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. Lacroix \book Spectral Theory of Random Schr\"odinger Operators \yr 1990 \publ Birkh\"auser \publaddr Boston Basel Berlin \endref \ref \key Co-Th \by J. M. Combes, L. Thomas \paper Asymptotic behavior of eigenfunctions for multi-particle Schr\"odinger operators \yr 1973 \jour Commun. Math. Phys. \vol 34 \pages 251--270 \endref \ref \key H-Ki \by R. Hempel, W. Kirsch \paper On the Integrated Density of States for Crystals with Randomly Distributed Impurities \yr 1994 \jour Commun. Math. Phys. \vol 159 \pages 459--469 \endref \ref \key He-Sj \by B. Helffer, J. Sj\"ostrand \paper On Diamagnetism and the De Haas-Van Alphen effect \yr 1990 \jour Ann. Inst. Henri Poincar\'e s\'er. Phys. Th\'eor. \vol 52 \page 303 \endref \ref \key Ki-Ma \by W. Kirsch, F. Martinelli \paper On the Spectrum of Schr\"odinger Operators with a Random Potential\yr 1982 \jour Commun. Math. Phys. \vol 85 \pages 329--350 \endref \ref \key Mat \by J. N. Mather \paper On Nirenberg's proof of Malgrange's Preparation Theorem \inbook Proceedings of Liverpool Singu-larities-Symposium I \bookinfo Lect. Notes Math. 192 \yr 1971 \publ Springer \publaddr Heidelberg New-York Berlin \endref \ref \key Ni-Lu \by Th. M. Nieuwenhuizen, J. M. Luck \paper Singular Behavior of the Density of States and the Lyapunov Coefficient in Binary Random Harmonic Chains \jour Jour. of Stat. Phys. \vol 41 \yr 1985 \pages 745--771 \endref \ref \key Pa-Fi \by L. Pastur, A. Figotin \book Spectra of Random and Almost-Periodic Operators \yr 1992 \publ Springer \publaddr Berlin Heidelberg New-York \endref \ref \key Re-Si \by M. Reed, B. Simon \book Methods of Modern Mathematical Physics, Vol IV: Analysis of Operators \yr 1978 \publ Academic Press \publaddr New-York \endref \ref \key Si \by B. Simon \paper Schr\"odinger Semigroups \yr 1982 \jour Bull. Am. Math. Soc. \vol 7 \pages 447--526 \endref \ref \key Si-Ta \by B. Simon, M. Taylor \paper Harmonic Analysis on $SL(2,\R)$ and Smoothness of the Density of States in the One-Dimensional Anderson Model \yr 1985 \jour Commun. Math. Phys. \vol 101 \pages 1--20 \endref \ref \key Sj \by J. Sj{\"o}strand \paper Microlocal analysis for periodic magnetic Schr{\"o}dinger equation and related questions \inbook Microlocal analysis and applications \bookinfo Lect. Notes Math. 1495 \yr 1989 \publ Springer \publaddr Heidelberg New-York Berlin \endref \ref \key Ya \by D. R. Yafaev \book Mathematical Scattering Theory \bookinfo Trans. of Mathematical Monographs \vol 105 \yr 1992 \publ A.M.S \publaddr Providence, Rhode Island \endref \endRefs \enddocument ENDBODY