\magnification=1200 \overfullrule=0pt \def\l{\ell} \def\parno{\par \noindent} \def\ref#1{\lbrack {#1}\rbrack} \def\leq#1#2{$${#2}\leqno(#1)$$} \def\vvekv#1#2#3{$$\leqalignno{&{#2}&({#1})\cr &{#3}\cr}$$} \def\vvvekv#1#2#3#4{$$\leqalignno{&{#2}&({#1})\cr &{#3}\cr &{#4}\cr}$$} \def\vvvvekv#1#2#3#4#5{$$\leqalignno{&{#2}&({#1})\cr &{#3}\cr &{#4}\cr &{#5}\cr}$$} \def\ekv#1#2{$${#2}\eqno(#1)$$} \def\eekv#1#2#3{$$\eqalignno{&{#2}&({#1})\cr &{#3}\cr}$$} \def\eeekv#1#2#3#4{$$\eqalignno{&{#2}&({#1})\cr &{#3}\cr &{#4}\cr}$$} \def\eeeekv#1#2#3#4#5{$$\eqalignno{&{#2}&({#1})\cr &{#3}\cr &{#4}\cr &{#5}\cr}$$} \def\eeeeekv#1#2#3#4#5#6{$$\eqalignno{&{#2}&({#1})\cr &{#3}\cr &{#4}\cr &{#5}\cr &{#6}\cr}$$} \def\eeeeeekv#1#2#3#4#5#6#7{$$\eqalignno{&{#2}&({#1})\cr &{#3}\cr &{#4}\cr &{#5}\cr&{#6}\cr&{#7}\cr}$$} \def\iint{\int\hskip -2mm\int} \def\iiint{\int\hskip -2mm\int\hskip -2mm\int} \def\buildover#1#2{\buildrel#1\over#2} \font \mittel=cmbx10 scaled \magstep1 \font \gross=cmbx10 scaled \magstep2 \font \klo=cmsl8 \font\liten=cmr10 at 8pt \font\stor=cmr10 at 12pt \font\Stor=cmbx10 at 14pt \centerline{\stor Exponential decay of averaged Green functions for} \centerline{\stor random Schr\"odinger operators, a direct approach .} \medskip \centerline{ J. Sj\"ostrand\footnote{*}{Centre de Math\'ematiques, Ecole Polytechnique, F-91128 Palaiseau cedex, France and URA 169, CNRS} and W.-M. Wang\footnote{$^\#$}{D\'ept. de Math\'ematiques, Universit\'e de Paris Sud, F-91405 Orsay cedex, France and URA 760, CNRS}} \vskip 1cm \par\noindent \it R\'esum\'e. \liten Sous des hypoth\`eses d'analyticit\'e convenables sur la densit\'e de probabilit\'e, nous \'etudions l'esp\'erence de la fonction de Green. Nous donnons des r\'esultats pr\'ecis sur des domaines d'extension holomorphe en \'energie et sur la d\'ecroissance exponentielle. L'ingr\'edient principal (comme dans [SW]) est la construction d'une m\'esure de probabilit\'e dans le domaine complexe apr\`es d\'eformation de contour. Ceci nous permet d'\'eviter d'utiliser des s\'eries de perturbation. Compar\'ee a la m\'ethode de [SW], la variante ici semble limit\'ee \`a l'\'equation de Schr\"odinger al\'eatoire, dans quel cas elle permet cependant de traiter des distributions de probabilit\'e plus g\'en\'erales. \bigskip \par\noindent \it Abstract. \liten Under suitable analyticity conditions on the probability distribution, we study the expectation of the Green function. We give precise results about domains of holomorphic extensions in energy and exponential decay. The key ingredient (as in [SW]) is the construction of a probability measure in the complex domain after contour deformation. This permits us to avoid the use of perturbation series. Compared to the method in [SW], the variant here seems limited to the random Schr\"odinger equation, in which case however it permits to treat more general probability distributions \bigskip \par\noindent \it Acknowledgements. \liten We thank L. Pastur for suggesting us to add a result (Theorem 0.1') about the expectation of moments of the Green function, as well as J.Chaumat for indicating the reference [B]. This work was supported by TMR programme FMRX-CT 960001 of the European Commission- Network Postdoctoral training programme in partial differential equations and applications in quantum mechanics. \rm \vfill \eject \centerline{\bf 0. Introduction.} \medskip \par The purpose of this work is to present a variant of the method in [SW], which gives a more direct approach and which permits to treat more general probability distributions; not only perturbations of the Cauchy distributions but also for instance Gaussian ones. The main idea of the proof is the same as in [SW], namely to replace a certain complex density by a probability measure, but here we exploit in a more essential way some special structure in the problem and avoid the use of Fourier transform. On the other hand, we feel that the method of [SW] is of a more general nature and is likely to have applications to analyticity problems in statistical mechanics. Though the results below are more general (in the random Schr\"odinger case) they permit to recover only a slightly weakened version of the main result in [SW]. \par Let $\Delta $ be the discrete Schr\"odinger operator on $\ell ^2({\bf Z}^d)$, defined by \ekv{0.1} {\Delta u(\nu )=\sum_{\vert \mu -\nu \vert _1=1}u(\mu ),\,\,\vert \cdot \vert _1=\vert \cdot \vert _{\ell^1}.} When $\Lambda \subset {\bf Z}^d$ is a finite subset, we put \ekv{0.2} {\Delta _\Lambda =r _\Lambda \Delta r_\Lambda ^*,} where $r_\Lambda :\ell^2({\bf Z}^d)\to \ell^2(\Lambda )$ is the restriction operator, so that the adjoint $r _\Lambda ^*=\ell^2(\Lambda )\to \ell^2({\bf Z}^d)$ is the operator of extension by $0$: $(r_\Lambda ^*u)(\nu )=u(\nu )$, when $\nu \in \Lambda $, $=0$, when $\nu \in{\bf Z}^d\setminus\Lambda $. \par If $g(v)dv$ is a probability measure on ${\bf R}$ and $t>0$, we are interested in the expectation value of the Green function \ekv{0.3} {G_\Lambda (E)=(t\Delta_\Lambda +{\rm diag\,}(v_j)-E)^{-1},\,\, E\not\in \sigma (t\Delta _\Lambda +{\rm diag\,}(v_j)),} given by \ekv{0.4} {\langle G_\Lambda (E)(\mu ,\nu )\rangle _g=\int (t\Delta_\Lambda +{\rm diag\,}(v_j)-E)^{-1}(\mu ,\nu )\prod_{j\in\Lambda }(g(v_j)dv_j),} first for ${\rm Im\,}E>0$, and then wherever this expression can be extended holomorphically w.r.t. $E$. \par Let $K\subset {\bf C}$ be compact, symmetric around ${\bf R}$ and assume: \smallskip \par\noindent (H1) \sl $g$ extends to a holomorphic function on ${\bf C}\setminus K$ (that we also denote by $g$), which satisfies $$\vert g(z)\vert \le C(1+\vert z\vert )^{-2},\,\, \vert z\vert \ge C,$$ for some $C>0$ with $K\subset D(0,C)$.