notag\\
		& \hspace{.7cm} \times \cdots \Big(\sum_{a_{m-1}\in\Zd} \chi_{a_{m-1}}\Big) (H-E_0+1)^{-1}
			\Big(\sum_{a_m\in\Zd} \chi_{a_m}\Big) \Big\|_p^p \notag \\
		& \leq \sum_{a_1,...,a_m\in \Zd}  \Big\| \prod_{l=1}^{m} \big(\chi_{a_{l-1}}(H-E_0+1)^{-1}\chi_{a_l}\big) \Big\|_p^p.
	\end{align}
	H\"older's inequality for Schatten classes and the Combes-Thomas estimate in the form of
	Lemma \ref{lem:SchattenCombesThomas} with $pm>d/2$ and $E=E_{0}-1$ imply
	\begin{align}
		\label{eq:aPriori20}
 		\Big\| \prod_{l=1}^{m} (\chi_{a_{l-1}}(H-E_0+1)^{-1}\chi_{a_l}) \Big\|_p^p
		& \leq \prod_{l=1}^{m} \big\Vert \chi_{a_{l-1}}(H-E_0+1)^{-1}\chi_{a_l} \big\Vert_{pm}^{p} \notag \\
	  & \leq  C_{1}  \prod_{l=1}^{m} \e^{-\mu_{1}|a_{l-1}-a_l|}
	\end{align}
	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 \eqref{eq:aPriori20} into \eqref{eq:aPriori2} and repeatedly using
	\begin{equation}
		\sum_{a_1\in\Zd} \e^{-\mu_{1}|a_0-a_1|} \e^{-\mu_{1}|a_1-a_2|} \leq C_{2} \e^{-\mu_{1}/2 |a_0-a_2|},
	\end{equation}
	where $C_{2}=C_{2,p,m} >0$ is a finite constant,
	we obtain the lemma from \eqref{eq:aPriori1}.
\end{proof}

\section{The index of a pair of projections}
\label{App:SuppMater}

The results in this appendix are essentially contained in \cite{artIND1994AvSeSi}.
We collect them here together with their short proofs for the convenience of the reader.
They will mainly be needed in the proof of Theorem \ref{th:IntMainRes}.

We start with a representation for powers of the difference of orthogonal projections.

\begin{lemma}
	\label{th:SimonsTrace1}
	Let $P$ and $Q$ be two orthogonal projections and let $P^c:= \id-P$, $Q^c:=\id-Q$. Then, the formulas
 	\begin{align}
		\label{appClmeq1}
		(P-Q)^{2n-1} &= (PQ^c)^{n} - (P^cQ)^{n} \\[-1ex]
	 	\intertext{and} \notag \\[-5.5ex]
		\label{appClmeq2}
		(P-Q)^{2n} &= (PQ^cP)^n+(P^cQP^c)^n
 	\end{align}
	hold for each $n\in\N$.
\end{lemma}

\begin{proof}
	We compute
	\beq
		\label{pfappClmeq1}
		P - Q = P(Q^c + Q) - (P + P^c )Q = PQ^c - P^c Q
	\eeq
	and
	\beq
		\label{pfappClmeq2}
		(P - Q)^2  = (P-Q)(P-Q)^* = PQ^cP+P^cQP^c .
	\eeq
	Formula \eqref{appClmeq2} follows from iterated multiplications of \eqref{pfappClmeq2} with itself.
	Formula \eqref{appClmeq1} follows from multiplying the $(n-1)$st power of \eqref{pfappClmeq2} with
	\eqref{pfappClmeq1} .
\end{proof}

For the next lemma we recall the definition of the spectral shift operator $T(E,A,B)$ from \eqref{def:T(E)}
and of the index $\theta(E,A,B)$ from  \eqref{def:index}.

\begin{lemma}
	\label{th:SimonsTrace2}
	Let  $A,B$ be self-adjoint operators in a Hilbert space, which are both bounded from below. Assume that
	the perturbation $B-A$ is sign-definite, that is,
	there exists $\alpha\in\{\pm1\}$ such that $\alpha(B-A) \ge 0$.
	Assume further that the spectral shift operator obeys $T(E,A,B)\in \mathcal{S}^{1}$ for some  $E\in\mathbb{R}$.
	Then, we have
  \begin{equation}
 		\label{eq:SimonsTrace2stat1}
  	\dim \ker \big( T(E,A,B) + \alpha\id \big)= 0
	\end{equation}
 	and
  \begin{equation}
 		\label{eq:SimonsTrace2stat2}
		\theta(E,A,B) =	\alpha\dim \ker \big( T(E,A,B) - \alpha\id \big).
	\end{equation}
	In particular, we have
	\beq
  	\label{eq:SimonsTrace2stat3}
 		1\in\sigma\big(T(E,A,B)^2\big) \iff \theta(E,A,B) \neq 0.
 	\eeq
\end{lemma}

\begin{proof}[Proof of Lemma \ref{th:SimonsTrace2}]
	The implication ``$\Leftarrow$'' in \eqref{eq:SimonsTrace2stat3} follows immediately from \eqref{eq:SimonsTrace2stat2};
	for ``$\Rightarrow$'', use \eqref{eq:SimonsTrace2stat2} and \eqref{eq:SimonsTrace2stat1}.
	The equality \eqref{eq:SimonsTrace2stat2} follows from the definition \eqref{def:index} of the index
	and \eqref{eq:SimonsTrace2stat1}. Thus, it remains to prove \eqref{eq:SimonsTrace2stat1}.
	We restrict ourselves to the case $B-A\geq 0$ and define the linear subspace
 	\beq
		\mathcal{V}_- := \ker \big( \id_{(-\infty,E]}(A)  \big)\cap \ran\big( \id_{(-\infty,E]}(B) \big).
 	\eeq
 	Clearly, $\mathcal{V}_{-} \subseteq\ker \big( T(E,A,B) + \id\big)$. Conversely, let
 	$\psi\in \ker \big( T(E,A,B) + \id\big)$, then
 	\beq
 		\<\psi, \id_{(-\infty,E]} (A) \psi\> + \<\psi, (1-\id_{(-\infty,E]}(B))\psi \>  = 0,
 	\eeq
 	which implies $\psi\in  \mathcal{V}_-$. This proves
	$\mathcal V_- = \ker \big( T(E,A,B) + \id\big)$.
  It remains to show that $\mathcal{V}_-=\{0\}$. Pick $E_{0}\in\R$ such that $A,B>E_0$. If $E \le E_{0}$ then
  $\ran\big( \id_{(-\infty,E]}(B) \big) =\{0\}$ and so $\mathcal V_-=\{0\}$. Now, consider the case $E>E_{0}$.
	We assume that there exists $\psi\in \mathcal{V}_-$
 	with $\Vert\psi\Vert=1$.
	Thus we have both $\psi\in \ker \big( \id_{(-\infty,E]} (A)\big) = \ran \big(\id_{(E,\infty)} (A)\big)$ and
 	$\psi\in \ran \big(\id_{(-\infty,E]} (B)\big)$. Using operator-monotonicity of the resolvent below the spectrum,
	we conclude
	\begin{equation}
		(E-E_0)^{-1} \leq \langle \psi,R_{E_0}(B)\psi \rangle \leq \langle \psi,R_{E_0}(A) \psi \rangle
   	< (E-E_0)^{-1},
  \end{equation}
 	which is a contradiction.
\end{proof}

\end{appendix}

redpajama_set_name: RedPajamaArXiv