BODY \input amstex \documentstyle{amsppt} \NoRunningHeads \pageheight{20cm} %hauteur de la page d'impression \pagewidth{15cm} %largeur de la page d'impression \hcorrection{0.8cm} %deplace le centre de la page horizontalement; negatif vers la gauche; positif vers la droite %\vcorrection{-1cm} %deplace le centre de la page verticalement; negatif vers le bas positif vers le haut \topmatter \title An asymptotic expansion for the density of states of a random Schr\"odinger operator with Bernoulli disorder \endtitle \author Fr\'ed\'eric Klopp %\footnotemark \endauthor %\footnotetext{U.R.A 760 C.N.R.S} \affil Department of Mathematics, The Johns Hopkins University, 3400 N. Charles St., Baltimore, 21218 MD, U.S.A \\ \\ D\'epartement de Math\'ematique, B\^at. 425, Universit\'e de Paris-Sud, Centre d'Orsay, 91405 Orsay C\'edex, France \endaffil \email kloppf@playmate.mat.jhu.edu \endemail \date February 1995 \enddate \keywords random Schr\"odinger operators, density of states, Bernoulli random variables \endkeywords \subjclass 35 Q 40, 47 B 80, 81 Q 10, 81 Q 20, 82 B 44 \endsubjclass \abstract In this paper, we study the density of states of a random Schr\"odinger operators of the form $H(t)=H+\sum_{\gamma\in{\Bbb Z}^d}t_\gamma V_\gamma$. Here $H$ is a periodic Schr\"odinger operator, $V$ is an exponentially decreasing function and $V_\gamma$, its translate by $\gamma$; the random variables $(t_\gamma)_{\gamma\in{\Bbb Z}^d}$ are chosen i. i. d. with the following common Bernoulli probability measure: $t_\gamma=1$ with probability $p$, and $t_\gamma=0$ with probability $1-p$. We show that $N_p(d\lambda)$, the density of states of $H(t)$, has an asymptotic expansion in $p$ when $p\to0$. Then, we use this expansion to deduce the behaviour of the integrated density of states of $H(t)$ in the gaps of $H$ when $p$ goes to 0. \bigskip \par\noindent {\smc R\'esum\'e.} Dans ce travail, nous \'etudions la densit\'e d'\'etats d'un op\'erateur de Schr\"odinger al\'eatoire de la forme $H(t)=H+\sum_{\gamma\in{\Bbb Z}^d}t_\gamma V_\gamma$. $H$ est un op\'erateur de Schr\"odinger p\'eriodique, $V$ est un potentiel exponentiellement d\'ecroissant et $V_\gamma$, son translat\'e par $\gamma$; les variables al\'eatoires $(t_\gamma)_{\gamma\in{\Bbb Z}^d}$ sont suppos\'ees i.i.d avec pour loi de probabilit\'e commune la loi de Bernoulli suivante: $t_\gamma=1$ avec probabilit\'e $p$, et $t_\gamma=0$ avec probabilit\'e $1-p$. Nous d\'emontrons l'existence d'un d\'eveloppement asymptotique en $p$ pour $N_p(d\lambda)$, ceci quand $p\to0$. Puis nous utilisons ce d\'eveloppement pour estimer la taille de la densit\'e d'\'etats int\'egr\'ee de $H(t)$ dans les lacunes du spectre de $H$ quand $p$ tend vers 0. \endabstract \endtopmatter \document %\hsize=40pc %\vsize=50pc %\magnification=1200 \loadbold \def\L2{L^2({\Bbb R}^d)} \def\Rd{{\Bbb R}^d} \def\Zd{{\Bbb Z}^d} \def\R{{\Bbb R}} \def\tr{\text{Tr}} \def\Sp{{\Cal S}'(\R)} \def\S{{\Cal S}(\R)} \def\C{{\Bbb C}} \define\equ{\operatornamewithlimits{\sim}} \subhead 0) Introduction \endsubhead \medskip The present paper is devoted to the study of the denstity of states of a simple model of random Schr\"odinger operators. The model (denoted by $H(t)$) consists in a periodic Schr\"odinger operator (denoted by $H$) to which impurities (represented by replicas of an exponentially decreasing potential $V$) have been added in the following way: at each site (represented by a point in $\Zd$), there is a probability $p$ of finding an impurity, and probability $1-p$ of finding none. $p$ is a parameter controlling the concentration of impurities in our model. The same model has been studied by R. Hempel and W. Kirsch in \cite{H-Ki} which inspired the present work. \par Let us call $N_p(d\lambda)$, the density of states for our model. Then $N_p$ is supported by the almost sure spectrum of $H(t)$, which does not depend on $p$, but, in the gaps of the spectrum of $H$, the density of states tends to 0 (in a weak sense) as $p$ tends to 0 (cf \cite{H-Ki}). As our main result, we prove that $N_p(d\lambda)$ admits an asymptotic expansion (in the distributional sense) in powers of $p$ when $p$ tends to 0, i.e, there exists a sequence of distributions $(n_k)_{k\geq0}$ such that $$N_p(d\lambda)\equ\Sb p\to0\\p>0\endSb\sum_{k\geq0}p^kn_k, \tag 0.1$$ the precise meaning of this asymptotic expansion being given in Theorem 1.1. \par We compute the distributions $(n_k)_{k\geq0}$. $n_0$ is the density of states of the periodic Schr\"odinger operator $H$. For $k\geq1$, let $\Lambda$ be a set of $k$ sites, and $H_\Lambda$ be the hamiltonian $H$ perturbed by putting exactly one impurity at each site of $\Lambda$. This is a relatively compact perturbation of $H$; moreover one can define $\zeta(\lambda;\Lambda)$, the spectral shift function for the pair $(H_\Lambda,H)$ (cf \cite{Ya}). Then we get the following formula for $n_k$ $$n_k(\lambda)=-\frac1{k!}\sum\Sb \Lambda\subset\Zd \\ \#\Lambda=k\text{ and }0\in\Lambda \endSb \sum_{A\subset\Lambda}(-1)^{\#\Lambda-\#A}\zeta'(\lambda;A).\tag 0.2$$ In the gaps of $H$, $H_\Lambda$ has only discrete eigenvalues. We say that $\lambda$ is a $k$-eigenvalue if it is an eigenvalue for some $H_\Lambda$ with $\#\Lambda=k$. Then, using (0.1) and (0.2), we show that, for $I$ an open interval strictly contained in a gap of $H$, \smallskip \par (a) $\dsize N_p(I)\sim\frac{{\frak n}_k(I)}{k!