\rm \smallskip \noindent Here $D(0,C)$ denotes the open disc of center $0$ and radius $C$. \par Let $K_-=\{ z\in K;\,{\rm Im\,}z\le 0\}$, and let ${\rm ch\,}(K_-)$ denote the convex hull of $K_-$. Our second assumption will need some further discussion: \smallskip \par\noindent (H2) \sl For every simple closed smooth negatively oriented curve $\gamma $ in ${\bf C}_-=\{ z\in{\bf C};\,{\rm Im}z\le 0\}$ (with non-vanishing derivative) which is real in a neighborhood of all real points of ${\rm ch\,}(K_-)$ and with $\{ z\in{\rm ch\,}(K_-); {\rm Im\,}z<0\} \subset {\rm int\,}(\gamma )$, there exists a probability measure $\mu _\gamma (dz)$ supported in $\gamma $, such that $$\int_\gamma f(z)g(z)dz=\int_\gamma f(z)\mu_\gamma (dz),$$ for all functions $f$ which are holomorphic in ${\rm int\,}(\gamma )$ and smooth in $\overline{{\rm int\,}(\gamma )}$.\rm \smallskip \par\noindent Here ${\rm int }(\gamma )$ denotes the bounded open set which has $\gamma $ as its (smooth) boundary. \par We notice that (H2) is equivalent to the apparently weaker assumption that for every neighborhood $V$ of ${\rm ch\,}(K_-)$ there exists a closed contour $\gamma =\gamma _V$ contained in $V$ and a probability measure $\mu _\gamma $ with support in $\gamma $, with the properties described in (H2). In fact, if $\gamma $ is a simple smooth closed loop (with non-vanishing derivative), let $P_\gamma :C(\gamma )\to C(\overline{{\rm int\,}(\gamma )})$ be the Poisson operator determined by ${P_\gamma u_\vert }_\gamma =u$, $P_\gamma u$ harmonic in ${\rm int\,}(\gamma )$. Since $P_\gamma $ is positivity preserving, the adjoint $P_\gamma ^*$ maps a positive measure $\mu $ supported in $\overline{{\rm int\,}(\gamma )}$ to a positive measure $P_\gamma ^*\mu $, supported on $\gamma $, and if $f$ is holomorphic in ${\rm int\,}(\gamma )$, or more generally harmonic, and in $C(\overline{{\rm int\,}(\gamma )})$, then \ekv{0.5} {\int f(z)\mu (dz)=\int P_\gamma ({f_\vert }_\gamma ) \mu (dz)=\int {f_\vert}_\gamma (z)(P_\gamma ^*\mu )(dz) .} Let $\gamma $, $\widetilde{\gamma }$ be two curves with the geometric properties described in (H2) and such that $\gamma $ is contained in ${\rm int\,}\widetilde{\gamma }$ and carries a probability measure $\mu _\gamma $ with the properties in (H2). Then $P_{\widetilde{\gamma }}^*(\mu _\gamma )$ is carried by $\widetilde{\gamma }$ and has the properties of (H2), and we have proven the claim. \par For $\lambda >2d$, let \ekv{0.6} { W(\lambda )=\{ \eta \in{\bf R}^d;\,2\sum_1^d\cosh \eta _j<\lambda \} .} This is a strictly convex bounded open neighborhood of $0$ with smooth boundary, which is symmetric around $0$. Let \ekv{0.7} { p_\lambda (x)=\sup_{\eta \in W(\lambda )}x\cdot \eta, \,\, x\in{\bf R}^d} be the support function of $W(\lambda )$. $p_\lambda $ is smooth outside $x=0$, convex, even, positively homogeneous of degree 1, and $p_\lambda (x)>0$ for $x\ne 0$. Let $D(0,r)$ denote the open disc of center $0$ and radius $r$. A main result of this work is: \medskip \par\noindent \bf Theorem 0.1. \sl The expectation value (0.4) extends from the open upper half plane to a holomorphic function on ${\bf C}\setminus ({\rm ch\,}(K_-)+\overline{D(0,2td)})$, that we continue to denote by $\langle G_\Lambda (E)(\mu ,\nu )\rangle $. If $E\in {\bf C}\setminus({\rm ch\,}(K_-)+\overline{D(0,2td)})$, let ${\rm dist\,}(E,{\rm ch\,}(K_-))=\inf_{F\in{\rm ch\,}(K_-)}\vert E-F\vert $ be the distance from $E$ to ${\rm ch\,}(K_-)$, so that ${1\over t}{\rm dist\,}(E,{\rm ch\,}(K_-))>2d$. Then for every $\lambda \in ]2d,{1\over t}{\rm dist\,}(E,{\rm ch\,}(K_-))[$, we have \ekv{0.8} { \Vert e^{\eta \cdot (\cdot )}\langle G_\Lambda (E)\rangle e^{-\eta \cdot (\cdot )}\Vert _{{\cal L}(\ell^2,\ell^2)}\le {1\over {\rm dist\,}(E,{\rm ch\,}(K_-))-t\lambda }, } for every $\eta \in W(\lambda )$, and in particular, \ekv{0.9} { \vert \langle G_\Lambda (E)(\mu ,\nu )\rangle\vert \le {1\over {\rm dist\,}(E,{\rm ch\,}(K_-))-t\lambda }\, e^{-p_\lambda (\mu -\nu )}. } \rm \medskip \par The proof will also show that the limit of $\langle G_\Lambda (E)(\mu ,\nu )\rangle $ exists when $\Lambda \nearrow{\bf Z}^d$, for the complex values of $E$ in the theorem, and we then get (0.8), (0.9) also for the limit. \par The first step in the proof is to notice that if $\gamma $ is a curve as in (H2): \ekv{0.10} { \langle G_\Lambda (E)\rangle =\int_{\gamma ^\Lambda }(t\Delta _\Lambda +{\rm diag\,}(v_j)-E)^{-1}\prod_{j\in\Lambda }g(v_j)dv_j, } for ${\rm Im\,}E>0$. Then since $(t\Delta _\Lambda +{\rm diag\,}(v_j)-E)^{-1}$ depends holomorphically on $v=\{ v_j\} _{j\in\Lambda }$ in ${\Omega _E}^\Lambda $, where $\Omega _E$ is an $E$-dependent neighborhood of the closed lower half plane, we can apply Fubini's theorem and (H2), to get: \ekv{0.11} {\langle G_\Lambda (E)\rangle =\int_{\gamma ^\Lambda }(t\Delta +{\rm diag\,}(v_j)-E)^{-1}\prod_{j\in\Lambda }\mu _\gamma (dv_j). } The remainder of the proof is given in section 2, and consists in showing that $(t\Delta _\Lambda +{\rm diag\,}(v_j)-E)^{-1}$ exists when $v_j\in\gamma $, $\forall j\in\Lambda $, when $\gamma $ is as above and convex, and $E\in{\bf C}\setminus(\overline{{\rm int\,}(\gamma )}+\overline{D(0,2td)})$, and satisfies (0.8), (0.9), with $K_-$ replaced by $\overline{{\rm int\,}(\gamma ))}$. It is clear that the proof gives a more general result. For $N\in\{ 1,2,..\}$ and $E_k\in{\bf C}$ with ${\rm Im\,}E_k>0$, consider \ekv{0.4'}{\langle \prod_{k=1}^N(G_\Lambda (E_k)(\mu ,\nu ))\rangle _g=\int\prod_{k=1}^N((t\Delta_\Lambda +{\rm diag\,}(v_j)-E_k)^{-1}(\mu ,\nu ))\prod_{j\in\Lambda }(g(v_j)dv_j).