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Improved version; typos corrected. ---------------0804151536432 Content-Type: text/plain; name="08-78.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="08-78.keywords" Gross-Pitaevskii equation, BBGKY hierarchy, Bose-Einstein condensates ---------------0804151536432 Content-Type: application/x-tex; name="gpV15.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="gpV15.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 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\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{Rigorous Derivation of the Gross-Pitaevskii Equation \\ with a Large Interaction Potential} \author{L\'aszl\'o Erd\H os${}^1$\thanks{Partially supported by SFB/TR12 Project from DFG} \; , Benjamin Schlein${}^1$\thanks{Supported by a Kovalevskaja Award from the Humboldt Foundation. On leave from Cambridge University}, \, and Horng-Tzer Yau${}^2$\thanks{Partially supported by NSF grant DMS-0602038} \\ \\ Institute of Mathematics, University of Munich, \\ Theresienstr. 39, D-80333 Munich, Germany${}^1$ \\ Department of Mathematics, Harvard University\\ Cambridge, MA 02138, USA${}^2$\\ } \maketitle \begin{abstract} Consider a system of $N$ bosons in three dimensions interacting via a repulsive short range pair potential $N^2V(N(x_i-x_j))$, where $\bx=(x_1, \ldots, x_N)$ denotes the positions of the particles. Let $H_N$ denote the Hamiltonian of the system and let $\psi_{N,t}$ be the solution to the Schr\"odinger equation. Suppose that the initial data $\psi_{N,0}$ satisfies the energy condition \[ \langle \psi_{N,0}, H_N \psi_{N,0} \rangle \leq C N \, . \] and that the one-particle density matrix converges to a projection as $N \to \infty$. Then, we prove that the $k$-particle density matrices of $\psi_{N,t}$ factorize in the limit $N \to \infty$. Moreover, the one particle orbital wave function solves the time-dependent Gross-Pitaevskii equation, a cubic non-linear Schr\"odinger equation with the coupling constant proportional to the scattering length of the potential $V$. In \cite{ESY}, we proved the same statement under the condition that the interaction potential $V$ is sufficiently small; in the present work we develop a new approach that requires no restriction on the size of the potential. \end{abstract} \section{Introduction} \setcounter{equation}{0} We consider a bosonic system of $N$ particles with a repulsive interaction. The states of the system are given by elements of the Hilbert space $L^2_s (\bR^{3N})$, the subspace of $L^2 (\bR^{3N})$ consisting of permutation symmetric wave functions. We are interested in describing the time evolution of special initial wave functions $\psi_N \in L^2_s (\bR^{3N})$ that exhibit complete Bose-Einstein condensation. \medskip For a given wave function $\psi_N$, we define the density matrix $\gamma_N = |\psi_N \rangle \langle \psi_N|$ associated with $\psi_N$ as the orthogonal projection onto $\psi_N$. Moreover, for $k =1, \dots, N$, we define the $k$-particle marginal density $\gamma^{(k)}_N$, associated with $\psi_N$, by taking the partial trace of $\gamma_N$ over the last $(N-k)$ variables. In other words, $\gamma^{(k)}_N$ is defined as a positive trace-class operator on $L^2 (\bR^{3k})$ with kernel given by \[ \gamma^{(k)}_N (\bx_k; \bx'_k) = \int \rd \bx_{N-k} \, \psi_N (\bx_k, \bx_{N-k}) \overline{\psi}_N (\bx'_k, \bx_{N-k})\,. \] Here and in the following we use the notation $\bx_k = (x_1, \dots ,x_k)$, $\bx'_k = (x'_1, \dots, x'_k)$, and $\bx_{N-k} = (x_{k+1}, \dots, x_N)$; we will also denote $\bx = (x_1, x_2, \dots ,x_N)$. A sequence $\{ \psi_N \}_{N \in \bN}$, with $\psi_N \in L^2_s (\bR^{3N})$ for all $N$, is said to exhibit complete Bose-Einstein