}p^k$ when $p\to0$, if $I$ contains $k$-eigenvalues but no $j$-eigenvalues for $1\leq j0$ (just by shifting $H$ by a constant). We recall that the spectrum of $H$ is made of bands of purely absolutely continuous spectrum (cf \cite{Re-Si} or \cite{Sj}). \par Let $V$ be a function that is not identically 0 and that satisfies \medskip \noindent(H.1) for some $\eta>0$, and some $C>0$, $\dsize\vert V(x)\vert\leq Ce^{-\eta\vert x\vert}$ for $x\in\Rd$. \medskip \noindent Let $(t_\gamma)_{\gamma\in\Zd}$ be independently identically distributed Bernoulli random variables taking value 1 with probability $p$ and value 0 with probability $1-p$; in other words, their common probability measure is defined by $P(t_0=1)=p$ and $P(t_0=0)=1-p$. \par We now consider the following random Schr\"odinger operator $$ H(t)=H+\sum_{\gamma\in\Zd}t_\gamma V_\gamma\qquad\text{ where }V_\gamma(x)=V(x-\gamma).\tag 1.1$$ $H(t)$ is a lower semi-bounded, essentially self-adjoint, ergodic random Schr\"odinger operator. Its domain is $H^2(\Rd)$. As $H(t)$ is ergodic, there exists a closed set $\Sigma$ such that, with probability 1, $\Sigma=\sigma(H(t))$ (cf \cite{Pa-Fi}, \cite{Ca-La}). As the potentials $\sum_{\gamma\in\Zd}t_\gamma V_\gamma$ are uniformly bounded for all realizations of $(t_\gamma)_{\gamma\in\Zd}$, by shifting $H$ by a constant, we may assume that $\inf(\Sigma)>0$. Moreover, as noted in \cite{H-Ki}, $\Sigma$ is independent of $p$ the parameter defining the probability measure of the $(t_\gamma)_{\gamma\in\Zd}$. \smallskip \subhead a) The density of states \endsubhead \smallskip \par Let $\Lambda_l$ be a cube in $\Rd$ centered at 0 and of sidelentgh $l$. Define $H_l^D(t)=H(t)_{|\Lambda_l}$ i.e. $H(t)$ restricted to $\Lambda_l$ with Dirichlet boundary conditions. Define, for $\lambda\in\R$, $${\Cal N}_l(\lambda)=\frac1{\text{Vol}(\Lambda_l)}\#\{\lambda_n;\ \lambda_n\text{ is an eigenvalue of }H_l^D(t)\text{ and }\lambda_n\leq\lambda\}$$ and $N_l(d\lambda)=\partial_\lambda{\Cal N}_l$, the corresponding discrete measure on $\R$. Then, e.g. by Theorem 5.20 of \cite{Pa-Fi}, there exists a non-random measure $N_p(d\lambda)$ such that, with probability 1, $$\lim_{l\to+\infty}N_l(d\lambda)=N_p(d\lambda),$$ and $$N_p(d\lambda)={\Bbb E}_p\{\tr(\chi_0 E_t(d\lambda)\chi_0)\}$$ where $E_t(d\lambda)$ is the spectral resolution of $H(t)$, $\chi_0$ is the characteristic function of the cube centered in 0 with sidelength 1, ${\Bbb E}_p$ is the average with respect to the probability measure defined by the $(t_\gamma)_{\gamma\in\Zd}$ and $\tr(A)$ is the trace of $A$. \par $N_p(d\lambda)$ is the {\it density of states} of $H(t)$. It is a positive measure supported in $\Sigma$. One has $$\int_{\R}\lambda^{-q}N_p(d\lambda)<+\infty\quad\text{for}\quad q>d/2$$ as $\chi_0(1-\Delta)^{-q}$ is trace class for $q>d/2$ and $H(t)$ is a uniformly bounded perturbation of $-\Delta$ over all realizations of $t$. Hence $N_p(d\lambda)$ is rapidly decreasing test function,a tempered distribution, that is an element of the Schwartz space $\Sp$. \par Before stating our main result let us introduce a last definition. Let $A\subset\Zd$ such that $A$ is finite, and define $\dsize H_A=H+\sum_{\gamma\in A}V_\gamma$. $H_A$ is relatively compact perturbation of $H$; moreover, for $q>d/2$ and $z\not\in\sigma(H_A)\cup\sigma(H)$, $(z-H_A)^{-q}-(z-H)^{-q}$ is trace class. By the arguments of \cite{Ki-Ma}, we know that $\inf(\sigma(H_A))\geq\inf(\Sigma)>0$. Hence, we define $\zeta(\lambda;A)$ to be the {\it spectral shift function} for the pair of operators $H_A$ and $H$; $\zeta(\lambda;A)$ is the distribution in $\Sp$ defined by, for $\phi\in\S$, $$Tr(\phi(H_A)-\phi(H))=\int_0^{+\infty}\zeta(\lambda;A)\phi'(\lambda)d\lambda$$ (see \cite{Ya} chapter 8 section 9). \par Let us now state our main result that is \proclaim{Theorem 1.1} $N_p(d\lambda)$ admits an asymptotic expansion in $\Sp$ when $p$ tends to 0 i.e, there exists a sequence of distributions $(n_k)_{k\geq0}$ such that \par (a) for any $k\geq0$, $n_k\in\Sp$, \par (b) for any $N>0$, there existe $\mid\cdot\mid_N$, a semi-norm in $\Sp$ such that, for any $\varphi\in\S$, rapidly decreasing test function, one has $$\vert\langle N_p(d\lambda),\varphi\rangle-\sum_{k=0}^N p^k\langle n_k,\varphi\rangle\vert\leq p^{N+1}\mid\varphi\mid_N\quad\text{for}\quad p\in[0,1].$$ Moreover the distributions $(n_k)_{k\in{\Bbb N}}$ are given by the following formulae: \par (c) if $k=0$, $n_0$ is the density of states for the unperturbed operator $H$, \par (d) if $k\geq1$, $n_k$ is given by the following convergent (in $\Sp$) series: $$n_k(\lambda)=-\frac1{k!}\sum\Sb \Lambda\subset\Zd \\ \#\Lambda=k\text{ and }0\in\Lambda \endSb \sum_{A\subset\Lambda}(-1)^{\#\Lambda-\#A}\zeta'(\lambda;A),$$ where $\zeta'(\lambda;A)$ is the derivative of $\zeta(\lambda;A)$ with respect to $\lambda$. \endproclaim \remark{Remark} Notice that, as supp$N_p(d\lambda)\subset\Sigma$, for any $k\geq 0$, one has supp$n_k\subset\Sigma$. In the proof of Theorem 1.1, we also obtain an upper bound on the order of $n_k$ which is ord$(n_k)\leq (k+1)(d+1)+2$. The same way one can give an estimate upon the order of the semi-norm used to control the remainder in the asymptotic expansion; if the expansion is made up to order $N$, then the remainder is at most of order $(N+2)(d+1)+2$.