} Then we have the following generalization of Theorem 0.1: \medskip \par\noindent \bf Theorem 0.1'. \sl The expectation value (0.4') extends to a holomorphic function on $({\bf C}\setminus ({\rm ch\,}(K_-)+\overline{D(0,2td)}))^N$, that we denote by the LHS of (0.4'). For $(E_1,..,E_N)$ in this product domain, let $\lambda _k\in ]2d,{1\over t}{\rm dist\,}(E_k,{\rm ch\,}K_-)[$. Then \ekv{0.9'}{\vert \langle \prod_{k=1}^N(G_\Lambda (E_k)(\mu ,\nu ))\rangle _g\vert \le \prod_{k=1}^N{e^{-p_{\lambda _k}(\mu -\nu )}\over ({\rm dist\,}(E_k,{\rm ch\,}(K_-))-t\lambda _k)}.} \rm\medskip \par\noindent \it Remark. \rm Using the fact that $g$ is holomorphic, to show that $\langle G_\Lambda (E)\rangle $ is holomorphic, has already been done before Constantinescu, Fr\"ohlich, Spencer [CFS]. Their proof uses the Neumann series of $(t\Delta +V-E)^{-1}$ for small $t$ and contour deformation in $v_j$ to show that the expectation value of the resulting series converges. However they did not try to show that the resulting measure on the new contour can be made positive. The method in [CFS] is effective when $g$ decays fast enough at infinity. The measures considered in the present paper do not necessarily have this property. \medskip \par The assumption (H2) is rather implicit and in order to have more applications, we shall prove in section 1 the following stability result: \medskip \par\noindent \bf Proposition 0.2. \sl Let $g_0(v)dv$ be a probability measure on ${\bf R}$ which satisfies (H1), (H2), with $K=K_0$. With $\gamma $ as in (H2), we denote by $\mu _{g_0,\gamma }$ the corresponding probability measure given in (H2). Assume that for every $\gamma $ as in (H2), there is a constant $C_\gamma >0$ such that $\mu _{g_0,\gamma }\ge {1\over C_\gamma }dt$ on ${\bf R}\cap \gamma \cap V_\gamma $, where $V_\gamma $ is some neighborhood of ${\rm ch\,}(K_-)$. \par Let $g_j(v)dv$, $j=1,2,..$ be a sequence of probability measures on ${\bf R}$ such that $g_j-g_0$ is of class $C^2$ and tends to 0 in $C^2$ when $j\to \infty $. Further, we assume that each $g_j$ satisfies (H1) with $K=K_j\to K_0$ and that $g_j\to g_0$ on every compact subset of ${\bf C}\setminus K_0$. \par Let $\widetilde{K}\subset{\bf C}$ be compact, symmetric around ${\bf R}$ and containing $K_0$ in its interior, and such that $\widetilde{K}_-=\{ z\in z\in \widetilde{K};\,{\rm Im\,}z\le 0\}$ is convex. Then for $j$ sufficiently large, $g_j$ satisfies (H1), (H2) with $K=\widetilde{K}$.\rm \medskip \par An immediate consequence of this result and Theorem 0.1 is: \medskip \par\noindent \bf Corollary 0.3. \sl We make the assumptions of Proposition 0.2. For every $\epsilon >0$, there is a $j(\epsilon )\in{\bf N}$ such that the following holds for $j\ge j(\epsilon )$: $\langle G_\Lambda (E)\rangle _{g_j}$ extends holomorphically to ${\bf C}\setminus (\widetilde{K}_-+\overline{D(0,2td+ \epsilon )})$ and so that for every $E$ in the latter set and every $\lambda \in ]2d,{1\over t}({\rm dist\,}(E,\widetilde{K}_-)-\epsilon )[:$ \ekv{0.12} { \Vert e^{\eta \cdot (\cdot )}\langle G_\Lambda (E)\rangle _{g_j}e^{-\eta \cdot (\cdot )}\Vert _{{\cal L}(\ell^2,\ell^2)}\le {1\over {\rm dist\,}(E,\widetilde{K}_-)-\epsilon -t\lambda }, } for all $\eta \in W(\lambda )$, and in particular, \ekv{0.13} { \vert \langle G_\Lambda (E)(\mu ,\nu )\rangle _{g_j}\vert \le {1\over {\rm dist\,}(E,\widetilde{K}_-)-\epsilon -t\lambda }\,e^{-p_\lambda (\mu -\nu )}.} (0.13) can be generalized as in Theorem 0.1'. \rm \medskip \par The following special case was considered in [SW]. Let \ekv{0.14}{g_0(v)={1\over\pi }\,{1\over v^2+1},} so that (following Economou [E],) by the residue method, using the pole at $v=-i$: \ekv{0.15} {\langle G_\Lambda (E)(\mu ,\nu )\rangle _{g_0}=(t\Delta _\Lambda -(E+i))^{-1}(\mu ,\nu ).} Let $\widetilde{K}\subset{\bf C}$ be compact, symmetric around ${\bf R}$, with $-i\not\in{\rm ch\,}\widetilde{K}_-$ and let $g_j$ have the properties in Corollary 0.3 with $K=K_j$ there equal to $\widetilde{K}\cup\{ -i,i\}$. We then recover the main result of [SW] in a slightly weakened form, by letting $E\in{\bf R}$ have the property that ${\rm dist\,}(E,{\rm ch\,}(\widetilde{K}_-\cup\{ -i\} ))=\vert E+i\vert >\vert E-F\vert $, for all $F\in({\rm ch\,}(\widetilde{K}_-\cup\{ -i\} ))\setminus\{ -i\} )$. \par The preceding example may be generalized. Take for instance \ekv{0.16} { g_0(v)={C_1\over\pi }\,{\beta _1\over (v-\alpha _1)^2+\beta _1^2}+{C_2\over\pi }\,{\beta _2\over (v-\alpha _2)^2+\beta _2^2}, } where $\alpha _1,\alpha _2\in{\bf R}$, $\beta _1,\beta _2>0$, $C_1,C_2\ge 0$, $C_1+C_2=1$. Then $$\langle G_\Lambda (E)(\mu ,\nu )\rangle _{g_0}=\langle G_\Lambda (E)(\mu ,\nu )\rangle _{C_1\delta _{\alpha _1-i\beta _1}+C_2\delta _{\alpha _2-i\beta _2}}$$ becomes the expectation value for a complex Bernoulli distribution, and we may consider perturbations as in Theorem 0.3. \par The plan of the paper is the following: In section 1 we discuss the assumption (H2), prove Proposition 0.2 and check that the $g_0$'s in (0.14) and in (0.16) satisfy (H2). In section 2, we complete the proof of Theorem 0.1. In section 3, we genealize our results further, by showing how to avoid making the assumption (H2). \bigskip \centerline{\bf 1. More about the assumption \rm (H2)\bf .