Content-Type: multipart/mixed; boundary="-------------0804231353868" This is a multi-part message in MIME format. ---------------0804231353868 Content-Type: text/plain; name="08-81.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="08-81.comments" 11 pages ---------------0804231353868 Content-Type: text/plain; name="08-81.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="08-81.keywords" many body quantum dynamics; mean field limit; Hartree equation ---------------0804231353868 Content-Type: application/x-tex; name="mf.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="mf.tex" \documentclass[11pt]{article} \usepackage{amsmath, amssymb, amsfonts, amsthm} \setlength{\topmargin}{-0.5in} \setlength{\textheight}{9in} \setlength{\oddsidemargin}{-.1in} \setlength{\textwidth}{6.6in} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \renewcommand{\theequation}{\thesection.\arabic{equation}} \newcommand{\myendproof}{\hspace*{\fill}{{\bf \small Q.E.D.}} \vspace{10pt}} \newcommand{\tri}{| \! | \! |} \newcommand{\rd}{{\rm d}} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\bey}{\begin{eqnarray}} \newcommand{\eey}{\end{eqnarray}} \newcommand{\Htrap}{H_N^{\text{trap}}} \newcommand{\Vtrap}{V_{\text{ext}}} \newcommand{\psitrap}{\psi_N^{\text{trap}}} \newcommand{\phitrap}{\phi_{\text{GP}}^{\text{trap}}} \newcommand{\uy}{\underline{y}} \newcommand{\sfrac}[2]{{\textstyle \frac{#1}{#2}}} \newcommand{\eps}{\varepsilon} \newcommand{\bc}{{\bf c}} \newcommand{\bw}{{\bf w}} \newcommand{\bp}{{\bf p}} \newcommand{\bq}{{\bf q}} \newcommand{\br}{{\bf r}} \newcommand{\bv}{{\bf v}} \newcommand{\bu}{{\bf u}} \newcommand{\bk}{{\bf k}} \newcommand{\bm}{{\bf m}} \newcommand{\bx}{{\bf x}} \newcommand{\by}{{\bf y}} \newcommand{\bz}{{\bf z}} \newcommand{\bh}{{\bf h}} \newcommand{\bK}{{\bf K}} \newcommand{\bn}{{\bf n}} \newcommand{\bX}{{\bf X}} \newcommand{\bV}{{\bf V}} \newcommand{\bY}{{\bf Y}} \newcommand{\bA}{{\bf A}} \newcommand{\ve}[1]{\bf #1} \newcommand{\ax}{{\langle x \rangle}} \newcommand{\axi}{{\langle \xi \rangle}} \newcommand{\axil}{{\langle \xi_l \rangle}} \newcommand{\axij}{{\langle \xi_j \rangle}} \newcommand{\axik}{{\langle \xi_k \rangle}} \newcommand{\aeta}{{\langle \eta \rangle}} \newcommand{\aetal}{{\langle \eta_l \rangle}} \newcommand{\aetaj}{{\langle \eta_j \rangle}} \newcommand{\aetak}{{\langle \eta_k \rangle}} \newcommand{\ay}{{\langle y \rangle}} \newcommand{\aq}{{\langle q \rangle}} \newcommand{\at}{{\langle t \rangle}} \newcommand{\tU}{{\widetilde U}} \newcommand{\tJ}{{\widetilde J}} \newcommand{\tA}{{\widetilde A}} \newcommand{\tb}{{\tilde b}} \newcommand{\tc}{{\tilde c}} \newcommand{\tk}{{\tilde k}} \newcommand{\tp}{{\tilde p}} \newcommand{\tq}{{\tilde q}} \newcommand{\tu}{{\tilde u}} \newcommand{\tv}{{\tilde v}} \newcommand{\ts}{{\tilde s}} \newcommand{\tn}{{\tilde n}} \newcommand{\tell}{{\tilde\ell}} \newcommand{\tI}{{\tilde I}} \newcommand{\tN}{{\tilde N}} \newcommand{\tm}{{\widetilde m}} \renewcommand{\tt}{t} \newcommand{\ttt}{{\tilde