\endremark \smallskip \subhead b) The behaviour of the integrated density of states in the gaps of the spectrum of $\bold H$ when $\bold p$ tend to 0 \endsubhead \smallskip \par Let $\Lambda\subset\Zd$, $\Lambda$ finite, and define $H_\Lambda=H+\sum_{\gamma\in\Lambda}V_\gamma$. Let $\sigma(H_\Lambda)$ be the spectrum of $H_\Lambda$. We know that $\dsize \inf_{\Lambda\subset\Zd}\inf(\sigma(H_\Lambda))>0$. By our assumptions on $V$, $H_\Lambda$ is relatively compact perturbation of $H$; hence, by Weyl's Theorem, for any $\Lambda$ finite, the essential spectrum of $H_\Lambda$ is the essential spectrum of $H$ that is the spectrum of $H$. By $\Sigma_{\text{disc}}(\Lambda)$, we denote the closure of discrete spectrum of $H_\Lambda$ (i.e. the eigenvalues of $H_\Lambda$ contained in the gaps of $H$ as well as, possibly, the edges of the gaps). Then we define \definition{Definition} For $k\geq 1$, we define the $k$-eigenvalues of $(H,V)$ to be the elements of the set $\dsize{\Cal E}_k=\bigcup_{\#\Lambda=k}\Sigma_{\text{disc}}(H_\Lambda)$. \enddefinition \remark{Remark} For $A\in\Zd$, $A$ finite, we know that $\zeta(\lambda;A)$ is constant in the gaps of $\sigma(H)$ except at the eigenvalues of $H_A$ where it has a jump discontinuity (see \cite{Ya}). Hence, formula (d) of Theorem 1.1 tells us that, for $k\geq 1$, supp$\dsize (n_k)\subset\sigma(H)\bigcup\bigcup_{1\leq j\leq k}{\Cal E_k}$. \endremark One shows \proclaim{Proposition 1.2} $\overline{{\Cal E}_1}={\Cal E}_1$ and for any $k\geq2$, $\dsize \overline{{\Cal E}_k}={\Cal E}_k\bigcup\overline{{\Cal E}_{k-1}}=\bigcup_{j=1}^k{\Cal E}_j$. Moreover, for $I$, a closed interval contained in $\R\setminus\sigma(H)$, the points of $\overline{{\Cal E}_k\cap I}\setminus\overline{{\Cal E}_{k-1}\cap I}$ are isolated in $\overline{{\Cal E}_k\cap I}$. Here $\overline A$ denotes the closure of $A\subset\R$. \endproclaim For $I\subset\R\setminus\sigma(H)$, we define ${\frak n}_k(I)$ to be the number of $k$-eigenvalues of $(H,V)$ in $I$ counted with multiplicities, that is, if we define $\Pi_\Lambda(\lambda)$ to be the eigenprojector of $H_\Lambda$ associated to the eigenvalue $\lambda$, then $$ {\frak n}_k(I)=\sum\Sb \Lambda\subset\Zd \\ \#\Lambda=k\text{ and }0\in\Lambda \endSb\sum_{\lambda\in I\cap\sigma(H_\Lambda)}\text{rank}(\Pi_\Lambda(\lambda)).$$ \remark{Remark} Notice that this number may be infinite. In the definition of ${\frak n}_k(I)$, we restrict ourselves to counting the eigenvalues of $H_\Lambda$ for $\#\Lambda=k$ and $0\in\Lambda$. The condition ``$0\in\Lambda$'' permits us to get rid of a natural infinite multiplicity coming from the fact that, as $H$ is $\Zd$-periodic, $H_\Lambda$ and $H_{\Lambda+\gamma}$ are unitarily equivalent for any $\gamma\in\Zd$. \endremark The behaviour of the integrated density of states in the gaps of the spectrum of $H$ when $p$ tend to 0 is given by \proclaim{Theorem 1.3} Let $I$ be a open interval such that $\overline I\subset\R\setminus\sigma(H)$. Then, \par (a) if $I\cap{\Cal E}_k\not=\emptyset$ and for $1\leq j\leq k-1$, $\overline I\cap{\Cal E}_j=\emptyset$ then, $0<{\frak n}_k(I)<+\infty$, $$N_p(I)=\int_IN_p(d\lambda)=\frac{{\frak n}_k(I)}{k!}p^k(1+O(p))\quad\text{when }p\to 0,\ p>0,$$ \par (b) if for $1\leq j\leq k$, $\overline I\cap {\Cal E}_j=\emptyset$ then, $$N_p(I)=\int_I N_p(d\lambda)=O(p^{k+1})\quad\text{when }p\to 0,\ p>0.$$ \endproclaim \remark{Remark} If the $(n_k)_{k\geq0}$ were distributions of order 0, we could easily deduce the non-regularity of the density of states from Theorem 1.1. Here this is not possible because of the bad estimates we have on the remainder term; bad here means that we do not control the order. \endremark We recover in a more precise form and extend the results of \cite{H-Ki}. As noted in that paper, Theorem 1.3 proves that the integrated density of states concentrates around the $k$-eigenvalues of $(H,V)$. This notion of concentration can be made more precise for we have \proclaim{Theorem 1.4} Let $\mu$ be an isolated $k$-eigenvalue i.e. $\mu\in\dsize{\Cal E}_k$ and $\mu\not\in\dsize{\Cal E}_j$ for $1\leq j\leq k-1$. Then, for any $\dsize 0<\epsilon< \frac1{(k+2)(d+1)+2}$, $$p^{-k}N_p([\mu-p^{\epsilon},\mu+p^{\epsilon}])\to \frac{{\frak n}_k(\{\mu\})}{k!}\quad\text{when }p\to0,\ p>0.$$ \endproclaim \remark{Remark} So we see that, in the limit $p\to0$, the $k$-eigenvalues are ``asymptotic'' singular points of the density of states. In \cite{Ni-Lu}, these points were found to be singularities of the density of states for some related 1-dimensional model. \endremark \medskip \subhead II) The asymptotic expansion for the density of states \endsubhead \medskip By Theorem 5.20 of \cite{Pa-Fi}, we know that, for $\varphi\in\S$, $$\int_\R\varphi(\lambda)N_p(d\lambda)={\Bbb E}_p\{\tr(\chi_0\varphi(H(t))\chi_0)\}. \tag 2.1$$ \smallskip \subhead a) A most useful formula \endsubhead \smallskip For any realization of $t$, $H(t)$ is essentially self-adjoint, and for a real function $\varphi\in\S$, $\varphi(H(t))$ is well defined by the Spectral Theorem and can expressed (see \cite{He-Sj}) by the following formula $$\varphi(H(t))=\frac i{2\pi}\int_\C\frac{\partial\tilde\varphi}{\partial\overline z}(z)(z-H(t))^{-1}d\overline z\wedge dz, \tag 2.2$$ where $\tilde\varphi:\ \C\to\C$ is an extension of $\varphi$ such that \par (a) for $z\in\R$, $\tilde\varphi(z)=\varphi(z)$, \par (b) supp$(\tilde\varphi)\subset\{z\in\C;\ \vert\text{Im}(z)\vert<1\}$, \par (c) $\tilde\varphi\in{\Cal S}(\{z\in\C;\ \vert\text{Im}(z)\vert<1\})$, \par (d) The family of functions $\dsize x\mapsto\frac{\partial\tilde\varphi}{\partial\overline z}(x+iy)\cdot\vert y\vert^{-n}$ (for $0<\vert y\vert<1$) is bounded in $\S$ for any \par\quad $n\in{\Bbb N}$ and, one has the following estimates: for $n\geq 0$, $\alpha\geq0$, $\beta\geq0$, there exists $C_{n,\alpha,\beta}>0$ \par\quad such that $$\sup_{0<\vert y\vert\leq 1}\sup_{x\in\R}\left\vert x^\alpha\frac{\partial^\beta}{\partial x^\beta}\left(\vert y\vert^{-n}\cdot\frac{\partial\tilde\varphi}{\partial\overline z}(x+iy)\right)\right\vert\leq C_{n,\alpha,\beta}\sup\Sb\beta'\leq n+\beta+2 \\ \alpha'\leq\alpha\endSb\sup_{x\in\R}\left\vert x^{\alpha'}\frac{\partial^{\beta'}\varphi}{\partial x^{\beta'}}(x)\right\vert.