} \medskip \par We first check that $g_0$ in (0.14) satisfies the assumptions (H1), (H2) with $K=\{ i,-i\} $: (H1) is obviously fulfilled and in order to check (H2), let $\gamma $ be a simple closed negatively oriented loop in the closed lower half-plane with $-i\in{\rm int\,}(\gamma )$. Let $f\in{\rm Hol\,}({\rm int\,}(\gamma ))\cap C^\infty (\overline{{\rm int\,}(\gamma )})$. Here we let ${\rm Hol\,}(\Omega )$ denote the space of holomorphic functions on $\Omega $, if $\Omega \subset{\bf C}$ is open. By the method of residues \ekv{1.1} { \int_\gamma f(z)g_0(z)dz=f(-i). } But $f$ is also harmonic, so \ekv{1.2} { f(-i)=\int_\gamma f(z)P_\gamma (-i,dz), } where $P_\gamma (-i,dz)=:\mu _\gamma (dz)$ is the harmonic measure (i.e. the Poisson kernel). This is a probability measure on $\gamma $, given by a strictly positive density, so (H2) is fulfilled. \par When $g_0$ is given by (0.16), we also have (H1), (H2) with $K=\{ \alpha _1+i\beta _1,\, \alpha _1-i\beta _1,\, \alpha _2+i\beta _2,\, \alpha _2-i\beta _2\}$, and ${\rm ch\,}(K_-)$ is the closed segment joining $\alpha _1-i\beta _1$ and $\alpha _2-i\beta _2$. \par We next prove Proposition 0.2. Let $g_0$, $\widetilde{K}$ be as in that proposition and choose $\gamma $ as in (H2), with $K$ there equal to $\widetilde{K}$. The discussion following (H2) shows that it suffices (for every such $\gamma $) to show that for $j$ large enough depending on $\gamma $: $$\int_\gamma f(z)g_j(z)dz=\int f(z)\mu _{\gamma ,j}(dz),\,\, f\in{\rm Hol\,}({\rm int\,}(\gamma ))\cap C^\infty (\overline{ {\rm int\,}(\gamma )}),$$ for some probability measure $\mu _{\gamma ,j}$ on $\gamma $. \par We already know that $$\int_\gamma f(z)g_0(z)dz=\int_\gamma f(z)\mu _{\gamma ,0}(dz),$$ where $\mu _{\gamma ,0}$ is a probability measure and from the discussion after (H2) and the positivity assumption on $\mu _{\gamma ,0}$ in ${\bf R}\cap{\rm neigh\,}(K)$, it follows that $\mu _{\gamma ,0}\ge {1\over C}\vert dz\vert $ everywhere on $\gamma $. It is then clear that Proposition 0.2 will follow from: \medskip \par\noindent \bf Proposition 1.1. \sl Let $\Omega \subset{\bf C}$ be open bounded, simply connected with positively oriented $C^\infty $ boundary $\partial \Omega =\gamma $. Let $g\in C^2(\gamma )$ be a \it complex-valued \sl function such that \ekv{1.3} { \int_\gamma g(z)dz\in{\bf R}. } Then there exists a \it real-valued \sl function $k\in C(\gamma )$, such that \ekv{1.4} { \int_\gamma f(z)g(z)dz=\int_\gamma f(z)k(z)\vert dz\vert , } for all $f\in{\rm Hol\,}(\Omega )\cap C^\infty (\overline{\Omega })$. \medskip \par\noindent \bf Proof. \rm The Riemann mapping theorem (see [B]) gives us a diffeomorphism $\kappa :\overline{\Omega }\to\overline{D(0,1)}$, holomorphic in the interior. By composing with $\kappa ^{-1}$, we can therefore reduce ourselves to the case when $\Omega =D(0,1)$. \par Expand in a Fourier series with $z=e^{it}$: \ekv{1.5} { g(z)dz=(\sum_{-\infty }^{+\infty }\widehat{g}(j)e^{i(j+1)t})idt=(\sum_{-\infty }^{+\infty }\widetilde{g}(j)e^{ijt})dt, } where $\widetilde{g}(j)=i\widehat{g}(j-1)$ and where the $C^2$ assumption assures normal convergence of the series. The assumption (1.3) tells us that \ekv{1.6} { \widetilde{g}(0)=i\widehat{g}(-1)\in{\bf R} .} For $f\in{\rm Hol\,}(D(0,1)\cap C^\infty (\overline{D(0,1)})$, we have $$f(e^{it})=\sum_{\ell =0}^\infty \widehat{f}(\ell )e^{i\ell t}$$ so \ekv{1.7} { \int_{\partial D(0,1)}f(z)g(z)dz=2\pi \sum_{\ell =0}^\infty \widehat{f}(\ell ) \widetilde{g}(-\ell ).}This expression does not change if we modify $\widetilde{g}(j)$ for $j\ge 1$ and we take \ekv{1.8} { k(t)dt=(\widetilde{g}(0)+\sum_{-\infty }^{-1}\widetilde{g}(j)e^{ijt}+\sum_1^\infty \overline{\widetilde{g}(-j)}e^{ijt})dt, } which is real thanks to (1.6).\hfill{$\#$} \medskip \par We also give a proof which avoids the use of the mapping theorem. For simplicity we assume that all objects are smooth. Let $\gamma $ be a positively oriented boundary of an open bounded simply connected set $\Omega \subset{\bf C}$ with smooth boundary. Let $f\in C^\infty (\gamma )$. \medskip \par\noindent \bf Proposition 1.2. \sl We have $\int_\gamma \phi (z)f(z)dz=0$ for all $\phi \in{\rm Hol\,}(\Omega )\cap C^\infty (\overline{\Omega })$ iff $f$ extends to an element in ${\rm Hol\,}(\Omega )\cap C^\infty (\overline{\Omega }).$ \medskip \bf \par\noindent Proof. \rm If $f$ extends to ${\rm Hol\,}(\Omega )\cap C^\infty (\overline{\Omega })$, then $\phi f$ extends to an element in the same space and $\int \phi fdz=0$. \par Before proving the converse statement, let $f\in C^\infty (\gamma ) $ and consider the two Cauchy integrals, $$C_{\rm int}f(z)={1\over 2\pi i}\int{f(\zeta )\over \zeta -z}\, d\zeta,\,\,\,z\in\Omega ,$$ $$C_{\rm ext}f(z)={1\over 2\pi i}\int{f(\zeta )\over\zeta -z}\, d\zeta ,\,\,\, z\in{\bf C}\setminus\overline{\Omega }.$$ Then $C_{\rm int}f\in{\rm Hol\,}(\Omega )\cap C^\infty (\overline{\Omega })$, $C_{\rm ext}f\in{\rm Hol\,}({\bf C}\setminus\overline{\Omega })\cap C^\infty ({\bf C}\setminus \Omega )$, and $$f={C_{\rm int}f_\vert }_\gamma -{C_{\rm ext}f_\vert }_\gamma .$$ \par If we now assume that $\int_\gamma \phi (z)f(z)dz=0$ for all $\phi \in{\rm Hol\,}(\Omega )\cap C^\infty (\overline{\Omega })$, then $C_{\rm ext}f=0$, so $f={C_{\rm int}f_\vert }_\gamma $ and $f$ has the extension ${C_{\rm int}f}$ in ${\rm Hol\,}(\Omega )\cap C^\infty (\overline{\Omega })$.\hfill{$\#$} \medskip \par\noindent \bf Proposition 1.3. \sl For every $g\in C^\infty (\gamma )$ with $\int_\gamma g(z)dz\in{\bf R}$, there is a unique $f\in{\rm Hol\,}(\Omega )\cap C^\infty (\overline{\Omega })$ such that ${\rm Im\,}({fdz_\vert }_\gamma )={\rm Im\,}({gdz_\vert }_\gamma )$. \medskip \bf \par\noindent Proof. \rm We first discuss uniqueness. It suffices to show that if $f\in{\rm Hol\,}(\Omega )\cap C^\infty (\overline{\Omega })$, ${\rm Im\,}({fdz_\vert }_\gamma )=0$, then $f=0$. Let $F\in{\rm Hol\,}(\Omega )\cap C^\infty (\overline{\Omega })$ be a primitive of $f$: ${\partial F\over\partial z}=f$, or equivalently: $dF=fdz$. Hence $d{\rm Im\,}F={\rm Im\,}(fdz)$, so ${\rm Im\,}F$ is a harmonic function on $\Omega $ with $$d({{\rm Im\,}F_\vert }_\gamma )={{\rm Im\,}fdz_\vert }_\gamma =0.$$ In other words, ${{\rm Im\,}F_\vert }_\gamma $ is constant, so the uniqueness in the standard Dirichlet problem implies that ${\rm Im\,}F={\rm Const.