t}} \newcommand{\ta}{{\tilde\alpha}} \newcommand{\tbeta}{{\tilde\beta}} \newcommand{\tsi}{{\tilde\sigma}} \newcommand{\ttau}{{\tilde\tau}} \newcommand{\tka}{{\tilde\kappa}} \newcommand{\teta}{{\tilde\eta}} \newcommand{\tchi}{{\tilde\chi}} \newcommand{\ph}{\varphi} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\tbk}{{\tilde{\bf k}}} \newcommand{\tbp}{{\tilde{\bf p}}} \newcommand{\tbq}{{\tilde{\bf q}}} \newcommand{\tbr}{{\tilde{\bf r}}} \newcommand{\tbv}{{\tilde{\bf v}}} \newcommand{\tbu}{{\tilde{\bf u}}} \newcommand{\tbK}{{\widetilde{\bf K}}} \newcommand{\tmu}{{\tilde\mu}} \newcommand{\tbm}{{\widetilde{\bf m}}} \newcommand{\tbh}{{\tilde{\bf h}}} \newcommand{\balpha}{{\boldsymbol \alpha}} \renewcommand{\a}{\alpha} \renewcommand{\b}{\beta} \newcommand{\g}{\gamma} \newcommand{\e}{\varepsilon} \newcommand{\s}{\sigma} \newcommand{\om}{{\omega}} \newcommand{\ka}{\kappa} \newcommand{\bU}{{\bf U}} \newcommand{\bE}{{\bf E}} \newcommand{\cU}{{\cal U}} \newcommand{\cM}{{\cal M}} \newcommand{\cX}{{\cal X}} \newcommand{\bR}{{\mathbb R}} \newcommand{\bC}{{\mathbb C}} \newcommand{\bN}{{\mathbb N}} \newcommand{\bZ}{{\mathbb Z}} \newcommand{\bi}{\bigskip} \newcommand{\noi}{\noindent} \newcommand{\tr}{\mbox{Tr}} \newcommand{\sgn}{\mbox{sgn}} \newcommand{\wt}{\widetilde} \newcommand{\wh}{\widehat} \newcommand{\ov}{\overline} \newcommand{\bxi}{{\boldsymbol \xi}} \newcommand{\bzeta}{\mbox{\boldmath $\zeta$}} \newcommand{\bpsi}{\mbox{\boldmath $\psi$}} \newcommand{\bnabla}{\mbox{\boldmath $\nabla$}} \newcommand{\btau}{\mbox{\boldmath $\tau$}} \newcommand{\tbtau}{\widetilde{\btau}} \newcommand{\const}{\mathrm{const}} \newcommand{\cG}{{\cal G}} \newcommand{\cS}{{\cal S}} \newcommand{\cC}{{\cal C}} \newcommand{\cF}{{\cal F}} \newcommand{\cA}{{\cal A}} \newcommand{\cB}{{\cal B}} \newcommand{\cE}{{\cal E}} \newcommand{\cP}{{\cal P}} \newcommand{\cD}{{\cal D}} \newcommand{\cV}{{\cal V}} \newcommand{\cW}{{\cal W}} \newcommand{\cK}{{\cal K}} \newcommand{\cH}{{\cal H}} \newcommand{\cL}{{\cal L}} \newcommand{\cN}{{\cal N}} \newcommand{\cO}{{\cal O}} \newcommand{\cJ}{{\cal J}} \newcommand{\A}{\left( \int \, W^2 | \nabla_m \nabla_k \phi|^2 \, + N \int \, W^2 |\nabla_m \phi|^2 + N^2 \int \, W^2 |\phi|^2 \right)} \newcommand{\B}{\left(\int \, W^2 |\nabla_m \phi|^2 + N \int \, W^2 |\phi|^2 \right)} \newcommand{\C}{\left( \int \, W^2 | \nabla_m \nabla_k \phi|^2 \, + N \int \, W^2 |\nabla_m \phi|^2 \right)} \newcommand{\Ci}{\left( \int \, W^2 | \nabla_i \nabla_j \phi|^2 \, + N \int \, W^2 |\nabla_i \phi|^2 \right)} \newcommand{\usi}{\underline{\sigma}} \newcommand{\ubp}{\underline{\bp}} \newcommand{\ubk}{\underline{\bk}} \newcommand{\utbp}{\underline{\tbp}} \newcommand{\utbk}{\underline{\tbk}} \newcommand{\um}{\underline{m}} \newcommand{\utm}{\underline{\tm}} \newcommand{\utau}{\underline{\tau}} \newcommand{\uttau}{\underline{\ttau}} \newcommand{\uxi}{\underline{\xi}} \newcommand{\ueta}{\underline{\eta}} \newcommand{\ux}{\underline{x}} \newcommand{\uv}{\underline{v}} \newcommand{\ualpha}{\underline{\alpha}} \newcommand{\Om}{\Omega} \newcommand{\Hn}{\cH^{\otimes n}} \newcommand{\Hsn}{\cH^{\otimes_s n}} \newcommand{\thi}{ \; | \!