$$ \par\noindent Such extensions always exist for $\varphi\in\S$ (see \cite{Mat}). \par Hence, for $\varphi\in\S$ such that supp$\varphi\subset(0,+\infty)$, for $q>d/2$, using (2.1) and (2.2) for the function $\psi(x)=x^q\varphi(x)$ (as $\varphi(H(t))=\psi(H(t))H(t)^{-q}$), we get $$\int_\R\varphi(\lambda)N_p(d\lambda)=\frac i{2\pi}{\Bbb E}_p\left\{\tr(\chi_0\left(\int_\C\frac{\partial\tilde\varphi}{\partial\overline z}(z)(z-H(t))^{-1}H(t)^{-q}d\overline z\wedge dz\right)\chi_0)\right\} \tag 2.3$$ as $\chi_0(z-H(t))^{-1}H(t)^{-q}\chi_0$ is trace class. Here $\tilde\varphi$ is an analytic extension of $\psi$. \par By Proposition 4.3, we know that, there exists $C>0$ such that for any realization of $t$, and for $z\in\C\setminus\R$ and $\vert$Im$(z)\vert\leq 1$, $$\vert\tr(\chi_0(z-H(t))^{-1}H(t)^{-q}\chi_0)\vert\leq C \left(\frac{1+\vert z\vert}{\vert\text{Im}(z)\vert}\right)^2.$$ Hence using (2.3), estimate (d) for analytic extensions and Fubini's Theorem, we get $$\int_\R\varphi(\lambda)N_p(d\lambda)=\frac i{2\pi}\int_\C\frac{\partial\tilde\varphi}{\partial\overline z}(z) {\Bbb E}_p\left\{\tr(\chi_0(z-H(t))^{-1}H(t)^{-q}\chi_0)\right\}d\overline z\wedge dz. \tag 2.4 $$ Finally, we are just left with finding an asymptotic expansion for ${\Bbb E}_p\{\tr(\chi_0(z-H(t))^{-1}H(t)^{-q}\chi_0)\}$. \smallskip \subhead b) The asymptotic expansion for ${\bold{\Bbb E}_{\bold p}\boldsymbol\{\tr({\boldsymbol\chi}_{\bold 0}\bold{(z-H(t))}^{\bold{-1}}\bold{H(t)}^{\bold{-q}}{\boldsymbol\chi}_{\bold 0}\bold )\boldsymbol\}}$ \endsubhead \smallskip For $l>0$ and a realization of $t$, we define $H_l(t)=H+\sum_{\gamma\in\Lambda_l}t_\gamma V_\gamma$ (here $\Lambda_l$ is cube in $\Zd$ of center 0 and sidelength $l$). By Proposition 4.3, we know that, for any configuration of $t$ and for $z\in\C\setminus\R$, $$ \Vert\chi_0(z-H(t))^{-1}H(t)^{-q}\chi_0-\chi_0(z-H_l(t))^{-1}H_l(t)^{-q}\chi_0\Vert_{{\Cal T}_1}\to0\quad\text{when }l\to+\infty$$ where $\Vert\cdot\Vert_{{\Cal T}_1}$ denotes the trace class norm. \par Hence using Lebesgue's Dominated Convergence Theorem, $${\Bbb E}_p\{\tr(\chi_0(z-H(t))^{-1}H(t)^{-q}\chi_0)\}=\lim_{l\to+\infty}{\Bbb E}_p\{\tr(\chi_0(z-H_l(t))^{-1}H_l(t)^{-q}\chi_0)\}.$$ By definition $H_l(t)$ depends only on finitely many random variables (i.e. the one indexed by $\gamma\in\Lambda_l$). Hence $$\aligned f_l(z,p)&:={\Bbb E}_p\{\tr(\chi_0(z-H_l(t))^{-1}H_l(t)^{-q}\chi_0)\} \\ &=\int_{[0,1]^{\Lambda_l}}\tr(\chi_0(z-H_l(t))^{-1}H_l(t)^{-q}\chi_0)\bigotimes_{\Lambda_l}(p\delta_1+(1-p)\delta_0).\endaligned$$ $f_l(z,p)$ obviously is a polynomial in $p$ of degree $\#\Lambda_l$. Hence we may expand it using Taylor's formula to an arbitrary order $N$ and get $$f_l(z,p)=\sum_{k=0}^N\frac{f^{(k)}_l(z,0)}{k!}p^k+\frac{p^{N+1}}{N!}\int_0^1f^{(N+1)}_l(z,pu)(1-u)^Ndu.$$ To compute the coefficients in this expansion, we notice that we are dealing with a measure on a finite set that depend analytically in the parameter $p$; so we just have to write the Taylor expansion in $p$ for this measure and integrate the trace against it; we compute $$\frac{d^k}{dp^k}\left(\bigotimes_{\Lambda_l}(p\delta_1+(1-p)\delta_0)\right)=\sum\Sb \Lambda\subset\Lambda_l \\ \#\Lambda=k\endSb\sum\Sb A\cup B=\Lambda \\ A\cap B=\emptyset\endSb(-1)^{\#B}\bigotimes_A\delta_1\bigotimes_B\delta_0\bigotimes_{\Lambda_l\setminus\Lambda}\left(p\delta_1+(1-p)\delta_0\right).$$ Hence, for $k\geq 1$, we get $$f^{(k)}_l(z,p)=\sum\Sb \Lambda\subset\Lambda_l \\ \#\Lambda=k\endSb\sum\Sb A\subset\Lambda\endSb(-1)^{\#(\Lambda\setminus A)}{\Bbb E}_p\{\tr(\chi_0(z-H_{\Lambda,A}(u))^{-1}H_{\Lambda,A}(u)^{-q}\chi_0)\},$$ where: \par (a) $H_{\Lambda,A}(u)=H+\sum_{\gamma\in A}V_\gamma+\sum_{\gamma\in(\Lambda_l\setminus\Lambda)}u_\gamma V_\gamma$, \par (b) $(u_\gamma)_{\gamma\in\Zd}$ are i.i.d random variables with common probability distribution $p\delta_1+(1-p)\delta_0$, \par (c) ${\Bbb E}_p$ is the expectation taken with respect to these random variables. \par\noindent One easily checks that this equality may be rewritten in the following form $$f^{(k)}_l(z,p)=\sum\Sb \Lambda\subset\Lambda_l \\ \#\Lambda=k\endSb\int_{[0,1]^{\Lambda}}{\Bbb E}_p\left\{\tr\left(\chi_0\partial_\Lambda\left[(z-H_{\Lambda}(t,u))^{-1}H_{\Lambda}(t,u)^{-q}\right]\chi_0\right)\right\}dt_\Lambda,$$ where: \par (a) $H_{\Lambda}(t,u)=H+\sum_{\gamma\in \Lambda}t_\gamma V_\gamma+\sum_{\gamma\in(\Lambda_l\setminus\Lambda)}u_\gamma V_\gamma$, \par (b) $(t_\gamma)_{\gamma\in\Lambda}$ are variables taking value in $[0,1]$, $\dsize\partial_\Lambda=\bigotimes_{\gamma\in\Lambda}\partial_{t_\gamma}$ et $\dsize dt_\Lambda=\bigotimes_{\gamma\in\Lambda}dt_\gamma$, \par (c) $(u_\gamma)_{\gamma\in\Zd}$ are defined as above. \par\noindent Define $$a_{\Lambda,l}(z,p)=\int_{[0,1]^{\Lambda}}{\Bbb E}_p\left\{\tr\left(\chi_0\partial_\Lambda\left[(z-H_{\Lambda}(t,u))^{-1}H_{\Lambda}(t,u)^{-q}\right]\chi_0\right)\right\}dt_\Lambda. \tag 2.5$$ We show the \proclaim{Proposition 2.1} (a) For any $\Lambda\subset\Zd$ and $K$, an arbitrary compact of $\C\setminus\R$, the sequence of functions $((z,p)\mapsto a_{\Lambda,l}(z,p))_{l\geq0}$ converges uniformly in $K\times[0,1]$ to a function $(z,p)\mapsto a_\Lambda(z,p)$ when $l$ tends to $+\infty$. \par\noindent (b) $a_\Lambda(z,0)$ is given by $$a_\Lambda(z,0)=\int_{[0,1]^{\Lambda}}\tr\left(\chi_0\partial_\Lambda\left[(z-H_{\Lambda}(t))^{-1}H_{\Lambda}(t)^{-q}\right]\chi_0\right)dt_\Lambda,$$ \par\noindent (c) There exists $C>0$ such that for $z\in\C\setminus\R$, $$\sum\Sb \Lambda\subset\Zd \\ \#\Lambda=k\endSb \sup\Sb p\in[0,1] \\ l>0 \endSb\vert a_{\Lambda,l}(z,p)\vert \leq C \left(\frac{1+\vert z\vert}{\vert\text{Im}(z)\vert}\right)^{(d+1)(k+1)}.$$ \endproclaim Then, using Lebesgue's dominated convergence theorem, we get that, for any $N>0$, $p\in[0,1]$ and $z\in\C\setminus\R$, $${\Bbb E}_p\{\tr(\chi_0(z-H(t))^{-1}H(t)^{-q}\chi_0)\}=\sum_{k=0}^N\frac{f_k(z)}{k!