condensation in $\ph \in L^2 (\bR^3)$ if \begin{equation}\label{eq:1tok} \gamma^{(1)}_N \to |\ph \rangle \langle \ph| \qquad \text{as } N \to \infty \end{equation} in the trace-norm topology. Physically, complete Bose-Einstein condensation means that all particles in the system, apart from a fraction vanishing as $N \to \infty$, are described by the same one-particle wave function $\ph$, known as the condensate wave function. Note that (\ref{eq:1tok}) implies that \begin{equation} \label{eq:1tok2} \gamma^{(k)}_N \to |\ph \rangle \langle \ph |^{\otimes k} \end{equation} for all $k \geq 1$, as was first proven by Lieb and Seiringer in \cite{LS}. The time-evolution of $N$ boson systems is governed by the Schr\"odinger equation \begin{equation}\label{eq:schr} i \partial_t \psi_{N,t} = H_N \psi_{N,t} \, \end{equation} with the Hamiltonian operator \begin{equation}\label{eq:ham1} H_N = \sum_{j=1}^N -\Delta_{j} + \sum_{i < j} V_N (x_i -x_j). \end{equation} Here and in the following we are going to use the convention $\Delta_j = \Delta_{x_j}$ and $\nabla_{x_j} = \nabla_j$. We consider the scaling introduced by Lieb, Seiringer and Yngvason in \cite{LSY} for the interaction potential $V_N$, i.e. we fix a repulsive potential $V$, and then we rescale it by defining $V_N (x) = N^2 V (Nx)$. This scaling is chosen so that the scattering length of $V_N$ is of the order $1/N$. We recall that the scattering length associated with a potential $V$ decaying sufficiently fast at infinity ($V$ has to be integrable at infinity) is defined through the solution of the zero-energy scattering equation \be \left(-\Delta + \frac{1}{2} V \right) f = 0 \label{eq:scatt} \ee with boundary condition $f(x) \to 1$ as $|x| \to \infty$. We usually write $f=1-w$. The scattering length of $V$, which is a measure of the effective range of the interaction, is defined by \[ a_0 = \lim_{|x| \to \infty} |x| w(x). \] An equivalent definition of the scattering length is given by the formula \begin{equation}\label{eq:a0} \int \rd x \, V(x) (1-w(x)) = 8 \pi a_0\, . \end{equation} By these definitions, it is clear that, if $a_0$ denotes the scattering length of the potential $V$, then the scattering length of the scaled potential $V_N$ is given by $a= a_0/N$. Our main result is as follows; suppose that the family of wave functions $\{ \psi_N \}_{N \in \bN}$ exhibits complete Bose-Einstein condensation \eqref{eq:1tok} with some $\ph \in H^1 (\bR^3)$ and assume its energy per particle to be bounded (in the sense that $\langle \psi_N, H_N \psi_N \rangle \leq C N$ for all $N$). Denote by $\psi_{N,t}$ the solution of the Schr\"odinger equation (\ref{eq:schr}) with initial data $\psi_{N,0}= \psi_N$. Under appropriate conditions on the potential $V$, we show that, for every time $t\in \bR$, the family $\{ \psi_{N,t} \}_{N \in \bN}$ still exhibits complete condensation, and that the condensate wave function evolves according to a cubic nonlinear Schr\"odinger equation known as the Gross-Pitaevskii equation. In other words, if $\gamma^{(1)}_{N,t}$ denotes the one-particle marginal density associated with $\psi_{N,t}$, we prove that \begin{equation}\label{eq:gammaNT1} \gamma_{N,t}^{(1)} \to |\ph_t \rangle \langle \ph_t| \end{equation} as $N \to \infty$, where $\ph_t$ is determined by the nonlinear Gross-Pitaevskii equation \begin{equation} \label{eq:GP1} i \partial_t \ph_t = -\Delta \ph_t + 8 \pi a_0 |\ph_t|^2 \ph_t \, \end{equation} with initial data $\ph_0 = \ph$ (here $a_0$ denotes the scattering length of the unscaled potential $V$). The cubic non-linear term expresses the local on-site self-interaction of the condensate wave function. Due to the strongly localized interaction, the many-body wave function develops a singular correlation structure