\! | \!\! | \;} \newcommand{\supp}{\operatorname{supp}} \newcommand{\bfeta}{{\boldsymbol \eta}} \newcommand{\no}{\nonumber} \renewcommand{\thefootnote}{\arabic{footnote}} \input epsf \def\req#1{\eqno(\hbox{Requirement #1})} \newcommand{\fh}{{\frak h}} \newcommand{\tfh}{\wt{\frak h}} \newcommand{\donothing}[1]{} \begin{document} \title{Quantum Dynamics with Mean Field Interactions: \\a New Approach} \author{L\'aszl\'o Erd\H os\thanks{Partially supported by SFB/TR12 Project from DFG} \; , Benjamin Schlein\thanks{Supported by a Kovalevskaja Award from the Humboldt Foundation. On leave from Cambridge University} \\ \\ Institute of Mathematics, University of Munich, \\ Theresienstr. 39, D-80333 Munich, Germany} \maketitle \begin{center} {\it {\large Dedicated to J\"urg Fr\"ohlich on the occasion of his 60th birthday, \\ with admiration and gratitude}} \end{center} \begin{abstract} We propose a new approach for the study of the time evolution of a factorized $N$-particle bosonic wave function with respect to a mean-field dynamics with a bounded interaction potential. The new technique, which is based on the control of the growth of the correlations among the particles, leads to quantitative bounds on the difference between the many-particle Schr\"odinger dynamics and the one-particle nonlinear Hartree dynamics. In particular the one-particle density matrix associated with the solution to the $N$-particle Schr\"odinger equation is shown to converge to the projection onto the one-dimensional subspace spanned by the solution to the Hartree equation with a speed of convergence of order $1/N$ for all fixed times. \end{abstract} \section{Introduction} We consider a system of $N$ interacting bosons in $\nu$ dimensions described on the Hilbert space $\cH_N = L^2_s (\bR^{N\nu}, \rd x_1 ,\dots \rd x_N)$, the subspace of $L^2 (\bR^{N\nu}, \rd x_1\dots \rd x_N)$ consisting of permutation symmetric wave functions. Here the variables $x_1, \dots, x_N \in \bR^{\nu}$ refer to the positions of the $N$ particles. The Hamiltonian is given by \begin{equation} H_N = \sum_{j=1}^N \left( -\Delta_{x_j} + U (x_j) \right) + \frac{1}{N} \sum_{in} V (x_{\ell} - x_j) \, . \] With respect to the dynamics generated by $H_N^{(n)}$, particles $1,\dots,n$ are decoupled from the rest of the system. In particular this implies that $e^{-i H_N^{(n)} t} B^{[1,\dots ,n]} e^{i H_N^{(n)} t}$ is still an operator acting only on the degrees of freedom of particles $1, \dots ,n$, with norm equal to the norm of $B$. Therefore, we have \begin{equation}\label{eq:f2} f_{m,n} (t) = \sup_{A \in \cB_m , B \in \cB_n} \frac{\left\| \left[ A^{[n+1,\dots,n+m]} , e^{iH_N t} e^{-i H^{(n)}_N t} B^{[1, \dots, n]} e^{iH_N^{(n)} t} e^{-iH_N t} \right] \right\|}{\| A \| \| B \|} \, . \end{equation} For given $A \in \cB_m$ and $B \in \cB_n$ we define the time-dependent bounded operator acting on $L^2 (\bR^{\nu N})$ \[ g_{A,B} (t) = \left[ A^{[n+1, \dots ,n+m]} , e^{iH_N t} e^{-i H^{(n)}_N t} B^{[1,\dots ,n]} e^{iH_N^{(n)} t} e^{-iH_N t} \right] \] and we compute its time-derivative \begin{equation*} \begin{split} \frac{\rd}{\rd t} g_{A,B} (t) = \; &\left[ A^{[n+1, \dots ,n+m]} , e^{iH_N t} \left[ i (H_N - H_N^{(n)}) , e^{-i H^{(n)}_N t} B^{[1,\dots , n]} e^{iH_N^{(n)} t}\right] e^{-iH_N t} \right] \\ = \; & \left[ A^{[n+1, \dots ,n+m]} , \left[ i \,e^{iH_N t} (H_N - H_N^{(n)}) e^{-iH_N t} , e^{iH_N t} e^{-i H^{(n)}_N t} B^{([1,\dots ,n]} e^{iH_N^{(n)} t} e^{-iH_N t} \right] \right] \\ = \; & \left[ i \, e^{iH_N t} (H_N - H_N^{(n)}) e^{-iH_N t} , g_{A,B} (t) \right] \\ & + \left[ e^{iH_N t} e^{-i H^{(n)}_N t} B^{[1,\dots,n]} e^{iH_N^{(n)} t}e^{-iH_N t} , \left[ A^{[n+1, \dots,n+m]}, e^{iH_N t} (H_N - H_N^{(n)}) e^{-iH_N t} \right] \right] \end{split} \end{equation*} where, in the last step we used the Jacobi identity. Next,we define \[ \cH^{(n)} (t) = e^{iH_N t} (H_N - H_N^{(n)}) e^{-iH_N t} \, . \] It is simple to see that $\cH^{(n)} (t)$ generates a two-parameter group of unitary transformations $\cU^{(n)} (t,s)$ satisfying \[ i\partial_t \cU^{(n)} (t,s) = \cH^{(n)} (t) \cU^{(n)} (t,s), \qquad \text{with } \cU^{(n)} (s,s) = 1 \quad \text{for all } s \in \bR \, .\] Therefore, we obtain \begin{equation} \begin{split} \frac{\rd}{\rd t} &\; \cU^{(n)} (0,t) \, g_{A,B} (t) \, \cU^{(n)} (t,0) \\ &= \cU^{(n)} (0,t) \, \left[ e^{iH_N t} e^{-i H^{(n)}_N t} B^{[1,\dots ,n]} e^{iH_N^{(n)} t}e^{-iH_N t} , \right. \\ &\left. \hspace{3cm} \left[ A^{[n+1, \dots ,n+m]}, e^{iH_N t} (H_N - H_N^{(n)}) e^{-iH_N t} \right] \right] \, \cU^{(n)} (t,0) \, . \end{split} \end{equation} Integrating this identity from time $0$ to time $t$, using that $g_{A,B} (0) = 0$ and the definition of $H^{(n)}_N$, we find that \begin{equation} \begin{split} g_{A,B} (t) = \frac{1}{N} \sum_{\ell =1}^n \sum_{j > n} \int_0^t \rd s \; \cU^{(n)} &(t,s) \left[ e^{iH_N s} e^{-i H^{(n)}_N s} B^{[1,\dots ,n]} e^{iH_N^{(n)} s} e^{-iH_N s} , \right. \\ &\left. \left[ A^{[n+1, \dots, n+m]}, e^{iH_N s} V(x_{\ell} - x_j) e^{-iH_N s} \right] \right] \cU^{(n)} (s,t)\,. \end{split}\end{equation} Taking norms, we get \begin{equation*} \begin{split} \| g_{A,B} (t) \| \leq \; &\frac{2}{N} \sum_{\ell=1}^n \sum_{j > n} \int_0^t \rd s \,\| B \| \, \left\| \left[ A^{[n+1, \dots ,n+m]}, e^{iH_N s} V(x_{\ell} - x_j) e^{-iH_N s} \right] \right\| \\ \leq \; & \frac{4\, mn \, t \, \| A \| \, \| B \| \, \| V \|}{N} + 2n \| B \| \int_0^t \rd s \, \left\| \left[ A^{[n+1, \dots, n+m]} , e^{iH_N s} V (x_1 -x_{n+m+1}) e^{-iH_N s} \right] \right\| \end{split} \end{equation*} where the first term on the last line corresponds to the terms with $j=n+1, \dots , n+m$, while in the second term we used the permutation symmetry. {F}rom (\ref{eq:f1}) and (\ref{eq:f2}), it follows that \begin{equation} f_{m,n} (t) \leq \frac{4\, m n \, t \, \| V \|}{N} + 2n \, \| V \| \int_0^t \rd s \; f_{m,2} (s) \, . \end{equation} Iterating this equation for $k$ times, we obtain