}$ on $\Omega $. Then $F$ is constant, and $f={\partial F\over\partial z}$ vanishes, as claimed. \par The proof of existence uses the same idea. Since $\int_\gamma {\rm Im\,}(gdz)=0$, there exists $G\in C^\infty (\gamma ;{\bf R})$ with $dG={\rm Im\,}({gdz_\vert }_\gamma )$. Indeed the vanishing of the integral assures us that the primitive is single valued. Let ${\cal G}\in C^\infty (\overline{\Omega })$ be the solution of the Dirichlet problem: $$\Delta {\cal G}=0\hbox{ on }\Omega ,\hbox{ }{{\cal G}_\vert }_\gamma =G. $$ Since ${\cal G}$ is harmonic and $\Omega $ simply connected, it is equal to the imaginary part of a holomorphic function $F$ which is easily seen to belong to ${\rm Hol\,}(\Omega )\cap C^\infty (\overline{\Omega })$. Let $f={\partial F\over\partial z}$. Then, $${\rm Im\,}{fdz_\vert }_\gamma ={\rm Im\,}{dF_\vert }_\gamma =d({{\rm Im\,}F_\vert }_\gamma )=dG={\rm Im\,}({gdz_\vert }_\gamma ).$$ \hfill{$\#$} \medskip \par Combining this result and the easy part of Proposition 1.2, we get a new proof of Proposition 1.1. Indeed, if $g$ is given as in Proposition 1.1, then let $f$ be as in Proposition 1.3. It follows from the proof that $f$ is of class $C^1$. According to Proposition 1.2, the real measure ${(gdz-fdz)_\vert }_\gamma $ has the required properties and can be written as $k\vert dz\vert $ with $k$ of class $C^1$. \par We end this section by linking the above discussion to the Neumann problem. Let $g\in C^\infty (\gamma )$ with \ekv{1.9} { \int g(z)dz=1. } As above, we look for $f\in{\rm Hol\,}(\Omega )\cap C^\infty (\overline{\Omega })$ with \ekv{1.10} { {gdz_\vert }_\gamma ={fdz_\vert }_\gamma +{k\vert dz\vert _\vert }_\gamma , } with the last term real. The unique solution of this problem is given by the function $f$ in Proposition 1.3, which is of the form $f(z)={\partial F\over\partial z}$, where $F$ is holomorphic in $\Omega $ and solves the Dirichlet problem \ekv{1.11} { \Delta {\rm Im\,}F=0\hbox{ in }\Omega ,\hbox{ }{{\rm Im\,}F_\vert }_\gamma (z)=\int_{\gamma _{z_0,z}}{\rm Im\,}(g(z)dz),\,\,z\in\gamma , } where $\gamma _{z_0,z}$ denotes the positively oriented segment of $\gamma $ which starts at $z_0$ and ends at $z$. \par We observe that the conjugated function ${\rm Re\,}F$ satisfies on $\gamma $: $${\partial \over\partial\nu _{\rm int} }{\rm Re\,}F=-{\partial \over\partial t}{\rm Im\,}F(\gamma (t)),\hbox{ }\nu _{\rm int}=\hbox{ interior unit normal,}$$ provided that we parametrize $\gamma $ by arc length with positive orientation, so that $\vert \gamma '(t)\vert =1$. But ${\partial \over\partial t}{\rm Im\,}F(\gamma (t))={\rm Im\,}(g(\gamma (t))\gamma '(t))$ according to the boundary condition in (1.11), so ${\rm Re\,}F$ is the solution (unique up to a constant) of the Neumann problem: \ekv{1.12} { \Delta {\rm Re\,}F=0\hbox{ in }\Omega ,\hbox{ }{\partial \over\partial \nu _{\rm int}}{\rm Re\,}F=-{\rm Im\,}(g\gamma ') \hbox{ on }\gamma .} \par Then we get ${\rm Re\,}({fdz_\vert }_\gamma )=d_t{\rm Re\,}F(\gamma (t))$, so \ekv{1.13} { k(t)={\rm Re\,}(g(\gamma (t))\gamma '(t))+{\partial \over\partial t}[(r_\gamma K_N({\rm Im\,}g\gamma '))(\gamma (t))]. } Here $K_N$ denotes the Poisson-Neumann operator which solves up to a constant the Neumann problem: $$\Delta K_Nv=0\hbox{ on }\Omega ,\hbox{ }{\partial \over\partial \nu _{\rm int}}K_Nv=v\hbox{ on }\gamma ,\hbox{ when }\int v(\gamma (t))dt=0,$$ and $r_\gamma $ is the restriction operator: $C^\infty (\overline{\Omega })\to C^\infty (\gamma )$. \bigskip \centerline{\bf 2. Coercivity and exponential decay.} \medskip \par For $\lambda >2d$, let \ekv{2.1} { W(\lambda )=\{ \eta \in{\bf R}^d;\,2\sum_1^d\cosh \eta _j<\lambda \} . } (We refer to [SW] for a more complete discussion, using also the Fourier transform.) $W(\lambda )$ is a convex bounded open set symmetric around $0$, and we let \ekv{2.2} { p_\lambda (x)=\sup_{\eta \in W(\lambda )}x\cdot \eta , } be the corresponding support function. $p_\lambda $ is convex, smooth outside $0$, and positively homogeneous of degree 1. Moreover $p_\lambda (x)>0$ for $x\ne 0$. In [SW], we observed that for $\eta \in W(\lambda )$: \ekv{2.3} {\Vert e^{\eta \cdot (\cdot )}\Delta _\Lambda e^{-\eta \cdot (\cdot )}\Vert _{{\cal L}(\ell^2,\ell^2)}\le \lambda ,} and similarly for $\Delta =\Delta _{{\bf Z}^d}$. \par Let $E\in{\bf C}$ with $\vert E\vert >2d$ and choose $\lambda \in ]2d,\vert E\vert [$. Then for $\eta \in W(\lambda )$, we have in the sense of self-adjoint operators: \ekv{2.4} { {\rm Re\,} e^{-i\arg E}(e^{\eta \cdot (\cdot )}(E-\Delta _\Lambda )e^{-\eta \cdot (\cdot )})\ge \vert E\vert -\lambda , } and similarly with $\Delta _\Lambda $ replaced by $\Delta $. As noticed in [SW] with a slightly different method, it follows that \ekv{2.5} { \Vert e^{\eta \cdot (\cdot )}(E-\Delta _\Lambda )^{-1}e^{-\eta \cdot (\cdot )}\Vert _{{\cal L}(\ell^2,\ell^2)}\le {1\over \vert E\vert -\lambda }, } and in particular for the matrix of the inverse: \ekv{2.6} { \vert (E-\Delta _\Lambda )^{-1}(\mu ,\nu )\vert \le {1\over \vert E\vert -\lambda }\, e^{-p_\lambda (\mu -\nu )}. } Using an argument of regularization of the weights near infinity, we obtain (2.5), (2.6) also for $\Delta $. In the present paper, we shall not use (2.5), (2.6) but rather (2.4) and establish: \medskip \par\noindent \bf Proposition 2.1. \sl Let $E\in{\bf C}$ with $\vert E\vert >2d$ and let $v_j$, $j\in\Lambda $ satisfy ${\rm Re\,}e^{-i\arg E}v_j\le 0$. Then $E-(\Delta _\Lambda +{\rm diag\,}(v_j))$ has a bounded inverse such that for every $\lambda \in ]2d,\vert E\vert [$ and every $\eta \in W(\lambda )$: \ekv{2.7} { \Vert e^{\eta \cdot (\cdot )}(E-(\Delta _\Lambda +{\rm diag\,}(v_j)))^{-1}e^{-\eta \cdot (\cdot )}\Vert _{{\cal L}(\ell^2,\ell^2)}\le {1\over \vert E\vert -\lambda }. } In particular, \ekv{2.8} { \vert (E-(\Delta _\Lambda +{\rm diag\,}(v_j)))^{-1}(\mu ,\nu )\vert \le {1\over \vert E\vert -\lambda }\, e^{-p_\lambda (\mu -\nu )}. } The result remains valid with $\Lambda $ replaced by ${\bf Z}^d$ if we assume that $\{ v_j\} _{j\in{\bf Z}^d}$ is bounded. \medskip \par\noindent \bf Proof. \rm Using (2.4), we get \ekv{2.9} { {\rm Re\,}e^{-i\arg E}(e^{\eta \cdot (\cdot )}(E-(\Delta _\Lambda +{\rm diag\,}(v_j)))e^{-\eta \cdot (\cdot )})\ge \vert E\vert -\lambda } and (2.7) and (2.8) are obtained as (2.5) and (2.6). The extension to the case $\Lambda ={\bf Z}^d$ is treated using regularization of the weights at infinity.