Content-Type: multipart/mixed; boundary="-------------0603220803602" This is a multi-part message in MIME format. ---------------0603220803602 Content-Type: text/plain; name="06-87.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-87.comments" J\"urg Fr\"ohlich, Simon Schwarz: Theoretische Physik, ETH Z\"urich, Switzerland. (juerg@itp.phys.ethz.ch, sschwarz@itp.phys.ethz.ch); Sandro Graffi: Dipartimento di Matematica, Universit\`{a} di Bologna, Italy. (graffi@dm.unibo.it) ---------------0603220803602 Content-Type: text/plain; name="06-87.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-87.keywords" Mean-field limit, Hartree equation, uniform estimates ---------------0603220803602 Content-Type: application/x-tex; name="FGS13.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="FGS13.tex" \documentclass[draft,11pt]{article} \usepackage{amsmath,amssymb,epsfig,graphicx,,color,epsf} \setlength{\textheight}{23.0cm} \setlength{\textwidth}{15.2cm} \hoffset=-1.0cm \voffset=-1.0cm \newtheorem{theorem}{Theorem} \newtheorem{proposition}{Proposition} \newtheorem{lemma}{Lemma} \newtheorem{corollary}{Corollary} \newtheorem{definition}{Definition} \renewcommand{\thesection} {\arabic{section}} \renewcommand{\thetheorem} {\thesection.\arabic{theorem}} \renewcommand{\theproposition} {\thesection.\arabic{proposition}} \renewcommand{\thelemma} {\thesection.\arabic{lemma}} \renewcommand{\thedefinition} {\thesection.\arabic{definition}} \renewcommand{\thecorollary} {\thesection.\arabic{corollary}} \renewcommand{\theequation} {\thesection.\arabic{equation}} \newcommand{\begsection}[1]{\setcounter{equation}{0}\section{#1}} \newcommand{\finsection}{\vskip20pt}\newcommand{\hindsp}{\hspace{2em}}% \def\C{{\mathcal C}} \def\L{{\mathcal L}} \def\B{{\mathcal B}}\def\D{{\mathcal D}}\def\H{{\mathcal H}}\def\P{{\mathcal P}} \def\G{{\mathcal G}} \def\vf{\varphi} \def\ve{\varepsilon} \def\psibx{\overline{\psi(x)}} \def\psiby{\overline{\psi(y)}} \def\As{{\cal A}_{\sigma,p}} \def\R{\mathbb R} \def\Z{\mathbb Z} \def\N{\mathbb N} \def\T{\mathbb T} \def\C{\mathbb C} \def\A{{\mathcal A}} \def\res{{\mathcal R}} \def\ha{Ha\-mil\-to\-nian} \def\Sc{Schr\"o\-din\-ger} \def\hp{{\hbar}} \def\la{\langle} \def\be{\begin{equation}} \def\ee{\end{equation}} \def\ra{\rangle} \def\ds{\displaystyle}\def\chit{\tilde{\chi}} \def\om{\omega} \def\Om{\Omega} \def\ep{\epsilon} \def\gk{\tilde{g}_k} \def\imma{{\rm Im}} \def\F{{\mathcal F}_{\rho,\sigma}} \def\limN{\lim_{N\to\infty}} \def\om{\omega} \def\be{\begin{equation}} \def\ee{\end{equation}} \def\ap{a^{(p)}} \def\alp{\alpha^{(p)}} \def\WAN{\widehat{A}_N} \def\HF{\widehat{H}} \def\w2{w^{(2)}} \def\ml{m^{(l)}} \def\ub{\overline{u}} \def\apn{A^{(p)}_N} %%%%%%%%%%%%%% %%%%%%%% begin %%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \baselineskip=20pt \date{15 February 2005} \begin{center} {\large\bf MEAN-FIELD- AND CLASSICAL LIMIT OF MANY-BODY SCHR\"ODINGER DYNAMICS FOR BOSONS } \end{center} \vskip 13pt \begin{center} J\"urg Fr\"ohlich\footnote{Theoretische Physik, ETH Z\"urich, Switzerland. (juerg@itp.phys.ethz.ch)}, Sandro Graffi\footnote{Dipartimento di Matematica, Universit\`{a} di Bologna, Italy. (graffi@dm.unibo.it)} , Simon Schwarz\footnote{Theoretische Physik, ETH Z\"urich, Switzerland. (sschwarz@itp.phys.ethz.ch)}\end{center} \begin{abstract} \noindent We