that \begin{equation} \begin{split} f_{m,n} (t) \leq \; &\frac{4\, mn \,\| V \|}{N} \, t + \frac{4 \, mn \, \| V \|}{N} \sum_{r=1}^{k-1} (4 \| V \|)^{r} \int_0^t \rd s_1 \dots \int_0^{s_{r-1}} \rd s_r \; s_r \\ &+ 2n \| V \| ( 4 \| V \|)^{k-1} \int_0^t \dots \int_0^{s_{k-1}} \rd s_k \; f_{m,2} (s_r) \,. \end{split} \end{equation} With the a-priori bound $f_{m,2} (s) \leq 2$, it follows that \[ f_{m,n} (t) \leq \frac{ mn \, }{N} \sum_{r=0}^{k-1} \frac{(4 \| V \| t)^{r+1}}{(r+1)!} + n \frac{(4 \| V \| t)^k}{k!} \leq \frac{mn}{N} \, \left( e^{4 \| V \| t} - 1 \right) + n \frac{(4 \| V \| t)^k}{k!} \] for all $k \in \bN$. Since the l.h.s. is independent of $k$, we obtain (\ref{eq:claim1}). \end{proof} The following proposition is a useful consequence of Theorem \ref{thm:comm}. \begin{proposition}\label{prop2} Suppose that $V \in L^{\infty} (\bR^{\nu})$. Denote $\psi_{N,t} = e^{-iH_N t} \psi_N$ the solution to the $N$-particle Schr\"odinger equation with factorized initial data $\psi_N = \ph^{\otimes N}$, for some $\ph \in L^2 (\bR^3)$ with $\| \ph \|_2 = 1$. Then, for any $A \in \cB_m$, $B \in \cB_n$, we have \begin{equation} \begin{split} \Big| \langle \psi_{N,t} , A^{[i_1, \dots , i_m]} B^{[j_1, \dots , j_n]} \psi_{N,t} \rangle - \langle \psi_{N,t} , A^{[i_1, \dots ,i_m]} \psi_{N,t} \rangle \langle &\psi_{N,t}, B^{[j_1, \dots ,j_n]} \psi_{N,t} \rangle \Big| \\ & \leq \frac{mn \| A \| \| B \|}{N} \, \left(e^{8\| V \| t} -1\right) \, \end{split} \end{equation} for arbitrary integers $1 \leq i_1 < \dots < i_m \leq N$, $1 \leq j_1 < \dots < j_n \leq N$ with $\{ i_1, \dots , i_m \} \cap \{ j_1, \dots ,j_n \} = \emptyset$. In particular, if $\gamma^{(k)}_{N,t}$ denotes the $k$-particle marginal associated with $\psi_{N,t}$, we have \[ \Big| \tr \; \left(A^{[1,\dots ,m]} \otimes B^{[m+1, \dots ,m+n]} \right) \left( \gamma^{(m+n)}_{N,t} - \gamma^{(m)}_{N,t} \otimes \gamma^{(n)}_{N,t} \right) \Big| \leq \frac{mn \| A \| \| B \|}{N} \, \left(e^{8\| V \| t} -1\right) \, .\] \end{proposition} \begin{proof} Because of the permutation symmetry we have \begin{multline} \langle \psi_{N,t} , A^{[i_1, \dots , i_m]} B^{[j_1, \dots , j_n]} \psi_{N,t} \rangle - \langle \psi_{N,t} , A^{[i_1, \dots ,i_m]} \psi_{N,t} \rangle \langle \psi_{N,t}, B^{[j_1, \dots ,j_n]} \psi_{N,t} \rangle \\ = \langle \psi_{N,t} , A^{[1, \dots , m]} B^{[m+1, \dots , m+n]} \psi_{N,t} \rangle - \langle \psi_{N,t} , A^{[1, \dots ,m]} \psi_{N,t} \rangle \langle \psi_{N,t}, B^{[m+1, \dots ,m+n]} \psi_{N,t} \rangle \, .\end{multline} We observe that \begin{equation} \begin{split} \langle \psi_{N,t} , &A^{[1,\dots,m]} B^{[m+1, \dots,m+n]} \psi_{N,t} \rangle \\ = \; &\langle \psi_{N} , e^{i H_N t} A^{[1,\dots,m]} e^{-iH_N t} e^{iH_N t} B^{[m+1, \dots,m+n]} e^{-iH_N t} \psi_N \rangle \\ = \; &\langle \psi_{N} , e^{i H_N t} A^{[1,\dots,m]} e^{-iH_N t} |\ph \rangle \langle \ph|^{\otimes N} e^{iH_N t} B^{[m+1, \dots, m+n]} e^{-iH_N t} \psi_N \rangle \\ &+ \sum_{j=1}^N \langle \psi_{N} , e^{i H_N t} A^{[1,\dots,m]} e^{-iH_N