\hfill{$\#$} \medskip \par We now return to the situation in the sections 0, 1. Let $\gamma $ be a curve as in (H2) so that \ekv{2.10} { \langle G(E)(\mu ,\nu )\rangle =\int (t\Delta _\Lambda +{\rm diag\,}(v_j)-E)^{-1}(\mu ,\nu )\prod _{j\in\Lambda }\mu _\gamma (dv_j). } We may assume without loss of generality that $\gamma $ is convex. Let $E\in{\bf C}$ belong to the exterior of $\gamma $, and let $\pi _\gamma (E)\in\gamma $ be the point on $\gamma $, closest to $E$: $\vert \pi _\gamma (E)-E\vert ={\rm dist\,}(E,\gamma )$. Then the line through $\pi _\gamma (E)$ which is perpendicular to $E-\pi _\gamma (E)$ separates $\gamma $ and $E$, and we have: \ekv{2.11} { {\rm Re\,}(e^{-i{\rm arg}(E-\pi _\gamma (E))}(\pi _\gamma (E)-v))\ge 0,\,\,\forall v\in\gamma . } Writing $$\eqalign{& t\Delta +{\rm diag\,}(v_j)-E=-(E-\pi _\gamma (E))+(t\Delta +{\rm diag\,}(v_j-\pi _\gamma (E)))= \cr &-t[{1\over t}(E-\pi _\gamma (E))-(\Delta +{\rm diag\,}({1\over t}(v_j-\pi _\gamma (E)))], }$$ we apply Proposition 2.1 with $E$ there replaced by ${1\over t}(E-\pi _\gamma (E))$ and $v_j$ there replaced by ${1\over t}(v_j-\pi _\gamma (E))$, and get for $\eta \in W(\lambda )$: \eekv{2.12} { \Vert e^{\eta \cdot (\cdot )}(t\Delta _\Lambda +{\rm diag\,}(v_j)-E)^{-1}e^{-\eta \cdot (\cdot )}\Vert _{{\cal L}(\ell^2,\ell^2)}\le} {\hskip 2cm {1\over t}\,{1\over ({1\over t}\vert E-\pi _\gamma (E)\vert -\lambda )}={1\over \vert E-\pi _\gamma (E)\vert -t\lambda }, } if ${1\over t}\,\vert E-\pi _\gamma (E)\vert >2d$ and $\lambda \in ]2d,{1\over t}\vert E-\pi _\gamma (E)\vert [$, and in particular, \ekv{2.13} { \vert (t\Delta _\Lambda +{\rm diag\,}(v_j)-E)^{-1}(\mu ,\nu )\vert \le {1\over \vert E-\pi _\gamma (E)\vert -t\lambda }\, e^{-p_\lambda (\mu -\nu )}. } Since $\mu _\gamma $ is a probability measure, we get from this and (2.10): \ekv{2.14} { \Vert e^{\eta \cdot (\cdot )}\langle G(E)\rangle e^{-\eta \cdot (\cdot )}\Vert _{{\cal L}(\ell^2,\ell^2)}\le {1\over \vert E-\pi _\gamma (E)\vert -t\lambda },\,\, \eta \in W(\lambda ), } \ekv{2.15} { \vert \langle G(E)(\mu ,\nu )\vert \le {1\over \vert E-\pi _\gamma (E)\vert -t\lambda }\, e^{-p_\lambda (\mu -\nu )}, } for $\lambda \in ]2d,{1\over t}\vert E-\pi _\gamma (E)\vert [$. \par Since we can choose $\gamma $ in an arbitrarily small neighborhood of ${\rm ch\,}(K_-)$, we may arrange so that $\vert E-\pi _\gamma (E)\vert \to {\rm dist\,}(E,{\rm ch\,}(K_-))$ and Theorem 0.1 follows.\hfill{$\#$} \bigskip \centerline{\bf 3. A more general result.} \medskip \par In this section we shall relax the assumption (H1) about holomorphic extendability, and suppress the assumption (H2). The price we have to pay, is that the result will be valid only for $E$ sufficiently far away from the region of non-analyticity in some precise sense that we will explain. We start by proving some auxiliary results which contain the essential ideas. Write $\langle z\rangle =\sqrt{(1+\vert z\vert ^2)}$. \medskip \par\noindent \bf Lemma 3.1. \sl Let ${\bf R}\ni t\mapsto \gamma (t)$ be a smooth curve in the open upper half plane ${\bf C}^+$, with $\gamma '(t)\ne 0$ and without self-intersections. Assume that $\gamma (t)=C_\pm +e^{i\theta _\pm }t$, $\pm t>>0$, where $\theta _-<0<\theta _+<\pi +\theta _-$, and let $\Omega \subset{\bf C}^+$ be the open set with $\partial \Omega =\gamma $. Let $g(z)$ be a continuous function on $\gamma $ with $g(z)={\cal O}(\langle z\rangle ^{-2-\epsilon })$, for some $\epsilon >0$, and with \ekv{3.1}{\int_\gamma g(z)dz\in{\bf R}} Then there exists a real measure $\mu $ on $\gamma $ of the form $\mu (dz)=m(z)\vert dz\vert $ with $m$ continuous and ${\cal O}(\langle z\rangle ^{-2-\epsilon })$, for some, possibly new $\epsilon >0$, such that \ekv{3.2}{\int_\gamma \phi (z)g(z)dz=\int_\gamma \phi (z)\mu (dz),\,\,\forall \phi \in{\rm Hol\,}(\Omega )\cap C(\overline{\Omega })\cap L^\infty (\Omega ).} \medskip \par\noindent \bf Proof. \rm Let $G\in C^1(\gamma )$ be a primitive of ${\rm Im\,}(gdz)$ with $G={\cal O}(\langle z\rangle ^{-1-\epsilon })$. Considering rotations of the functions ${\rm Re\,}z^{-1-\epsilon }$, we see that we can find a positive harmonic function $q$, defined near $\overline{\Omega }$, of the order of magnitude $\vert z\vert ^{-1-\epsilon }$, for sufficiently small $\epsilon >0$, depending on $\theta _+$, $\theta _-$. Consider the Dirichlet problem for ${\cal G}\in C^1(\overline{\Omega })$: \ekv{3.3} {\Delta {\cal G}=0\hbox{ in }\Omega ,\,\,{{\cal G}_\vert }_\gamma =G.} Approaching this problem with suitable problems on $\Omega _R:=\Omega \cap D(0,R)$, using the maximum principle with $q$ as a comparison function, and letting $R\to \infty $, we see that (3.3) has a unique solution with ${\cal G}={\cal O}(\langle z\rangle ^{-1-\epsilon })$, for some $\epsilon >0$. \par Using a scaling argument we also see that \ekv{3.4}{\nabla {\cal G}={\cal O}(\langle z\rangle ^{-2-\epsilon }).} Since ${\cal G}$ is harmonic in $\Omega $, we have \ekv{3.5} {{\cal G}={\rm Im\,}{\cal F},\,\, {\cal F}\in{\rm Hol\,}(\Omega ).} >From the Cauchy-Riemann equations, we see that ${\cal F}\in C^1(\overline{\Omega })$, and \ekv{3.6} {\nabla {\cal F}={\cal O}(\langle z\rangle ^{-2-\epsilon }).