}p^k+p^{N+1}G_N(z,p), \tag 2.6$$ where $(f_k(z))_{0\leq k\leq N}$ and $G_N(z,p)$ are analytic in $z$ for $z\in\C\setminus\R$. The $(f_k)_{0\leq k\leq N}$ are defined by the following formulae $$ f_0(z)=\tr((z-H)^{-1}H^{-q}\chi_0), \tag 2.7.a$$ and, for $1\leq k\leq N$, $$%\aligned f_k(z)%& =\sum\Sb \Lambda\subset\Zd\\ \#\Lambda=k\endSb\int_{[0,1]^{\Lambda}}\tr\left(\chi_0\partial_\Lambda\left[(z-H_{\Lambda}(t))^{-1}H_{\Lambda}(t)^{-q}\right]\chi_0\right)dt_\Lambda %\\ &=\sum\Sb \Lambda\subset\Zd\\ \#\Lambda=k\endSb\sum\Sb A\subset\Lambda\endSb(-1)^{\#(\Lambda\setminus A)}\tr(\chi_0(z-H_{A})^{-1}H_{A}^{-q}\chi_0). \endaligned \tag 2.7.b$$ Moreover one has the following growth estimates at infinity and close to the real axis, for $0\leq k\leq N$, $$\vert f_k(z)\vert\leq C \left(\frac{1+\vert z\vert}{\vert\text{Im}(z)\vert}\right)^{(d+1)(k+1)}\quad\text{and}\quad\sup_{p\in[0,1]}\vert G_N(z,p)\vert\leq C \left(\frac{1+\vert z\vert}{\vert\text{Im}(z)\vert}\right)^{(d+1)(N+2)}. \tag 2.8$$ Using the translation invariance of $H$, for $k\geq 1$, we may rewrite (2.7.b) in the following form $$\aligned f_k(z)&=\sum\Sb \Lambda\subset\Zd\\ \#\Lambda=k\endSb\sum\Sb A\subset\Lambda\endSb(-1)^{\#(\Lambda\setminus A)}\tr(\left[(z-H_{A})^{-1}H_{A}^{-q}-(z-H)^{-1}H^{-q}\right]\chi_0) \\ &= \sum\Sb \Lambda\subset\Zd\\ \#\Lambda=k\text{ and }0\in\Lambda\endSb\sum_{\gamma\in\Zd}\sum\Sb A\subset\Lambda+\gamma\endSb(-1)^{\#(\Lambda\setminus A)}\tr(\left[(z-H_{A})^{-1}H_{A}^{-q}-(z-H)^{-1}H^{-q}\right]\chi_0) \\ &=\sum\Sb \Lambda\subset\Zd\\ \#\Lambda=k\text{ and }0\in\Lambda\endSb\sum\Sb A\subset\Lambda\endSb(-1)^{\#(\Lambda\setminus A)}\sum_{\gamma\in\Zd}\tr(\left[(z-H_{A-\gamma})^{-1}H_{A-\gamma}^{-q}-(z-H)^{-1}H^{-q}\right]\chi_0) \\ &=\sum\Sb \Lambda\subset\Zd\\ \#\Lambda=k\text{ and }0\in\Lambda\endSb\sum\Sb A\subset\Lambda\endSb(-1)^{\#(\Lambda\setminus A)}\sum_{\gamma\in\Zd}\tr(\left[(z-H_{A})^{-1}H_{A}^{-q}-(z-H)^{-1}H^{-q}\right]\chi_\gamma) \\ &=\sum\Sb \Lambda\subset\Zd\\ \#\Lambda=k\text{ and }0\in\Lambda\endSb\sum\Sb A\subset\Lambda\endSb(-1)^{\#(\Lambda\setminu! s A)}\tr((z-H_{A})^{-1}H_{A}^{-q} Let us now complete the proof of Theorem 1.1. By the estimate (d) for the analytic extension of $\psi$, by (2.7.b) and estimate (c) of Proposition 2.1, we know that, for $k\geq 1$, $$\aligned &\sum\Sb A\subset\Lambda\endSb(-1)^{\#(\Lambda\setminus A)}\tr((z-H_{A})^{-1}H_{A}^{-q}-(z-H)^{-1}H^{-q})\frac{\partial\tilde\varphi}{\partial\overline z}(z)\in L^1(\C; dzd\overline z)\text{ for any }\Lambda\subset\Zd \\ &\text{ and }\quad G_N(z,p)\frac{\partial\tilde\varphi}{\partial\overline z}(z)\in L^1(\C; dzd\overline z), \endaligned$$ and $$\left(\sum\Sb \Lambda\subset\Zd\\ \#\Lambda=k\text{ and }0\in\Lambda\endSb\left\vert\sum\Sb A\subset\Lambda\endSb(-1)^{\#(\Lambda\setminus A)}\tr((z-H_{A})^{-1}H_{A}^{-q}-(z-H)^{-1}H^{-q})\right\vert\right)\cdot\vert\frac{\partial\tilde\varphi}{\partial\overline z}(z)\vert\in L^1(\C; dzd\overline z).$$ Hence we may apply Lebesgue's Dominated Convergence Theorem to (2.4) and (2.6) to get $$\int_\R\varphi(\lambda)N_p(d\lambda)=\sum_{k=0}^N\frac{p^k}{k!}\left(\frac i{2\pi}\int_\C f_k(z)\frac{\partial\tilde\varphi}{\partial\overline z}(z)d\overline z\wedge dz\right)+p^{N+1}\left(\frac i{2\pi}\int_\C G_N(z,p)\frac{\partial\tilde\varphi}{\partial\overline z}(z)d\overline z\wedge dz\right) $$ and, for $k\geq 1$, using Lebesgue's Dominated Convergence Theorem and (2.2), $$\aligned &\frac i{2\pi}\int_\C f_k(z)\frac{\partial\tilde\varphi}{\partial\overline z}(z)d\overline z\wedge dz \\ &=\sum\Sb \Lambda\subset\Zd\\ \#\Lambda=k\text{ and }0\in\Lambda\endSb\sum\Sb A\subset\Lambda\endSb(-1)^{\#(\Lambda\setminus A)}\left\{\frac i{2\pi}\int_\C \tr((z-H_{A})^{-1}H_{A}^{-q}-(z-H)^{-1}H^{-q})\frac{\partial\tilde\varphi}{\partial\overline z}(z)d\overline z\wedge dz\right\} \\ &=\sum\Sb \Lambda\subset\Zd \\ \#\Lambda=k\text{ and }0\in\Lambda\endSb\sum\Sb A\subset\Lambda\endSb(-1)^{\#(\Lambda\setminus A)}\tr\left(\frac i{2\pi}\int_\C(z-H_{A})^{-1}H_{A}^{-q}\frac{\partial\tilde\varphi}{\partial\overline z}(z)d\overline z\wedge dz \right.\\ &\hskip 7cm \left.-\frac i{2\pi}\int_\C(z-H)^{-1}H^{-q}\frac{\partial\tilde\varphi}{\partial\overline z}(z)d\overline z\wedge dz\right) \\ &=\sum\Sb \Lambda\subset\Zd\\ \#\Lambda=k\text{ and }0\in\Lambda\endSb\sum\Sb A\subset\Lambda\endSb(-1)^{\#(\Lambda\setminus A)}\tr(\varphi(H_A)-\varphi(H)). \endaligned \tag 2.10$$ Moreover, using estimate (2.8) and estimate (d) for analytic extensions, we see that the distribution $n_k$ defined by $\dsize\varphi\mapsto\langle n_k,\varphi\rangle=\frac i{2\pi}\int_\C f_k(z)\frac{\partial\tilde\varphi}{\partial\overline z}(z)d\overline z\wedge dz$ is of order at most $(k+1)(d+1)+2$. \par By (2.8) and estimate (d) for analytic extensions, $\dsize\varphi\mapsto\frac i{2\pi}\int_\C G_N(z,p)\frac{\partial\tilde\varphi}{\partial\overline z}(z)d\overline z\wedge dz$ defines a distribution in $\Sp$ (of order at most $(N+2)(d+1)+2$) that satisfies the following estimate: there exists $C_N>0$ such that, for $\varphi\in\S$, $$\left\vert\frac i{2\pi}\int_\C G_N(z,p)\frac{\partial\tilde\varphi}{\partial\overline z}(z)d\overline z\wedge dz\right\vert\leq C_{N,\alpha,\beta}\sup\Sb \alpha'\leq (N+2)(d+1)+q \\ \beta'\leq (N+2)(d+1)+2 \endSb\sup_{x\in\R}\left\vert x^{\alpha'}\frac{\partial^{\beta'}\varphi}{\partial x^{\beta'}}(x)\right\vert. \tag 2.11$$ Hence, we proved points (a) and (b) of Theorem 1.1. By the definition of the spectral shift function, (2.10) gives us the formula for $n_k$ for $k\geq 1$. For $k=0$, by (2.7.a), we know that $f_0(z)=\tr(\chi_0(z-H)^{-1}H^{-q}\chi_0)$, so, using the definition of $n_0(d\lambda)$, the density of states for $H$ (see e.g. \cite{Re-Si} or \cite{Sj}), we get $$\frac i{2\pi}\int_\C f_0(z)\frac{\partial\tilde\varphi}{\partial\overline z}(z)d\overline z\wedge dz=\int_{\R} \varphi(\lambda)n_0(d\lambda).