present a new proof of the convergence of the $N-$particle \Sc\ dynamics for bosons towards the dynamics generated by the Hartree equation in the mean-field limit. For a restricted class of two-body interactions, we obtain convergence estimates uniform in $\hbar$, up to an exponentially small remainder. For $\hbar = 0$, the classical dynamics in the mean-field limit is given by the Vlasov equation. \end{abstract} \vskip 1cm \date{\today} %%%%%%%%%%%%%%%%%%%%% \begsection{Introduction and statement of results} \setcounter{equation}{0}% \setcounter{theorem}{0}% \setcounter{proposition}{0}% %\setcounter{lemma}{0}% \setcounter{corollary}{0}% \setcounter{definition}{0}% Consider the \Sc\ operator \begin{eqnarray} \label{hn} H_N&=&H_{N}^0+W_{N} \\ H_{N}^{0}&=&-\sum_{i=1}^N\frac{\hbar^2}{2}\Delta_i, \quad W_{N}=\frac1{N}\sum_{i0$ and $t\geq 0$ be fixed, and let $w\in L^\infty(\R^3)$. If $\Psi_{N,0}(x_1,\ldots,x_N)=\psi(x_1)\cdots\psi(x_N)$ is a normalized "coherent" (i.e., product) initial state, then \begin{eqnarray} \label{t11} \lim_{N\to\infty}\langle\Psi_{N,0},e^{iH_Nt/\hbar}A^{(p)}_Ne^{-iH_Nt/\hbar}\Psi_{N,0}\rangle = \qquad \\ \nonumber \limN\langle\Psi_{N,t},A^{(p)}_N\Psi_{N,t}\rangle =\la\Psi_{p,t},a^{(p)}\Psi_{p,t}\ra =:a^{(p)}(\psi_{t}) \end{eqnarray} Here $\Psi_{N,t}$ is again a coherent state, i.e., $\Psi_{N,t}(x_1,\ldots,x_N)= \psi_t(x_1)\cdots\psi_t(x_N)$, and $\Psi_{p,t} = \Psi_{N=p,t}$, where $\psi_t$ is a solution of the Hartree equation \be \label{Hartree} i\hbar\partial_t\psi_{t}=-\frac{\hbar^2}{2}\Delta\psi_{t}+(w\ast|\psi_{t}|^{2})\psi_{t} \ee with initial condition $\psi_{t=0} = \psi$. \end{theorem} {\bf Remarks} \begin{enumerate} \item For large $N$, the quantum evolution $e^{-iH_{N}t/\hbar}\Psi_{N,0}\quad$ can be replaced by the {\it nonlinear} single-particle evolution $\Psi_{N,t} (x_{1},\ldots,x_{N})$. Particle interaction effects are translated into the nonlinearity of this evolution. This justifies interpreting the limit $N\rightarrow\infty$ as a mean-field limit. \item The corrections to the limit in (\ref{t11}) are $O(1/N)$. \item Since $\ds \limN {N(N-1)\cdots (N-p+1)}/{N^p}=1$, the second equality in (\ref{t11}) follows easily from (\ref{A_N}), because $$ \limN \la\Psi_{N,t},A^{(p)}_N\Psi_{N,t}\ra= \la\Psi_{p,t}, a^{(p)}\Psi_{p,t}\ra_{L^2(\R^{3p})}\|\psi\|_{L^2(\R^{3})}^{2(N-p)} $$ \item Theorem \ref{mainth} was first proven in \cite{He}, see also [GiVe]. A new proof was given in \cite{Sp} and extended to more general classes of two-body potentials, including the Coulomb potential, in \cite{EY}, \cite{BGM}, \cite{BEGMY}. The proof in our paper is quite different. It is inspired by a second-quantization formalism to be published elsewhere. It enables us to tackle the problem of obtaining convergence estimates uniform in Planck's constant $\hbar$, as we now proceed to discuss. \end{enumerate} It is well known that, for $W_{N}$ as in (\ref{h0}), the classical dynamics of $N$ particles tends to the dynamics defined by the Vlasov equation, in the limit $N\to\infty$. More precisely, if $\rho_N$ denotes the empirical distribution, namely $$ \rho_N(dx,d\xi;t)=\frac1{N}\sum_{i=1}^N\,\delta(x-x_i(t))\delta(\xi-\xi_i(t))\,dxd\xi $$ where $(x_1(t),\ldots,x_N(t);\xi_1(t),\ldots,\xi_N(t))$ is a solution of the classical equations of motion, then, in the limit $N\to\infty$, $\rho_{N}$ tends weakly to $f_t(x,\xi) dx d\xi$, where $f_t(x,\xi)$ is a solution of the Vlasov equation: \begin{eqnarray} \label{Vlasov} \partial_{t} f_t & = & -\xi\cdot \nabla_{x} f_t + \nabla_{x} V_{eff} \cdot \nabla_{\xi} f_t\;\\ V_{eff} (x,t) & = &\int w(x-y) f_{t}(y,\xi) dyd\xi\;, \end{eqnarray} see \cite{BH}. It is natural to ask whether this convergence result is related to that of Theorem 1.1. Our next result provides, under very restrictive assumptions on the two-body interactions, a partial answer to this question. First, we define a restricted class of interactions. For $\sigma > 0$, we define the spaces \begin{eqnarray} L^1_{\sigma,p}&:=&\{f\in L^1(\R^{6p})\,|\,e^{\sigma|z|}f\in L^1(\R^{6p})\}, \\ {\cal A}_{\sigma,p}&:=&\{f\in L^1(\R^{6p})\,|\,e^{\sigma|s|}\widehat f \in L^1(\R^{6p})\}, \end{eqnarray} Here $x_j\in\R^3$, $\xi_j\in\R^3$, $j=1,\ldots,p$, and \newline $z:=(X_p,\Xi_p)\in\R^{3p}\times\R^{3p}$; $X_p:=(x_1,\ldots,x_p)$, $\Xi_p:=(\xi_1,\ldots,\xi_p)$, $$ |z|:=\sum_{j=1}^p(|x_j|+|\xi_j|); $$ $\widehat f(s), s:=(S,\Sigma)\in\R^{3p}\times\R^{3p}$ is the Fourier transform of $f$. We further denote by $\Phi^N_t: (X_N;\Xi_N)\mapsto (X_N(t);\Xi_N(t))$ the flow generated by $H_N^c$, where $H^c_N$ is the classical Hamilton function corresponding to the operator $H_N$. \begin{definition} We define by: \begin{enumerate} \item \begin{align} \label{W1} & W_N^{\Psi_N}(X_N,\Xi_N;t)= \\ \nonumber & (2\pi)^{-3N}\int_{\R^{3N}}e^{i\la Y_N,\Xi_N\ra}\Psi_N(X_N+\hbar Y_N/2,t) \overline{\Psi}_N(X_N-\hbar Y_N/2,t)\,dY_N \end{align} the {\rm Wigner distribution} of the $N$-particle normalized wave function $\Psi_N(X_N,t)$; \item \begin{eqnarray} \label{W2} W_j^{\Psi_N} (X_j,\Xi_j;t)=\int_{\R^{3(N-j)}}W_N^{\Psi_N} (X_N,\Xi_N;t)\,dX_{N-j}d\Xi_{N-j}, \end{eqnarray} the $j-${\rm particle Wigner function} ($(N-j)$-marginal distribution of the $N$-particle Wig\-ner distribution). \item \be \label{W3} W(\psi)(x,\xi;t)=(2\pi)^{-3}\int_{\R^{3}}e^{i\la y,\xi\ra}\psi_{t}(x+\hbar y/2)\overline{\psi_{t}}(x-\hbar y/2)\,dy, \ee the Wigner distribution of the solution $\psi_{t}(x)$ of the Hartree equation. \end{enumerate} \end{definition} \noindent Our second main result is \begin{theorem}. \label{unif} Let $w\in {\cal A}_{\sigma,1}$, for some $\sigma>0$. Let $\Psi_{N,0}$ be a product state. Set $\ep:=\|w\|_\infty \,t$. Then, for fixed $p$, there is a constant $C_p>0$ \underline{independent} of $\hbar$ such that, as an equality between tempered distributions, \be \label{uniff1} W_p^{\Psi_N} (X_p,\Xi_p;t)=\prod_{j=1}^pW(\psi)(x_j,\xi_j;t)+\frac{C_p}{N}+O\left(e^{-1/\sqrt{\ep}}\right), \ee as $N\to\infty$. \end{theorem} {\bf Remarks} \begin{enumerate} \item It is known that $W(\psi)(x,\xi;t)$ converges in ${\cal S}^\prime(\R^6)$ to a solution $f_{t}(x,\xi)$ of the Vlasov equation, as $\hbar\to 0$ \cite{NS}. It is also known that $$ \limN W_p^{\Psi_N} (X_p,\Xi_p;t)=\prod_{j=1}^p f_{t}(x_j,\xi_j) $$ whenever $N\to\infty$ entails $\hbar\to 0$, as in the case of the Kac potentials \cite{NS},\cite{GMP}. \item Result (\ref{uniff1}) shows that, up to an exponentially small error independent of $\hbar$, the mean-field convergence towards a single-particle nonlinear dynamics holds {\it uniformly} in $\hbar$. \item The classical limit is equivalent to the limit of heavy particles. We set $\hbar=1$ in (\ref{h0}), but let the particle mass $m$ become large. We impose the condition that the kinetic energy per particle be independent of $m$, namely $mv^2_i=O(1)$, i.e., $\ds |v_i|=O(1/\sqrt{m})$, for all $i$. This suggests to rescale time as $t=\sqrt{m}\tau$. Then the Schr\"odinger equation becomes $$ \frac{i}{\sqrt{m}}\partial_\tau\Psi_N=\sum_{j=1}^N -\frac{\Delta_j}{2m}\Psi_N+ \frac{1}{N}\sum_{i,j=1}^Nw(x_i-x_j)\Psi_N, $$ which is equivalent to (\ref{hn})-(\ref{sc}), for $\hbar=1/\sqrt{m}$. \end{enumerate} \vskip 1cm\noindent %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begsection{The $N\to\infty$ limit: convergence estimates} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Kinematical algebra of "observables"} The above systems can be described by a kinematical algebra of operators, the quantum mechanical analogue of the algebra of functions on phase space of a classical system. Let $\H^{(p)}:=L^2_S(\R^{3p})$, $0i=1}^{p+n-1} \int_0^t\ldots\int_0^{t_{n-1}} e^{iH_N^0 t_{n}/\hbar}e^{-iH_N t_{n}/\hbar} H(n-1;p;N) e^{iH_N t_{n}/\hbar}e^{-iH_N^0 t_{n}/\hbar} \,dt_{n}\ldots dt_1, \\ \label{Duhamel6} & R^{(p),k}_{t,N} = \int_0^t\ldots\int_0^{t_{k-1}} e^{iH_N^0 t_{n}/\hbar} e^{-iH_Nt_{n}/\hbar}H(k,p;N) e^{iH_N t_{n}/\hbar}e^{-iH_N^0 t_{n}/\hbar} \,dt_{k}\ldots dt_1, \end{align} with \be \label{HPN} H(s,p;N):=\sum_{ii=1}^{p+n-1}iA^{(p+n)}_{N}([w^{ij}_{t_n},g^{n-1,p}_{t_1,\ldots,t_{n-1}}])= A^{(p+n)}_{N}(g^{n,p}_{t_1,\ldots,t_{n}})+ \\ & \frac{1}{N\hbar}\sum_{j>i=1}^{p+n-1}iA^{(p+n)}_{N} ({[w_{t_n}^{ij},g^{(n-1;p)}_{t_1,\ldots,t_{n-1}}]}) \end{align*} %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% \subsection{Control of the expansion, small time, $\hbar$ fixed} First, we prove a bound on the norm of $A^{(p+n)}_{N}(g^{(n;p)}_{t_1,\ldots,t_{n}})$ \be \label{NormGn} \|A^{(p+n)}_{N}(g^{(n;p)}_{t_1,\ldots,t_{n}})\|_{\H^{(N)}}\leq \frac{2}{\hbar}(p+n-1)\|w\|_\infty\, \|A^{(p+n-1)}_{N}(g^{(n-1;p)}_{t_1,\ldots,t_{n-1}})\|_{\H^{(N)}} \ee This follows from the unitarity of the free time evolution and the boundedness of the interactions, $\|w^{ij}\|=\|w\|_\infty$. The bound (\ref{NormGn}) then yields recursively \begin{eqnarray} \label{normGn1} \nonumber \|A^{(p+n)}_{N}(g^{(n;p)}_{t_1,\ldots,t_{n}})\|_{\H^{(N)}}&\leq & (p+n-1)(p+n-2)\cdots (p+1)p \left(\frac{2}{\hbar}\|w\|_\infty\right)^n\|A^{(p)}_N\|_{\H^{(N)}} \\ &\leq & \frac{(p+n)!}{p!}\left(\frac{2}{\hbar}\|w\|_\infty\right)^n\|\ap\|_{\H^{(p)}}, \end{eqnarray} independently of all time indices. Considering the expansion (\ref{Duhamel3}), we have that \begin{align} \nonumber & \left\|\sum_{n=1}^\infty\int_0^t\ldots\int_0^{t_{n-1}} A^{(p+n)}_{N}(g^{(n;p)}_{t_1,\ldots,t_{n}})\,dt_n\cdots dt_1\right\|_{\H^{(N)}} \leq \\ \label{est1} & \sum_{n=1}^\infty\frac{t^n}{n!}\frac{(p+n)!}{p!}(\frac{2}{\hbar}\|w\|_\infty t)^n\|A^{(p)}_N\|_{\H^{(N)}} \leq 2^p \|\ap\|_{\H^{(p)}}\, \sum_{n=1}^\infty\left(\frac{4}{\hbar}\|w\|_\infty\,t\right)^n \end{align} because $\ds \frac{(p+n)!}{p!n!