on the scale $1/N$. As $N$ goes to infinity, this short scale structure produces the scattering length as a coupling constant in the limiting Gross-Pitaevskii equation. \medskip This result gives a mathematical description of the dynamics of initial data exhibiting complete Bose-Einstein condensation. The simplest example of such initial data are product states $\psi_N = \ph^{\otimes N}$, for arbitrary $\ph \in L^2 (\bR^{3})$. Physically more interesting examples of complete Bose-Einstein condensates are the ground states of trapped Bose gases. A trapped Bose gas in the Gross-Pitaevskii scaling is an $N$-boson system described by the Hamiltonian \begin{equation}\label{eq:hamtrap} H^{\text{trap}}_N = \sum_{j=1}^N \left(-\Delta_{x_j} + V_{\text{ext}} (x_j) \right) + \sum_{i5$ (here $\langle x \rangle = (1 + x^2)^{1/2}$). \medskip The removal of the smallness condition requires completely new ideas. The main challenge in the derivation of the Gross-Pitaevskii equation is to identify and resolve the short scale correlation structure in the $N$-body wave function. In \cite{ESY} we achieved this by locally factoring out the solution of the zero energy scattering equation \eqref{eq:scatt}. This approach, however, is not sufficiently precise for a large interaction potential. In the present paper we propose an {\it intrinsic} characterization of the correlation structure in terms of the two-particle scattering wave operator. More precisely, we prove that the action of the wave operator in the relative coordinate $x_i-x_j$ regularizes $\psi_{N,t}$ in this variable. This regularity is essential to control the convergence of the many-body interaction to the local on-site nonlinearity in the limiting equation \eqref{eq:GP1}. An a-priori estimate leading to this regularity will be obtained from the conservation of the second moment of the energy, i.e. from $\langle \psi_{N,t}, H_N^2 \psi_{N,t}\rangle$. In case of a large potential, however, this a-priori bound controls only a specific combination of two derivatives, $\nabla_{x_i}\cdot \nabla_{x_j}$, acting on the regularized wave function. We thus need to establish a new Poincar\'e-type inequality involving this combination of derivatives only. In the next section we discuss the main ideas of our new approach. \section{Resolution of the correlation structure for large potential} \setcounter{equation}{0} \medskip As in \cite{ESY}, the general strategy of our proof is based on the study of solutions of the BBGKY hierarchy of equations for the marginal densities $\gamma^{(k)}_{N,t}$ associated with the solution of the $N$-particle Schr\"odinger equation (\ref{eq:schr}): \begin{equation}\label{eq:BBGKY} \begin{split} i\partial_t \gamma_{N,t}^{(k)} = \; &\sum_{j=1}^k \left[-\Delta_j , \gamma_{N,t}^{(k)} \right] + \sum_{i5$, and for all $x \in \bR^3$. Assume that the family $\psi_N \in L_s^2 (\bR^{3N})$, with $\| \psi_N \| =1$ for all $N$, has finite energy per particle, in the sense that \begin{equation}\label{eq:assH1} \langle \psi_N, H_N \psi_N \rangle \leq C N \end{equation} and that it exhibits complete Bose-Einstein condensation in the sense that the one-particle marginal $\gamma_N^{(1)}$ associated with $\psi_N$ satisfies \begin{equation}\label{eq:asscond} \gamma_N^{(1)} \to |\ph \rangle \langle \ph| \qquad \text{in the trace norm topology as } \; N \to \infty \end{equation} for some $\ph \in H^1 (\bR^3)$. Then, for every $k \geq 1$ and $t \in \bR$, we have \[ \gamma^{(k)}_{N,t} \to |\ph_t \rangle \langle \ph_t|^{\otimes k} \] as $N \to \infty$, in the trace norm. Here $\ph_t$ is the solution of the nonlinear Gross-Pitaevskii equation \begin{equation}\label{eq:GPthm} i \partial_t \ph_t = -\Delta\ph_t + 8\pi a_0 |\ph_t|^2 \ph_t \end{equation} with initial data $\ph_{t=0} = \ph$. \end{theorem} {\it Remark.