$$ This ends the proof of Theorem 1.1. \smallskip \subhead c) Proof of Proposition 1.2, Theorems 1.3 and 1.4 \endsubhead \smallskip \demo{Proof of Proposition 1.2} Assume $(\lambda_n)_{n\geq 0}$ is a sequence in ${\Cal E}_k$ converging to $\lambda_\infty$. We may assume that $\lambda_n$ stays in $\overline G$, the closure of a fixed gap $G$ of $H$. We also may assume that $\overline G$ is compact as the operators we consider are uniformly semi-bounded. Then for any $n\geq 0$, there exists $\Lambda_n\subset\Zd$ such that $\#\Lambda_n=k$ and $\lambda_n$ is an eigenvalue of $H_{\Lambda_n}$ (in $G$). Define $\Lambda_n:=\{\gamma_n^1,\dots,\gamma_n^k\}$. \par Either $\lambda_\infty$ is the edge of $G$ in which case the result is proved. If not so, we may assume that $\lambda_n$ stays in a compact interval $I$ contained in the interior of $G$. \par We define the following equivalence relation on the set $\{1,\dots,k\}$: $j\sim l$ if $\gamma_n^j-\gamma_n^l$ stays bounded when $n\to+\infty$. This relation induces a partition $\{1,\dots,k\}=\cup_{1\leq a\leq b}U^a$. If we define $U_n^a=\{\gamma_n^j; j\in U^a\}$, then, when $n\to+\infty$, dist$(U_n^a,U_n^{a'})\to+\infty$ and, for any $j\in U^a$, $U_n^a-\gamma_n^j$ stays bounded. Using the fact that $H$ is translation invariant and that the eigenfunctions associated to eigenvalues of $H_{U^a_n}$ in $I$ are localized near $U_n^a$ (as $I$ is a compact set contained in the interior of the resolvent set of $H$), we get, that $$\text{dist}\left(\sigma(H_{\Lambda_n})\cap G,\bigcup_{1\leq a\leq b}\sigma(H_{U_n^a})\cap G\right)\to0\text{ when }n\to+\infty.$$ So, extracting a subsequence from $(\lambda_n)_{n\geq 0}$, we may assume that, for some $a$, dist$(\lambda_n,\sigma(H_{U_n^a}))\to0$ when $n\to+\infty$. By construction, the sets $(U_n^a)_{n\geq 0}$ are contained in a bounded set (modulo translation). So $\cup_{n\geq 0}\sigma(H_{U_n^a})\cap I$ contains at most a finite number of points. Hence $\lambda_n$ must converge to one of these points, which is an eigenvalue for some $H_{U_n^a}$. Hence $\lambda_n$ converges to some point in ${\Cal E}_j$ for $j0$ and $I'$ and $I''$ two open intervals such that, $I'+(-\delta,\delta)\subset I$, $I+(-\delta,\delta)\subset I''$, $I''\cap {\Cal E}_k=I'\cap {\Cal E}_k=I\cap {\Cal E}_k$ and $\overline{I''}\cap\overline{{\Cal E}_{k-1}}=\overline{I'}\cap\overline{{\Cal E}_{k-1}}=\emptyset$. Let $\chi_I$, $\chi_{I'}$ and $\chi_{I''}$ be respectively the characteristic functions of $I$, $I'$ and $I''$. Pick $0\leq\psi\in{\Cal C}_0^\infty(\R)$ such that $\psi(x)=1$ for $\vert x\vert<1/2$, $\psi(x)=0$ for $\vert x\vert>1$ and $\dsize \int_\R\psi(x)dx=1$. For $\epsilon>0$, set $\dsize\psi_\epsilon(x)=\frac1\epsilon\psi(\frac x\epsilon)$ and $\chi_\epsilon=\chi_I*\psi_\epsilon$, $\chi'_\epsilon=\chi_{I'}*\psi_\epsilon$ and $\chi''_\epsilon=\chi_{I''}*\psi_\epsilon$. Then $0\leq\chi'_\epsilon\leq\chi_I\leq\chi''_\epsilon$ for $0<\epsilon<\delta$. Moreover, we choose $\epsilon>0$ small enough such that supp$(\chi_\epsilon)\cap\overline{{\Cal E}_{k-1}}=\emptyset$, supp$(\chi_\epsilon)\cap{\Cal E}_k=I\cap{\Cal E}_k$, for $\lambda\in I\cap{\Cal E}_k$, $\chi_\epsilon(\lambda)=1$, and such that the same properties hold for $\chi'_\epsilon$ and $\chi''_\epsilon$. \par As $N_p(d\lambda)$ is a positive measure $$\langle N_p,\chi'_\epsilon\rangle\leq N_p(I)\leq\langle N_p,\chi''_\epsilon\rangle.$$ Let us compute the left and right hand side of this inequality. By Theorem 1.1, $$\langle N_p(d\lambda),\chi'_\epsilon\rangle=\sum_{j=0}^k p^j\langle n_j,\chi'_\epsilon\rangle+p^{k+1}\langle R_k(p),\chi'_\epsilon\rangle,\tag 2.12$$ where, for $\vert\cdot\vert_k$, some semi-norm in $\S$, we have for any $\varphi\in\S$, $$\sup_{p\in[0,1]}\vert\langle R_k(p),\varphi\rangle\vert\leq\vert\varphi\vert_k.\tag 2.13$$ As $\chi'_\epsilon$ is supported in a gap of $\sigma(H)$, we know that (cf \cite{Ya}) for $A\in\Zd$, $A$ finite, $$\langle\zeta'(\lambda;A),\chi'_\epsilon\rangle=-\sum_{\lambda\in\sigma(H_A)\cap\text{supp}(\chi_\epsilon)}\text{rank}(\Pi_\Lambda(\lambda))\cdot\chi'_\epsilon(\lambda).$$ Hence as supp$\chi'_\epsilon\cap\overline{{\Cal E}_{k-1}}=\emptyset$ and supp$(\chi'_\epsilon)\cap{\Cal E}_k=I\cap{\Cal E}_k$, by formula (d) of Theorem 1.1, we get $\langle n_j,\chi'_\epsilon\rangle=0$ for $0\leq j\leq k-1$, and $$ \langle n_k,\chi'_\epsilon\rangle=\frac1{k!}\sum\Sb \Lambda\subset\Zd \\ \#\Lambda=k\text{ and }0\in\Lambda \\ \sigma(H_\Lambda)\cap I\not=\emptyset\endSb\left(\sum_{\lambda\in\sigma(H_\Lambda)\cap I}\text{rank}(\Pi_\Lambda(\lambda))\right)=\frac1{k!}{\frak n}_k(I). \tag 2.14$$ The same way one proves that $\langle n_j,\chi''_\epsilon\rangle=0$ for $0\leq j\leq k-1$ and $\dsize\langle n_k,\chi''_\epsilon\rangle=\frac{{\frak n}_k(I)}{k!}$. Plugging this and (2.14) into (2.12), and using (2.13), we get point point (a) of Theorem 1.3. Point (b) is obtained by noticing that under the new assumptions, all $(\langle n_j,\chi'_\epsilon\rangle)_{0\leq j\leq k}$ and $(\langle n_j,\chi''_\epsilon\rangle)_{0\leq j\leq k}$ vanish. \enddemo \demo{Proof of Theorem 1.4} Without loss of generality we may assume $\mu=0$. Take $\psi$ as in the proof of Theorem 1.3, and set $\psi_\delta(x)=\psi(x/\delta)$. Then, for any $n\geq 0$, $\sup_{x\in\R}\vert(\partial^n\psi_\delta)(x)\vert\leq C_n\delta^{-n}$. Obviously $0\leq\psi_{\delta}\leq\chi_{[-\delta,\delta]}\leq\psi_{2\delta}$, hence $$\langle N_p,\psi_\delta\rangle\leq N_p([-\delta,\delta])\leq\langle N_p,\psi_{2\delta}\rangle.$$ Then by the same computations (and with the same notations) as in the proof of Theorem 1.3, using (2.11) to estimate the rest, we get $$ \vert p^{-k}N_p([-\delta,\delta])-{\frak n}(\{0\}) \vert\leq p\cdot\delta^{-(k+2)(d+1)+2}.