}\leq 2^{n+p}$. The series on the R.S. of (\ref{est1}) converges for $\ds |t|<\left(\frac{4}{\hbar}\|w\|_\infty\right)^{-1}$. The third term in (\ref{Duhamel3}) is bounded similarly. Let \be A^{(p+n)}_{N,I}(i,j):=e^{iH^0_Nt_n/\hbar}e^{-iH_Nt_n/\hbar}A^{(p+n)}_N([g^{(n-1;p)}_{t_1,\ldots,t_{n-1}},w^{ij}_{t_n}])e^{iH^0_Nt_n/\hbar}e^{-iH_Nt_n/\hbar} \ee Then \begin{align*} & \frac1{N}\sum_{n=1}^\infty\bigg \|\sum_{i0$, $\forall\delta>0$. Let now $g_r:=\{g_{r-1},w\}_M, r>1; g_1=\{g,w\}_M$. Then, applying (\ref{M1}) $r$ times, we can write: \be \label{M2} \|g_r\|_{\sigma-r\delta}\leq \left(\frac{1}{e^2\delta^2}\right)^r\|w\|_\sigma^r\,\|g\|_\sigma \ee These results immediately yield the following bound. \begin{lemma}. \label{lemma1} Let the operator $\ap$ be the Weyl quantization of a symbol $\tau_a(x,\xi)\in {\cal A}_{\sigma,p}$ for some $\sigma >0$. Then there is $L(p)>0$ independent of $\hbar$ such that \be \label{unif1} \|A^{(p+n)}_N(g^{(n;p)}_{t_1,\ldots,t_{n}})\|_{\H^{(N)}}\leq L^n\,n!^3\,(2\|w\|_\sigma)^n\|\ap\|_\sigma \ee \end{lemma} {\bf Proof} \noindent Denote by ${\cal G}^{(n,p)}_{t,t_1,\ldots,t_{n}}$ the symbol of $g^{(n;p)}_{t_1,\ldots,t_{n}}$. Using definition (\ref{treen}) and the estimate (\ref{M1}) we get the uniform estimate corresponding to (\ref{normGn1}): \be \label{unif2} \|{\cal G}^{(n,p)}_{t,t_1,\ldots,t_{n};N}\|_{\sigma-n\delta_n}\leq \frac{2(p+n-1)}{e^2\delta_n^2}\|w\|_\sigma\, \|{\cal G}^{(n-1,p)}_{t,t_1,\ldots,t_{n-1}}\|_\sigma , \;\;0<\delta_n<\sigma \ee The recursive definition (\ref{treen}) allows us to use the recursive estimate (\ref{M2}). We get \begin{align} \label{unif3} \|{\cal G}^{(n,p)}_{t,t_1,\ldots,t_{n}}\|_{\sigma-n\delta_n} & \leq 2^n ({e^2\delta_n^2})^{-n}\frac{(p+n)!}{p!}\|w\|_\sigma^n\|a^{(p)}\|_{\sigma} \end{align} Setting $\ds \delta_n:=\frac{1}{2n}$ we get the bound (\ref{unif1}) on account of the majorizations $$ \|A^{(p+n)}_N(g^{(n;p)}_{t_1,\ldots,t_{n}})\|_{\H^{(N)}}\leq \|{\cal G}^{(n,p)}_{t,t_1,\ldots,t_{n}}\|_{\sigma/2}, \quad \|w\|_{L^2\to L^2 }=\|w\|_\infty\leq \|w\|_\sigma. $$ This proves the Lemma. \vskip 0.3cm\noindent {\bf Remark} \newline The uniform control in $\hbar$ introduces an extra $n!^2$ divergence with respect to the fixed-$\hbar$ estimate (\ref{normGn1}). \vskip 0.3cm\noindent We now obtain uniform estimates of the three terms in expansion (\ref{Duhamel3}). \begin{lemma}. \label{lemma2} There exist constants $M_1>0, M_2>0, M_3>0, L_1>0, L_2>0, L_3>0$, independent of $(\hbar,t)$ and $N$, such that \begin{eqnarray} \label{unif4} \|B_{t,N}^{(p),k}\|_{\H^{(N)}}&\leq& M_1\|\ap\|_\sigma\sum_{n=1}^kL_1^n\,n!^2(\|w\|_\sigma t)^n \\ \label{unif5} \|Q_{t,N}^{(p),n} \|_{\H^{(N)}}&\leq& M_3\|\ap\|_\sigma L_2^n\,n!^2(\|w\|_\sigma\,t)^n \\ \label{unif6} \|R_{t,N}^{(p),k}\|_{\H^{(N)}} &\leq& M_3\|\ap\|_\sigma L_3^k\,k!^2(\|w\|_\sigma\,t)^k \end{eqnarray} \end{lemma} {\bf Proof} \noindent Inserting the estimate (\ref{unif1}) in the expressions (\ref{Duhamel4},\ref{Duhamel5}, \ref{Duhamel6}) we get, on account of unitarity of $U_0(t)$: \begin{eqnarray*} \|B_{t,N}^{(p),k}\|_{\H^{(N)}}&\leq & \|\ap\|_\sigma\sum_{n=1}^{k}\,(2L\|w\|_\sigma)^n\,n!^3\,\int_0^t\ldots\int_0^{t_{n-1}}\,dt_n\ldots dt_1 \\ &\leq & \|\ap\|_\sigma \,\sum_{n=1}^{k}\,(2L\|w\|_\sigma |t|)^n\,n!^2. \end{eqnarray*} The last inequality comes from performing the time integrations, which are majorized by a factor $|t|^n/n!$ in (\ref{unif4},\ref{unif5}) and by a factor $|t|^k/k!