} Put $\mu (dz)=gdz-d{{\cal F}_\vert }_\gamma =gdz-f{dz_\vert }_\gamma $, where $f={\partial {\cal F}\over\partial z}$. Then, \ekv{3.7} {{\rm Im\,}(gdz-{d{\cal F}_\vert }_\gamma )={\rm Im\,}(gdz)-dG=0,} so $\mu $ is real. Moreover, \ekv{3.8} {\mu (dz)=m(z)\vert dz\vert ,\,\, m\in C(\gamma ),\, m={\cal O}(\langle z\rangle ^{-2-\epsilon }).} Finally we have (3.2), since $$\int_\gamma \phi (z)f(z)dz=0,\,\, \phi \in{\rm Hol\,}(\Omega )\cap C (\overline{\Omega })\cap L^\infty (\Omega ), $$ by a standard argument for Cauchy integrals. \hfill{$\#$} \medskip \par\noindent \it Remark. \rm Assume that $g$ satisfies the regularity and growth assumptions of the Lemma outside some compact subset $K$ of $\gamma $ and that $g$ is a distribution near $K$. Then we can find a real distribution $\mu $ on $\gamma $ which is of the form $m(z)\vert dz\vert $ outside $K$ with $m$ as in the lemma, such that the identity in (3.2) holds for all $\phi \in{\rm Hol\,}(\Omega )\cap C^\infty (\overline{\Omega })\cap L^\infty (\Omega )$. To see this we repeat the proof. $G$ will now be a distribution near $K$, and with the same properties as before outside $K$. Then we can solve (3.3), where ${\cal G}$ is of temperate growth near $K$ and elsewhere $C^1$ up to the boundary, and $={\cal O}(\langle z\rangle ^{-1-\epsilon })$ far away. We still have (3.4) far away and can define ${\cal F}$ as before, holomorphic in $\Omega $, of temperate growth near $K$ and $C ^1$ up to the boundary away from $K$. Moreover $f:={\partial {\cal F}\over\partial z}={\cal O}(\langle z\rangle ^{-2-\epsilon })$ far away. Define $\mu (dz)$ as before, now with ${fdz_\vert }_\gamma $ interpreted as a boundary value in the sense of distributions. We get a real distribution, and our claim follows from the fact that $$\int_\gamma \phi (z)f(z)dz=0,\,\, \phi \in{\rm Hol\,}(\Omega )\cap C^\infty (\overline{\Omega })\cap L^\infty (\Omega ).$$ \medskip \par Let $\gamma $ be as in the lemma and let $\mu (dz)=m(z)\vert dz\vert $ with $m$ real, continuous and $={\cal O}(\langle z\rangle ^{-2-\epsilon })$. Let $p(z)={1\over\pi }{{\rm Im\,}z\over \vert z\vert ^2}$ and let $$Pu(z)=\int p(z-t)u(t)dt$$ be the Poisson operator for the upper half plane ${\bf C}^+$, mapping bounded continuous functions on ${\bf R}$ to bounded continuous functions on $\overline{{\bf C}^+}$. We recall that $P$ is positivity preserving. The adjoint $P^*$ maps bounded measures (i.e. with finite total mass) on ${\bf C}^+$ to bounded measures on ${\bf R}$, and in the case of positive measures, the total mass is conserved. We are interested in $P^*(\mu )$ which is of the form $k(t)dt$ with $k(t)=\int p(z-t)\mu (dz)$. \medskip \par\noindent \bf Lemma 3.2. \sl Let $\gamma _j$, $j=1,2$ be as in Lemma 3.1, and let $\mu _j(dz)$ be a real measure on $\gamma _j$ of the form $m_j(z)\vert dz\vert $ with $m_j$ continuous and ${\cal O}(\langle z\rangle ^{-2})$. Assume that \ekv{3.9} {\int_{\gamma _1}\phi (z)\mu _1(dz)=\int_{\gamma _2}\phi (z)\mu _2(dz),} for all $\phi \in{\rm Hol\,}({\bf C}^+)$, which are bounded on every half-plane: ${\rm Im\,}z>\epsilon $, $\epsilon >0$. Then \ekv{3.10} {P^*(\mu _1)=P^*(\mu _2).} \medskip \par\noindent \bf Proof. \rm Since $\mu _j$ are real, it follows that $$\int_{\gamma _1}{\rm Re\,}(\phi (z))\mu _1(dz)=\int_{\gamma _2}{\rm Re\,}(\phi (z))\mu _2(dz),$$ for all $\phi $ as in the lemma. It then suffices to notice that $p(z-t)={\rm Re\,}\phi _t$, where $\phi _t$ is as in the lemma.\hfill{$\#$} \medskip \par As in the remark after the proof of Lemma 3.1, we can relax the regularity assumptions on $\mu _j$ on some bounded part of $\gamma _j$. \medskip \par\noindent \bf Lemma 3.3. \sl Let $\gamma $ be as in Lemma 3.1 and let $\mu (dz)=m(z)\vert dz\vert $ be a real density on $\gamma $ with $m(z)$ continuous, $={\cal O}(\langle z\rangle ^{-2-\epsilon })$ for some $\epsilon >0$. \smallskip \par\noindent a) Then $P^*(\mu )=k(t)dt$ with $k(t)$ continuous and ${\cal O}={\cal O}(\langle t\rangle ^{-2})$. \smallskip \par\noindent b) For $T\ge 0$, let $\gamma _T(t)=\gamma (t)+iT$, and define $\mu _T$ on $\gamma _T$ as $m(z-iT)\vert dz\vert $. If \ekv{3.11} {\int _\gamma \mu (dz)=1,} then there exists $T_0\ge 0$, such that $P^*(\mu _T)\ge 0$ on ${\bf R}$, precisely for $T\ge T_0$.\rm \medskip \par\noindent \bf Proof. \rm We have $$\eqalign{&k(t)=\int_\gamma p(z-t)\mu (dt)={1\over\pi }\int_{-\infty }^{+\infty }{{\rm Im\,}\gamma (s)\over\vert \gamma (s)-t\vert ^2}m(\gamma (s))\gamma '(s)ds= \cr & {\cal O}(1)\int_{-\infty }^{+\infty }{\langle s\rangle \over (1+\vert t\vert +\vert s\vert )^2}\langle s\rangle ^{-2}ds={\cal O}(1)\int_0^\infty {1\over (1+\vert t\vert +s)^2}{1\over 1+s}ds={\cal O}({1\over (1+\vert t\vert )^2}),}$$ and we have proved a), using only the weaker assumption that $m={\cal O}(\langle z\rangle ^{-2})$. \par If $P^*(\mu _{T_0})\ge 0$, for some $T_0$ and $T>T_0$, then we can identify this measure with the measure $P_{T-T_0}^*(\mu _T)$ on the line ${\rm Im\,}z=T-T_0$, where $P_{T-T_0}$ is the Poisson operator for the half plane ${\rm Im\,}z>T-T_0$. It is easy to see that $P^*(\mu _T)=P^*P_{T-T_0}^*(\mu _T)\ge 0$, so to prove b), it suffices to find one $T>0$ such that $P^*(\mu _T)\ge 0$. Write $P^*(\mu _T)=k_Tdt$, where \eekv{3.12} {k_T(t)=\int_{\gamma _T}p(z-t)\mu _T(dz)=} {p(\gamma _T(0)-t)+\int_{\gamma _T}(p(z-t)-p(\gamma _T(0)-t))\mu _T(dz).} Here $p(\gamma _T(0)-t)\sim T/(T+\vert t\vert )^2$. \par When $z\in \gamma _T$ and $\vert z-\gamma _T(0)\vert \le T$, we have $$\vert p(z-t)-p(\gamma _T(0)-t)\vert \le C{\vert z-\gamma _T(0)\vert \over \vert \gamma _T(0)-t\vert ^2},$$ and the corresponding contribution to the last integral in (3.12) is \ekv{3.13} {{\cal O}(1)\int_{-T}^{T}{1+\vert s\vert \over (T+\vert t\vert )^2(1+\vert s\vert )^2}ds={\cal O}(1){\log T\over (T+\vert t\vert )^2}=o(1)p(\gamma _T(0)-t),} when $T\to \infty $, uniformly in $t$. \par The integral over $\vert z-\gamma _T(0)\vert >T$ can be split in two terms: \ekv{3.14} {{\cal O}(1)\int_{\vert s\vert \ge T}{\vert s\vert \over s^2+ t^2}{1\over s^{2+\epsilon }}ds,} and \ekv{3.15} {{\cal O}(1)\int_{\vert s\vert \ge T}{T\over T^2+t^2}{1\over s^{2+\epsilon }}ds.