$$ Hence, if we choose $\delta=p^\epsilon$ pour $\dsize 0<\epsilon<\frac1{(k+2)(d+1)+2}$, we get Theorem 1.4. \enddemo \medskip \subhead III) Proof of Proposition 2.1 \endsubhead \medskip \par Let us first prove point (c) of Proposition 2.1. Obviously, one has, by (2.5), $$\vert a_{\Lambda,l}(z,p)\vert \leq \sup_{t\in[0,1]^{\Lambda}}\sup_{u\in\{0,1\}^{\Lambda_l\setminus\Lambda}}\vert\tr\left(\chi_0\partial_\Lambda\left[(z-H_{\Lambda}(t,u))^{-1}H_{\Lambda}(t,u)^{-q}\right]\chi_0\right)\vert.$$ We compute $$\aligned &\partial_\Lambda\left[(z-H_{\Lambda}(t,u))^{-1}H_{\Lambda}(t,u)^{-q}\right]=\bigotimes_{\gamma\in\Lambda}\partial_{t_\gamma}\left[(z-H_{\Lambda}(t,u))^{-1}H_{\Lambda}(t,u)^{-q}\right] \\ &\qquad =\sum \Sb \bigcup_{j=1}^{q+1}A_j=\Lambda \\ A_j\cap A_l=\emptyset \text{ if }j\not=l\endSb \left(\sum_{\sigma_1\in\frak S(A_1)}(z-H_{\Lambda}(t,u))^{-1} \left(\prod_{\gamma_1\in A_1}V_{\sigma_1(\gamma_1)}(z-H_{\Lambda}(t,u))^{-1}\right)\right.\cdot \\ &\hskip5cm \cdot\prod_{j=2}^{q+1}\left(\sum_{\sigma_j\in\frak S(A_j)}H_{\Lambda}(t,u)^{-1} \left(\prod_{\gamma_j\in A_j}V_{\sigma_j(\gamma_j)}H_{\Lambda}(t,u)^{-1}\right)\right), \endaligned \tag 3.1$$ where $\frak S(A_j)$ is the group of permutations of $A_j$. \par\noindent Hence, using the estimates of Proposition 4.2 and the exponential decrease of $V$ to control the trace-class norm of the right hand side of (3.1), we get, for some $C>0$ and for any $z\in\C\setminus\R$, $$\multline \sup\Sb p\in[0,1] \\ l>0 \endSb \vert a_{\Lambda,l}(z,p) \vert \leq \\ \leq C^{q+1}\sum \Sb \bigcup_{j=1}^{q+1}A_j=\Lambda \\ A_j\cap A_l=\emptyset \text{ if }j\not=l\endSb\left( \sum\Sb \sigma_1\in\frak S(A_1),\dots \\ \dots,\sigma_{q+1}\in\frak S(A_{q+1})\endSb\left(1+\frac1{\eta(z)}\right)^{\#A_1}e^{-\text{diam}(z;A_1,\dots,A_{q+1};\sigma_1,\dots,\sigma_{q+1})}\right) \endmultline\tag 3.2$$ where $\eta(z)$ is given in Proposition 4.3 and diam$(z;A_1,\dots,A_{q+1};\sigma_1,\dots,\sigma_{q+1})$ is defined in the following way: if for $1\leq j\leq q+1$, we write $A_j=\{\gamma_1^j,\dots,\gamma_{m_j}^j\}$ (i.e $\#A_j=m_j$) then $$\text{diam}(z;A_1,\dots,A_{q+1};\sigma_1,\dots,\sigma_{q+1})=\eta(z)\sum_{l=0}^{m_1}\vert \sigma_1(\gamma_l^1)-\sigma_1(\gamma_{l+1}^1)\vert+\sum_{j=2}^{q+1}\frac1C\sum_{l=0}^{m_j}\vert \sigma_j(\gamma_l^j)-\sigma_j(\gamma_{l+1}^j)\vert,$$ with (a) $\sigma_j(\gamma_{m_j+1}^j)=\sigma_{j+1}(\gamma_1^{j+1})$ for $1\leq j\leq q$, \par\quad (b) $\sigma_{q+1}(\gamma_{m_{q+1}+1}^{q+1})=0$ and $\sigma_0(\gamma_0^1)=0$. \par\noindent Hence we get $$\aligned\text{diam}(z;A_1,\dots,A_{q+1};\sigma_1,\dots,\sigma_{q+1})&\geq\frac{\inf(\eta(z),1/C)}k\cdot\sum_{1\leq j\leq q+1}\sum_{l=1}^{m_j}\vert\sigma_j(\gamma_l^j)\vert \\&=\frac{\inf(\eta(z),1/C)}k\sum_{\gamma\in\Lambda}\vert\gamma\vert.\endaligned \tag 3.3$$ Summing the estimates (3.2) and (3.3) over all possible $\gamma_1,\dots,\gamma_k$ in $\Zd$ and replacing $\eta(z)$ by an obvious lower bound, we get, for some $C>0$ (depending on $k$ but not on $z\in\C\setminus\R$) $$\sum\Sb \Lambda\subset\Zd \\ \#\Lambda=k\endSb \sup\Sb p\in[0,1] \\ l>0 \endSb\vert a_{\Lambda,l}(z,p)\vert \leq C \left(\frac{1+\vert z\vert}{\vert\text{Im}(z)\vert}\right)^{(k+1)(d+1)}.$$ This proves point (c) of Proposition 2.1. \par Let us now prove the point (a). It suffices to show that $(a_{\Lambda,l}(z,p))_{l\geq 0}$ is a Cauchy sequence of continuous functions for $(z,p)$ in $K\times[0,1]$. Let $l'>l>0$ large such that $\Lambda\subset\Lambda_l$. Then $$\multline \vert a_{\Lambda,l'}(z,p)-a_{\Lambda,l}(z,p)\vert\leq\int_{[0,1]^{\Lambda}}{\Bbb E}_p\left\{\vert\tr\left(\chi_0\partial_\Lambda\left[(z-H_{\Lambda,l'}(t,u))^{-1}H_{\Lambda,l'}(t,u)^{-q}-\right.\right.\right. \\ \left.\left.\left.-(z-H_{\Lambda,l}(t,u))^{-1}H_{\Lambda,l}(t,u)^{-q}\right]\chi_0\right)\vert\right\}dt_\Lambda \endmultline \tag 3.4$$ where $H_{\Lambda,l'}(t,u)=H+\sum_{\gamma\in \Lambda}t_\gamma V_\gamma+\sum_{\gamma\in(\Lambda_{l'}\setminus\Lambda)}u_\gamma V_\gamma$ and the random variables $u=(u_\gamma)_{\gamma\in\Zd}$ and the variables $t=(t_\gamma)_{\gamma\in\Lambda}$ are defined as in section 2. \par\noindent By Proposition 4.3, for any realization of $u$ and $t$, for $z\in\C\setminus\R$, $$\multline \Vert\chi_0\left[(z-H_{\Lambda,l'}(t,u))^{-1}H_{\Lambda,l'}(t,u)^{-q}-(z-H_{\Lambda,l}(t,u))^{-1}H_{\Lambda,l}(t,)^{-q}\right]\chi_0\Vert_{{\Cal T}_1} \\ \leq C\left(1+\frac1{\eta(z)}\right)^2e^{-\epsilon\inf(\eta,\eta(z))\cdot\vert l\vert}. \endmultline$$ It is clear that the arguments given in section 4 stay valid for $z$ in $K$, some compact of $\C\setminus\R$ and the perturbations $V$ having a small imaginary part (the size depending on $K$). Hence as $(z-H_{\Lambda,l'}(t,u))^{-1}H_{\Lambda,l'}(t,u)^{-q}-(z-H_{\Lambda,l}(t,u))^{-1}H_{\Lambda,l}(t,u)^{-q}$ is analytic in $t$ (uniformly for $z\in K$ and $\vert t\vert$ small enough), we get, using a Cauchy estimate, for some $C>0$ (depending on $K$ and $\eta$), for $z\in K$ and $p\in[0,1]$, $$\Vert\partial_\Lambda\left(\chi_0\left[(z-H_{\Lambda,l'}(t,u))^{-1}H_{\Lambda,l'}(t,u)^{-q}-(z-H_{\Lambda,l}(t,u))^{-1}H_{\Lambda,l}(t,u)^{-q}\right]\chi_0\right)\Vert_{{\Cal T}_1}\leq Ce^{-\vert l\vert/C}.$$ This gives then by (3.4), for some $C>0$, for $z\in K$ and $p\in[0,1]$, $$\vert a_{\Lambda,l}(z,p)-a_{\Lambda,l'}(z,p)\vert\leq Ce^{-\vert l\vert/C}.