$ in (\ref{unif6}) (proven with the help of the same argument). This proves the lemma. \vskip 0.3cm\noindent Using this result, we can easily prove the uniform version of the expansion (\ref{Duhamel3}). \begin{proposition}. \label{repre} Let $\epsilon:=\|w\|_\infty t$. Then, in the same assumption of Lemma \ref{lemma1} on the operator $\ap$, there exists $k=k(\ep)$, $\Lambda=\Lambda(\ep)$ such that \begin{eqnarray} \label{rep1} e^{iH_Nt/\hbar}A^{(p)}_Ne^{-iH_Nt\hbar}&=&A^{(p)}_{t,N}+B_{t,N}^{(p),k}+ R_{t,N}^{(p),k} +\frac{\Lambda}{N}, \end{eqnarray} where \be \label{unifbis} B_{t,N}^{(p),k}=\sum_{n=1}^k\int_0^t\ldots\int_0^{t_{n-1}}A^{(p+n)}_N(g^{(n;p)}_{t_1,\ldots,t_{n}})\,dt_n\ldots dt_1. \ee Here $B_{t,N}^{(p),k}$ fulfills the majorization (\ref{unif4}), and \be \label{unif7} \|R_{t,N}^{(p),k}\|_{\H^{(N)}}\leq M_3e^{-L_3/\sqrt{\epsilon}} \ee \end{proposition} {\bf Proof} \noindent The estimate (\ref{unif5}) and a standard Nekhoroshev-type argument show that the choice \be \label{kopt} k(\ep):=\frac1{\sqrt{\ep}}=\frac{1}{\|w\|_\infty\,t} \ee minimizes the divergence of $R_{t,N}^{(p),k}$. A straightforward computation then yields (\ref{unif7}). By definition of $Q^{(p),N}_{t,N}$ we get the uniform version of the estimate (2.21), whence $$ \Lambda(\ep):=p2^p\|\ap\|_{\sigma}\sum_{n=1}^{k(\ep)}n!^2(2\ep)^n\leq p2^p\ep^{-1/2}(e/2)^{-1/\sqrt{\ep}} $$ %%%%%% %%%%%%%% \vskip 1.0cm\noindent \begsection{Connection with the Hartree equation and proof of the theorems} We wish to prove that the representation of the evolution obtained in Proposition \ref{conv1} coincides with the evolution generated by the Hartree equation in the limit $N\to\infty$ . \noindent For this purpose, we recall that the Hartree equation is Hamiltonian. We define the functional \be \label{Ham} {\cal H}(\psi,\overline\psi)=-\frac{\hbar^2}{2} \int_{\R^3}|\nabla\psi(x)|^2\,dx+ {\cal W}(\psi,\overline\psi), \ee for $\psi\in H^{1}(\R^{3})$, where \be {\cal W}(\psi,\overline\psi)=\frac12\int_{\R^3\times\R^3}\psibx\psiby w(x-y)\psi(x)\psi(y)\,dxdy \ee If $\psi(x), \psiby$ are considered as canonical variables with Poisson brackets $$ \{\psi(x),\psiby)\}=i\hbar\delta(x-y), \quad \{\psi(x),\psi (y)\}= \{\psibx,\psiby\}=0, $$ then (\ref{Ham}) is the Hamiltonian functional generating a time evolution of fun\-ctionals on phase space equivalent to the Hartree equation. Namely, if ${\cal A}(\psi)$ is a functional and ${\cal A}_t$ denotes its time evolution, one has that $$ \partial_t{\cal A}_t(\psi)=\frac{1}{\hbar}\{{\cal H},{\cal A}_t\}(\psi) $$ Choosing ${\cal A}=\langle\phi,\psi\rangle$, $\phi\in C_0^\infty(\R^3)$, then ${\cal A}_t(\psi)={\cal A}(\psi_t)$, where $\psi_t$ is a solution of the Hartree equation \be \label{Har2} i\hbar\partial_t \psi_{t}=-\frac{\hbar^2}{2}\Delta\psi_{t}+(w\ast|\psi_{t}|^{2})\psi_{t} \ee \noindent Define the free flow $\Phi_t^0({\cal A}):={\cal A}_t$ of ${\cal A}$ by $$ {\cal A}_t={\cal A}(e^{i\Delta t/\hbar}\psi) $$ and denote by $\Phi_t({\cal A})$ the interacting flow. Formally, the interacting flow is given by the Lie expansion in the interaction representation (analogous to the Schwinger-Dyson expansion of Section 2.2). Indeed we have the following result: \par\noindent \begin{lemma}. \label{lLie} $\Phi_t({\cal A})$ {\it admits the formal expansion} \be \label{Lie} \Phi_t({\cal A})={\cal A}_t+\Phi^0_t\left(\sum_{n=1}^\infty \left(\frac1{\hbar}\right)^n\int_0^t\ldots \int_0^{t_n}\{ {\cal W}_{t_n}\ldots \{{\cal W}_{t_1},{\cal A}\}\ldots\}\,dt_n\ldots dt_1\right) \ee \end{lemma} {\bf Proof.