t} \, \left[ |\ph \rangle \langle \ph|^{\otimes (j-1)} \otimes \left( 1-|\ph \rangle \langle \ph| \right) \otimes 1^{\otimes (N-j)} \right] \\ & \hspace{5cm} \times e^{iH_N t} B^{[m+1, \dots, m + n]} e^{-iH_N t} \psi_N \rangle\,. \end{split} \end{equation} Since $\psi_N = \ph^{\otimes N}$, we get \begin{equation} \begin{split} \langle \psi_{N,t} , &A^{[1,\dots,m]} B^{[m+1, \dots, m+n]} \psi_{N,t} \rangle \\= & \; \langle \psi_{N,t} , A^{[1,\dots,m]} \psi_{N,t} \rangle \langle \psi_{N,t}, B^{[m+1, \dots, m+n]} \psi_{N,t} \rangle \\ &+ \sum_{j=1}^m \left\langle \psi_{N} , e^{i H_N t} A^{[1, \dots ,m]} e^{-iH_N t} \, \left( |\ph \rangle \langle \ph|^{\otimes (j-1)} \otimes 1^{\otimes (N-j+1)}\right) \right. \\ & \left. \hspace{4cm} \times \left[ \left( 1-|\ph \rangle \langle \ph| \right)^{[j]}, e^{iH_N t} B^{[m+1, \dots, m+n]} e^{-iH_N t} \right] \psi_N \right\rangle \\ &+ \sum_{j=m+1}^{m+n} \left\langle \psi_{N} , \left[ e^{i H_N t} A^{[1,\dots,m]} e^{-iH_N t} , (1-|\ph\rangle \langle \ph|)^{[j]} \right] \, \left(\ |\ph \rangle \langle \ph|^{\otimes (j-1)} \otimes 1^{\otimes (N-j+1)} \right) \right. \\ &\left. \hspace{4cm} \times \; e^{iH_N t} B^{[m+1, \dots ,m+n]} e^{-iH_N t} \psi_N \right\rangle \\ &+ \sum_{j=m+n+1}^N \left\langle \psi_{N} , \left[ e^{i H_N t} A^{[1,\dots ,m]} e^{-iH_N t}, \left( 1-|\ph \rangle \langle \ph| \right)^{[j]} \right] \, \left( |\ph \rangle \langle \ph|^{\otimes (j-1)} \otimes 1^{\otimes (N-j+1)} \right)\, \right. \\ & \left.\hspace{4cm} \times \left[ \left( 1-|\ph \rangle \langle \ph| \right)^{[j]} , \, e^{iH_N t} B^{[m+1, \dots, m+n]} e^{-iH_N t} \right] \psi_N \right\rangle \,, \end{split} \end{equation} where we used the notation $(1-|\ph \rangle \langle \ph|)^{[j]} = 1^{(j-1)} \otimes (1-|\ph \rangle \langle \ph|) \otimes 1^{(N-j-1)}$ for the operator acting as the projection $(1-|\ph \rangle \langle \ph|)$ over the $j$-th particle, and as the identity over the other $(N-1)$ particles. Note that, by definition $((1-|\ph\rangle \langle \ph|)^{[j]} )^2 = 1-|\ph \rangle \langle \ph|$. {F}rom Theorem \ref{thm:comm}, we obtain \begin{equation*} \begin{split} \Big| \langle \psi_{N,t} , A^{[1, \dots,m]} B^{[m+1, \dots ,m+n]} \psi_{N,t} \rangle - &\langle \psi_{N,t} , A^{[1,\dots,m]} \psi_{N,t} \rangle \langle \psi_{N,t}, B^{[m+1, \dots , m+n]} \psi_{N,t} \rangle \Big| \\ \leq \; & \frac{2\, mn \, \| A \| \| B \|}{N} \, (e^{4\| V \| t} -1) + \frac{mn \| A \| \| B \|}{N} (e^{4\| V \| t} -1 )^2 \\ = \; &\frac{mn \| A \| \|B \|}{N} \, (e^{8\| V \| t} -1) \end{split} \end{equation*} which completes the proof of the proposition. \end{proof} \section{Derivation of the Hartree equation}\label{sec:proof} Using the bounds on the correlations obtained in the previous section, we can now state and prove our main result. \begin{proof}[Proof of Theorem \ref{thm:main}] {F}rom the BBGKY hierarchy, we obtain the integral equation \begin{equation}\begin{split} \gamma_{N,t}^{(k)} = \; &\cU^{(k)} (t) |\ph \rangle \langle \ph|^{\otimes k} - \frac{i}{N} \sum_{i