$$ Hence this ends the proof of point (a) of Proposition 2.1. The formula for $a_{\Lambda}(z,0)$ is obvious taking $p=0$. \medskip \subhead IV) Some estimates for the resolvent of $\bold{H(t)}$ \endsubhead \medskip Let $\chi_0$ be the characteristic function of the cube of center 0 and sidelength 1 in $\Rd$. Let $\chi_\alpha$ be its translated by the vector $\alpha$ i.e $\chi_\alpha(x)=\chi_0(x-\alpha)$. Then we have \proclaim{Proposition 4.1} For $q>d/2$, there exists $\epsilon>0$ such that, for any $V\in L^\infty(\Rd)$ and for any $z\not\in\sigma(-\Delta+V)$, for any $(\alpha,\beta)\in\Zd\times\Zd$, $$\multline \Vert\chi_\alpha\cdot(z-(-\Delta+V))^{-1}\cdot\chi_\beta\Vert_{{\Cal T}_q}\leq C\left(1+\frac1{\eta(z,V)}\right)e^{-\epsilon\eta(z,V)\cdot\vert\alpha-\beta\vert} \\ \text{ where }\eta(z,V)=\frac{\text{dist}(z,\sigma(-\Delta+V))}{\vert z\vert+\vert V\vert_\infty+1},\endmultline$$ here $\Vert\cdot\Vert_{{\Cal T}_q}$ denotes the norm in the $q$-th Schatten class, $\vert V\vert_\infty$, the supremum norm of $V$ and dist$(z,z')$ denotes the distance in $\C$. \endproclaim \demo{Proof} By \cite{Si} section B.9, we know that $\chi_\alpha\cdot(z-(-\Delta+V))^{-1}\cdot\chi_\beta\in{\Cal T}_q$. To prove the estimate on its norm we will use the idea used in \cite{Co-Th} or \cite{Si}. Let $H=-\Delta+V$, and for $a\in\Rd$, define $$H_a=e^{a\cdot x}He^{-a\cdot x}=(i\nabla-ia)^2+V.$$ Then, for $z\not\in\sigma(H)\cup\sigma(H_a)$, one has $$\chi_\alpha(z-H)^{-1}\chi_\beta=\left(e^{-ax}\chi_\alpha\right)\chi_\alpha(z-H_a)^{-1}\chi_\beta\left(e^{ax}\chi_\beta\right).\tag 4.1$$ Expanding $H_a$, we get $$(z-H_a)=z-H+\vert a\vert^2-2a\cdot\nabla=(z-H)(1+(z-H)^{-1}(-2a\cdot\nabla+\vert a\vert^2)).$$ We now estimate the last term of this product; first, for some $C>0$, independent of $z$ and $V$, $$\Vert (z-H)^{-1}\nabla\Vert=\Vert (\Delta-1)^{-1}\nabla-(z-H)^{-1}(z+1-V)(\Delta-1)^{-1}\nabla\Vert\leq C\left(1+\frac{\vert z\vert+\vert V\vert_\infty+1}{\text{dist}(z,\sigma(-\Delta+V))}\right)$$ (here $\Vert\cdot\Vert$ denotes the norm of bounded operators). \par Now, choose $a$ such that $\dsize \vert a\vert=\epsilon\left(1+\frac{\text{dist}(z,\sigma(-\Delta+V))}{\vert z\vert+\vert V\vert_\infty+1}\right)$ and $\epsilon>0$ such that $8\epsilon(C+1)<1$ then $$\Vert(z-H)^{-1}(-2a\cdot\nabla+\vert a\vert^2)\Vert<\frac34$$ hence $(z-H_a)$ is invertible and for some $C>0$, $$\aligned\Vert\chi_\alpha\cdot(z-H_a)^{-1}\cdot\chi_\beta\Vert_{{\Cal T}_q}&\leq 4\Vert (z-H)^{-1}\cdot\chi_\beta\Vert_{{\Cal T}_q}\\&\leq 4\Vert (\Delta-1)^{-1}\chi_\beta-(z-H)^{-1}(z+1-V)(\Delta-1)^{-1}\chi_\beta\Vert_{{\Cal T}_q} \\ &\leq C(1+ \Vert (z-H)^{-1}(z+1-V)\Vert).\endaligned$$ Hence, by (4.1) and \cite{Si} section B.9, we get, for some $C>0$, $$\Vert\chi_\alpha\cdot(z-(-\Delta+V))^{-1}\cdot\chi_\beta\Vert_{{\Cal T}_q}\leq C\left(1+\frac{\vert z\vert+\vert V\vert_\infty+1}{\text{dist}(z,\sigma(-\Delta+V))}\right)e^{a\cdot(\alpha-\beta)},$$ which becomes the estimate given in Proposition 4.1 if we choose $\dsize a=\frac{\vert a\vert}{\vert\alpha-\beta\vert}(\alpha-\beta)$. \enddemo We now can apply Proposition 4.1 to estimate powers of the resolvent and get \proclaim{Proposition 4.2} Let $p$, $q$ be integers such that $p\cdot q>d/2$. Then, there exists $\epsilon>0$ and $C_{p,q}>0$, such that for any $V\in L^\infty(\Rd)$ and for any $z\not\in\sigma(-\Delta+V)$, for any $(\alpha,\beta)\in\Zd\times\Zd$, $\chi_\alpha\cdot(z-(-\Delta+V))^{-p}\cdot\chi_\beta\in{\Cal T}_q$ and $$\Vert\chi_\alpha\cdot(z-(-\Delta+V))^{-p}\cdot\chi_\beta\Vert_{{\Cal T}_q}\leq C_{p,q}\left(1+\frac1{\eta(z,V)}\right)^{p(d+1)}e^{-\epsilon\eta(z,V)\cdot\vert\alpha-\beta\vert}$$ where $\eta(z,V)$ is given in Proposition 4.1. \endproclaim \demo{Proof} One writes $$\multline \chi_\alpha\cdot(z-H)^{-p}\cdot\chi_\beta=\sum\Sb\alpha_1,\dots,\alpha_{p-1} \\ \alpha_j\in\Zd\endSb \left(\chi_\alpha\cdot(z-H)^{-1}\cdot\chi_{\alpha_1}\right)\cdot\left(\chi_{\alpha_1}\cdot(z-H)^{-1}\cdots\right. \\ \left.\cdots\chi_{\alpha_{p-1}}\right)\left(\chi_{\alpha_{p-1}}\cdot(z-H)^{-1}\cdot\chi_\beta\right),\endmultline$$ and uses the norm estimates given in Proposition 4.1 and the product rule for elements in ${\Cal T}_{p\cdot q}$. \enddemo We apply this to the realization of the random Schr\"odinger operator we are studying to get \proclaim{Proposition 4.3} (1) Let $q>d/2$. There exists $C>0$ such that, for any realization of $t$, any $z\in\C\setminus\R$ and any $(\alpha,\beta)\in\Zd\times\Zd$, $$\Vert\chi_\alpha(z-H(t))^{-1}H(t)^{-q}\chi_\beta\Vert_{{\Cal T}_1}\leq C\left(1+\frac1{\eta(z)}\right)^2e^{-\epsilon\eta(z)\cdot\vert\alpha-\beta\vert}.$$ \par\noindent (2) Let $H_l(t)=H+\sum_{\gamma\in\Zd\cap\Lambda_l}t_\gamma V_\gamma$. Let $q>d/2$. There exists $C>0$ such that, for any realization of $t$, any $z\in\C\setminus\R$ and any $l>0$, $$\Vert\chi_0(z-H(t))^{-1}H(t)^{-q}\chi_0-\chi_0(z-H_l(t))^{-1}H_l(t)^{-q}\chi_0\Vert_{{\Cal T}_1}\leq C\left(1+\frac1{\eta(z)}\right)^2e^{-\epsilon\inf(\eta,\eta(z))\cdot\vert l\vert}.$$ Here \eta$ is the parameter controlling the exponential decrease of $V$ and $\eta(z)$ is chosen as the infimum of $\eta(z,V(t))$ (given in Proposition 4.1) when $t$ runs over all possible configurations. Notice that $\eta(z)>0$ for any $z\in\C\setminus\R$. \endproclaim \demo{Proof} Part (1) is clear by Proposition 4.2. To get part (2), we write $$\aligned (z-H(t))^{-1}H(t)^{-q}&-(z-H_l(t))^{-1}H_l(t)^{-q} \\ &=((z-H(t))^{-1}-(z-H_l(t))^{-1})H(t)^{-q}+ \\ &\hskip 5cm +(z-H_l(t))^{-1}(H(t)^{-q}-H_l(t)^{-q}) \\ &=(z-H(t))^{-1}(H(t)-H_l(t))(z-H_l(t))^{-1}H(t)^{-q}+ \\ &\hskip 3cm +(z-H_l(t))^{-1}\sum_{k=1}^q H_l(t)^{k-q}(H(t)-H_l(t))H(t)^{-k}.\endaligned \tag 4.2$$ Then, using the exponential decrease for $V$ and the boundedness of $t$, we get that $$\dsize\Vert H(t)-H_l(t)\Vert\leq C e^{-\eta(d(\alpha,\Zd\setminus\Lambda_l)+d(\beta,\Zd\setminus\Lambda_l))}.$$ Plugging this into (4.2) and using Proposition 4.2, we get the announced estimate. \enddemo \Refs\nofrills{References} \widestnumber\key{Ca-Kl-Ma} \ref \key Ca-Kl-Ma \by R. Carmona, A. Klein, F. Martinelli \paper Anderson localization for Bernoulli and other random potentials \yr 1987 \jour Commun. Math. Phys. \vol 108 \pages 41--67 \endref \ref \key Ca-La \by R. Carmona, J. 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