} To see this, we consider the dynamics in the interaction picture. We set $$ \tilde{{\cal A}}_t := \Phi_t\circ \Phi^0_{-t}({\cal A}) $$ Then $$ \partial_t \tilde{{\cal A}}_t=\frac{\{\tilde{\cal A}_t,{\cal W}_{t}\}}{\hbar} $$ where ${\cal W}_t:=\Phi^0_t({\cal W})$ is the free evolution of ${\cal W}$. After integrating in time we get $$ \tilde{{\cal A}}_t={\cal A}+ \int_0^t\frac{\{{\cal W}_{s},{\cal A}\}}{\hbar}\,ds $$ whence $$ \Phi_t({{\cal A}})={\cal A}_t+ \Phi^0_t\left(\int_0^t\frac{\{{\cal W}_{s},{\cal A}\}}{\hbar}\,ds\right) $$ Iterating this identity, we obtain the series (\ref{Lie}), and this concludes the proof of the Lemma. \par\noindent The desired identification is based on the following proposition \begin{proposition}. Let $\psi\in H^1(\R^3)$, and let $\Psi$ be a product state, i.e. $$ \Psi(x_1,\ldots,x_l)=\prod_{s=1}^l\psi(x_s) $$ Then, for all $N\geq p$, \begin{align} \label{equality} g^{(n;p)}_{t_1,\ldots,t_{n}}(\psi)&:=\frac{(p+n)^{p+n}}{(p+n)!}\la\Psi_{n+p},A^{(p+n)}_{p+n}(g^{(n;p)}_{t_1,\ldots,t_{n}})\Psi_{n+p}\ra_{\H^{(n+p)}} = \\ \nonumber & \left(\frac1{\hbar}\right)^n\{{\cal W}_{t_n}\ldots\{ {\cal W}_{t_1},{\cal A}\}\ldots\}(\psi), \end{align} where $$ {\mathcal A}=\ap(\psi):=\int\,\overline{\psi(x_1)}\cdots\overline{\psi(x_p)}\alpha^{(p)}(x_1,\ldots,x_p;y_1,\ldots,y_p)\overline{\psi(y_1)}\cdots\overline{\psi(y_p)}\prod_{k=1}^p\,dx_kdy_k $$ \end{proposition} {\bf Proof.} \noindent We have that $$ \left.\Phi_t\circ \Phi^0_{-t}(\{{\cal W}_t,{\cal A}\}/\hbar)\right|_{t=0}=\{{\cal W},{\cal A}\}/\hbar=\partial_t\tilde{{\cal A}}|_{t=0} $$ Define the projection $\rho:=|\psi\rangle\langle\psi|$. Then, denoting $\tilde{\rho}_t:=\Phi_t\circ \Phi^0_{-t}({\rho})$, we have that, for all $n\geq 1$, $$ \tilde{{\cal A}}_t={\rm Tr}({\cal A}\tilde{\rho}_t^{\otimes n});\quad \partial_t\tilde{{\cal A}}|_{t=0}=\partial_t{\rm Tr}({\cal A}\tilde{\rho}_t^{\otimes n})|_{t=0}. $$ Therefore, in the interaction picture $$ \partial_t\tilde{\rho}_t={\rm Tr}(\tilde{\rho}_t^{\otimes 2}{\cal W}^{12}_t- {\cal W}^{12}_t\tilde{\rho}_t^{\otimes 2})/\hbar $$ and in the same way we get $$ \frac1{\hbar}\{ {\cal W}_t, {\cal A}\}=\sum_{i=1}^n{\rm Tr}(({\cal W}_t^{in+1}{\cal A}-{\cal A}{\cal W}_t^{in+1})\rho^{\otimes\, n+1}) $$ It is then easy to check that \begin{eqnarray*} \frac1{\hbar^n}\{{\cal W}_{t_n}\ldots\{ {\cal W}_{t_1},{\cal A}\}\ldots\}(\psi)&=&{\rm Tr}(g^{(n;p)}_{t_1,\ldots,t_{n}} \cdot \rho^{\otimes\, n+p}) \\ &=&\frac{(p+n)^{p+n}}{(p+n)!} \langle\Psi_{n+p},A^{(p+n)}_{p+n}(g^{(n;p)}_{t_1,\ldots,t_{n}})\Psi_{n+p}\rangle_{\H^{(n+p)}} \\ &=& g^{(n;p)}_{t_1,\ldots,t_{n}}(\psi), \end{eqnarray*} and this concludes the proof of the Proposition. \vskip 0.3cm\noindent We are now in a position to prove our main results. \par\noindent {\bf Proof of Theorem \ref{mainth}} \noindent Consider the expectation value of the expansion (\ref{exp1}) in a coherent (i.e., product) state: \begin{align*} & \la \Psi_N,e^{iH_Nt/\hbar}A_N^{(p)}e^{-iH_Nt/\hbar}\Psi_N\ra_{\H^{(N)}}=\la\Psi_N,A_{t,N}^{(p)}\Psi_N\ra_{\H^{(N)}}+ \\ &\sum_{n=1}^{N}\int_0^t\ldots\int_0^{t_{n-1}}\la\Psi_N,A^{(p+n)}_N(g^{(n;p)}_{t_1,\ldots,t_{n}})\Psi_N\ra_{\H^{(N)}}\,dt_n \cdots dt_1+O(1/N). \end{align*} By definition of $\Psi_N$, \begin{align*} &\frac{N^p}{N(N-1)\ldots (N-p+1)}\la \Psi_N,e^{iH_Nt/\hbar}A_Ne^{-iH_Nt/\hbar}\Psi_N\ra_{\H^{(N)}} =\la\Psi_p,a_t^{(p)}\Psi_p\ra_{\H^{(p)}}+ \\ & +\sum_{n=1}^{\infty}\int_0^t\ldots\int_0^{t_{n-1}} \la\Psi_{n+p},g^{(n;p)}_{t_1,\ldots,t_{n}}\Psi_{n+p}\ra_{\H^{(n+p)}}\,dt_n\cdots dt_1+O(1/N). \end{align*} Since the series is norm- convergent, the limits $N\to\infty$ and $n\to\infty$ can be interchanged. Then \begin{eqnarray} \label{final} \limN \la\Psi_N,e^{iH_Nt/\hbar}A_Ne^{-iH_Nt/\hbar}\Psi_N\ra_{\H^{(N)}}= \la\Psi_p,a_t^{(p)}\Psi_p\ra_{\H^{(p)}}+\qquad\qquad\\ \nonumber \sum_{n=1}^{\infty}\int_0^t\ldots\int_0^{t_{n-1}} \la\Psi_{n+p},g^{(n;p)}_{t_1,\ldots,t_{n}}\Psi_{n+p}\ra_{\H^{(n+p)}}\,dt_n\cdots dt_1= \ap(\psi_t), \end{eqnarray} where the last equality follows from formula (\ref{equality}). \vskip 0.3cm\noindent {\bf Proof of Theorem \ref{unif}} \noindent If, instead of (\ref{exp1}), the representation (\ref{rep1}) is considered, the above argument yields \begin{align} \label{final1} &\frac{N^p}{N(N-1)\ldots (N-p+1)}\la \Psi_N,e^{iH_Nt/\hbar}A_N^{(p)}e^{-iH_Nt/\hbar}\Psi_N\ra= \la\Psi_p,a^{(p)}_t\Psi_p\ra+\la\Psi_N,R^{k,p}_{t,N}\psi_N\ra \\ \nonumber &+\sum_{n=1}^{k(\ep)}\int_0^t\ldots\int_0^{t_{n-1}} \la\Psi_{n+p},g^{(n;p)}_{t_1,\ldots,t_{n}}\Psi_{n+p}\ra\,dt_n\cdots dt_1= a^{(p)}(\psi_t)+O(e^{-1/\sqrt{\epsilon}}). \end{align} Given any bounded operator $A$ on $L^2(\R^{3l})$ with (Weyl) symbol $\sigma_A(x,\xi): {\cal S}(\R^{6l})\to\R$, where ${\mathcal S}$ is the Schwartz space of rapidly decreasing functions, its matrix elements can be expressed in terms of the symbol and of the Wigner function by the following well known formula (see e.g.\cite{Fo}): \be \label{W4} \la \Psi,A\Psi\ra=\int_{\R^{3l}\times\R^{3l}}\sigma_A(x,\xi)W_\Psi(x,\xi)\, dxd\xi \ee where $W_\Psi(x,\xi)$ is the Wigner function of the state $\Psi$. Therefore, in our case $$ \frac{N^p}{N(N-1)\ldots (N-p+1)}\la \Psi_N,A_N(t)\Psi_N\ra=\int_{\R^{3p}\times\R^{3p}}\sigma_A(X_p,\Xi_p)W_N^{\Psi_N}(X_p,\Xi_p,t)\, dX_pd\Xi_p, $$ where $\ds W_N^{\Psi_N}(X_p,\Xi_p,t)$ is the Wigner function corresponding to the time evolution, \linebreak $\ds e^{iH_N t/\hbar}\Psi_N$, of the product state $\Psi_{N,0}=\psi(x_1)\ldots\psi(x_N)$. The $N-p$ variables $(X_{N-p},\Xi_{N-p})$ are integrated out. By (\ref{final1}) and (\ref{W3}), we can take the $N\to\infty$ limit and write \begin{eqnarray*} \int_{\R^{3p}\times\R^{3p}}\sigma_A(X_p,\Xi_p)W_N^{\Psi_N}(X_p,\Xi_p,t)\, dX_pd\Xi_p= \\ \int_{\R^ {3p} \times \R^ {3p} } \, \sigma_A(X_p,\Xi_p) \prod_{l=1}^p W_{\psi}(x_l,\xi_l;t) \, dX_pd\Xi_p +O(e^{ -1/\sqrt{\ep}}). \end{eqnarray*} Since this formula holds for any $\sigma_A(X_p,\Xi_p)\in {\cal S}(\R^{3p}\times\R^{3p})\cap {\mathcal A}_{\sigma,p}$, the assertion is proved. \par\noindent {\bf Proof of formula (\ref{global})}. \par\noindent By (\ref{equality}), we have that \begin{align*} & \la\Psi_N,e^{iH_N T/\hbar} A^{(p+n)}_N(g^{(n;p)}_{t_1,\ldots,t_{n}})e^{-iH_N T/\hbar}\Psi_N\ra_{\H^{(N)}} = \\ & \left(\frac1{\hbar}\right)^n\{{\cal W}_{t_n}\ldots\{{\cal W}_{t_1},{\cal A}\}\ldots\}(\psi_T) \end{align*} which yields formula (\ref{global}), by Lemma \ref{lLie}, on account of the uniform convergence of the series. \vskip 1.0cm\noindent %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vfill\eject \begin{thebibliography}{DDDDD} \vphantom{Chhosing} {\small \bibitem[BGM]{BGM} C. 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