} the last expression is ${\cal O}(1){1\over T^2+t^2}=o(1)p(\gamma _T(0)-t)$. (3.14) is \ekv{3.16} {{\cal O}(1)\int_T^\infty {1\over s^2+t^2}{1\over s^{1+\epsilon }}ds.} If $T\ge \vert t\vert $, we estimate this by ${\cal O}(1)T^{-2}=o(1)p(\gamma _T(0)-t)$. If $T\le \vert t\vert $, then the change of variables $s=\vert t\vert \sigma $ gives $${{\cal O}(1)\over \vert t\vert ^{2+\epsilon }}\int_{T/\vert t\vert }^\infty {1\over 1+\sigma ^2}{1\over \sigma ^\epsilon }{d\sigma \over \sigma }={\cal O}(1){1\over T^\epsilon \vert t\vert ^2}=o(1)p(\gamma _T(0)-t).$$ Summing up, we get \ekv{3.17} {k_T(t)=(1+o(1))p(\gamma _T(0)-t),\,\,T\to\infty ,} uniformly in $t$, and the positivity follows, when $T$ is large enough. \hfill{$\#$} \medskip \par\noindent \it Remark. \rm a) We can relax the continuity assumption on $m$ and allow $\mu (dz)$ to be a distribution on some bounded set and elsewhere as in the lemma. Then a), b) still hold. In estimating for instance the last integral in (3.12), we decompose $\mu _T$ into a continuous density and a distribution with compact support. For the contribution of the latter, we only need to notice that for $k\ge 1$: $$\nabla _z^k(p(z-t)-p(\gamma _T(0)-t))={\cal O}_k(1){1\over (T+\vert t\vert )^{1+k}}=o(1)p(\gamma _T(0)-t).$$ b) If we map the upper half-plane conformally onto the unit disc, then the curves in the preceding lemmas close (at the image point of infinity), and we get a conceptual link with some of the arguments in section 1. \medskip \par We can now start to formulate the main result of this section. Let $g(x)dx$ be a probability measure on ${\bf R}$ and assume that for some $\epsilon >0$: \ekv{3.18} {g\hbox{ is continuous and }{\cal O}(\langle x\rangle ^{-2-\epsilon })\hbox{ outside some bounded set.}} \eekv{3.19} {g \hbox{ has a holomorphic extension to some set of the form }} {\vert {\rm Im\,}z\vert <{1\over C}({\rm Re\,}z-C),\hbox{ which satisfies }g={\cal O}(\langle z\rangle ^{-2-\epsilon }).} A straight line $L$ is called \it admissible \rm if $L$ is non-parallel to ${\bf R}$ and if $g$ has a holomorphic extension $g={\cal O}(\langle z\rangle ^{-2-\epsilon })$ to a set of the form $\{ z\in{\bf C};\, {\rm dist\,}(z,K_L)2dt\}$. If $E$ belongs to the latter set and $\lambda \in[2d,h(E)/t[$, then for every $\eta \in W(\lambda )$: $$\Vert e^{\eta \cdot (\cdot )}\langle (t\Delta +{\rm diag\,}(v_j)-E)^{-1}\rangle _ge^{-\eta \cdot (\cdot )}\Vert _{{\cal L}(\ell^2,\ell^2)}\le {1\over h(E)-t\lambda },$$ and in particular, $$\vert \langle (t\Delta _\Lambda +{\rm diag\,}(v_j)-E)^{-1}(\mu ,\nu )\rangle _g\vert \le {1\over h(E)-t\lambda }\, e^{-p_\lambda (\mu -\nu )}.$$ \medskip \par\noindent \bf Proof. \rm Let $L$ be an admitted line and choose $\gamma $ as above. For ${\rm Im\,}E>0$, we have \ekv{3.21} {\langle (t\Delta _\Lambda +{\rm diag\,}(v_j)-E)^{-1}\rangle _g=\int_{\gamma ^\Lambda } (t\Delta _\Lambda +{\rm diag\,}(v_j)-E)^{-1}\prod g(v_j)dv_j.} \par Let $E\in\Pi _L^+$ with ${\rm dist\,}(E,L)>2dt$ and let $v_j\in\Pi _L^-$, $\eta \in W(\lambda )$, $2d\le \lambda <{\rm dist\,}(E,L)/t$. Then $$-{\rm Re\,}(e^{\eta \cdot (\cdot )}e^{-i\arg (E-\pi _L(E))}(t\Delta _\Lambda +{\rm diag\,}(v_j)-E)e^{-\eta \cdot (\cdot )})\ge {\rm dist\,}(E,L)-t\lambda ,$$ where $\pi _L(E)$ is the point in $L$ which realizes the distance from $E$ to $L$. It follows that $$\Vert e^{\eta \cdot (\cdot )}(t\Delta _\Lambda +{\rm diag\,}(v_j)-E)^{-1}e^{-\eta \cdot (\cdot )}\Vert \le {1\over {\rm dist\,}(E,L)-t\lambda },$$ $$\vert (t\Delta _\Lambda +{\rm diag\,}(v_j)-E)^{-1}(\mu ,\nu )\vert \le {1\over {\rm dist\,}(E,L)-t\lambda }\, e^{-p_\lambda (\mu -\nu )}.$$ If in addition ${\rm Im\,}E>0$, then we can use (3.21) to get $$\eqalign{&\langle (t\Delta _\Lambda +{\rm diag\,}(v_j)-E)^{-1}\rangle _g=\int_{\gamma ^\Lambda }(t\Delta _\Lambda +{\rm diag\,}(v_j)-E)^{-1}\prod \mu _\gamma (dv_j)\cr &\hskip 3cm =\int_{L^\Lambda }(t\Delta _\Lambda +{\rm diag\,}(v_j)-E)^{-1}\prod P_L^*(\mu _\gamma )(dv_j).}$$ >From this identity and the preceding estimates, we see that $\langle (t\Delta _\Lambda +{\rm diag\,}(v_j)-E)^{-1}\rangle _g$ extends holomorphically to the set of $E$ in $\Pi_L^+ $ with ${\rm dist\,}(E,L)>t\lambda $ and satisfies the same estimates. It then suffices to vary $L$ among all admitted lines.\hfill{$\#$} \medskip We have not tried to formulate a result of maximal generality or sharpness. One obvious generalization would be to consider holomorphic extensions to some Riemann surface. Another would be to consider the situation in the introduction, and only make the hypothesis (H1). When $\gamma $ is a closed bounded curve, Lemma 3.1 reduces to Proposition 1.1, and Lemma 3.2, 3.3 remain valid. We leave the formulation of the analogue of Theorem 3.4 to the interested reader. \bigskip \centerline{\bf References.} \medskip \item{[B]} S.Bell, \it The Cauchy transform, potential theory, and conformal mapping, \rm Studies in advanced mathematics, CRC Press, Boca Raton, Ann Arbor, London, Tokyo 1992. \item{[CFS]} F.Constantinescu, J.Fr\"ohlich, T.Spencer, \it Analyticity of the density of states and replica methods for random Schr\"odinger operators on a lattice, \rm J. Stat. Phys. 34(1984), 571--596. \item{[E]} E.N.Economou, \it Green's functions in quantum physics, \rm Springer series in solid state physics 7, 1979. \item{[SW]} J.Sj\"ostrand, W.-M.Wang, \it Supersymmetric measures and maximum principles in complex space-- Exponential decay of Green's functions. \rm Preprint. \end