} Note that the condition $V(x) \leq C \langle x \rangle^{-\sigma}$ for some $\sigma >5$ and for all $x \in \bR^3$ is only required to apply the result of Yajima in \cite{Ya}, which guarantees that the wave operator $W$ associated with the Hamiltonian $\fh =-\Delta +(1/2) V$ maps $L^p (\bR^3)$ into itself, for all $1 \leq p \leq \infty$ (see Proposition \ref{prop:waveop}). If one knows, by different means, that $\| W \|_{L^p \to L^p} < \infty$ (it suffices to know it for $p=1$ and $p=\infty$), then it would be enough to assume that $V \in L^1 (\bR^3, (1+|x|^2) \rd x) \cap L^2 (\bR^3, \rd x)$. \medskip {\it Remark.} Compared with our previous result in \cite{ESY}, we do not require here the potential $V$ to be spherical symmetric (we only need that $V(-x) = V(x)$). \medskip {\it Remark.} The fact that $\ph \in H^1 (\bR^3)$ does not need to be assumed separately, since it already follows from the assumption (\ref{eq:assH1}). \medskip To prove this theorem we will use an approximation argument and the following theorem, which proves Theorem \ref{thm:main} for a smaller class of initial $N$-particle wave functions. \begin{theorem}\label{thm:main2} Assume the same conditions on the potential $V$ as in Theorem \ref{thm:main}. Suppose moreover that $|\nabla^{\alpha} V (x)| \leq C$ for all multi-indices $\alpha$ with $|\alpha| \leq 2$. Assume that the family $\psi_N \in L^2 (\bR^{3N})$, with $\| \psi_{N} \| =1$, is such that \begin{equation}\label{eq:enerk} \langle \psi_N , H_N^k \psi_N \rangle \leq C^k N^k \end{equation} for all $k \in \bN$, and that \begin{equation}\label{eq:init} \gamma_N^{(1)} \to |\ph \rangle \langle \ph| \qquad \text{in the trace norm topology as } N \to \infty \end{equation} for some $\ph \in H^1 (\bR^3)$. Then, for every $k \geq 1$ and $t \in \bR$ \[ \gamma^{(k)}_{N,t} \to |\ph_t \rangle \langle \ph_t|^{\otimes k} \] as $N \to \infty$, in the trace norm. Here $\ph_t$ is the solution of the nonlinear Gross-Pitaevskii equation (\ref{eq:GPthm}) with initial data $\ph_{t=0} = \ph$. \end{theorem} \section{Proof of the main theorem}\label{sec:outline} \setcounter{equation}{0} In this section we present the proof of Theorem \ref{thm:main2} and we show how it implies Theorem \ref{thm:main}, making use of several key proposition, whose proof is deferred to subsequent sections. \medskip We start by defining an appropriate space of time-dependent density matrices. To use Arzela-Ascoli compactness argument, we will need to establish the concept of uniform continuity in this space, thus we have to metrize the weak* topology. \medskip Let $\cK_k \equiv \cK (L^2 (\bR^{3k}))$ denote the space of compact operators on $L^2 (\bR^{3k})$ equipped with the operator norm topology and let $\cL^1_k \equiv \cL^1 (L^2 (\bR^{3k}))$ denote the space of trace class operators on $L^2 (\bR^{3k})$ equipped with the trace norm. It is well known that $\cL^1_k$ is the dual of $\cK_k$. Since $\cK_k$ is separable, we can fix a dense countable subset of the unit ball of $\cK_k$; we denote it by $\{J^{(k)}_i\}_{i \ge 1} \in \cK_k$, with $\| J^{(k)}_i \| \leq 1$ for all $i \ge 1$. Using the operators $J^{(k)}_i$ we define the following metric on the space $\cL^1_k \equiv \cL^1 (L^2 (\bR^{3k}))$: for $\gamma^{(k)}, \bar \gamma^{(k)} \in \cL^1_k$ we set \begin{equation}\label{eq:etak} \eta_k (\gamma^{(k)}, \bar \gamma^{(k)}) : = \sum_{i=1}^\infty 2^{-i} \left| \tr \; J^{(k)}_i \left( \gamma^{(k)} - \bar \gamma^{(k)} \right) \right| \, . \end{equation} Then the topology induced by the metric $\eta_k$ and the weak* topology are equivalent on the unit ball of $\cL^1_k$ (see \cite{Ru}, Theorem 3.16) and hence on any ball of finite radius as well. In other words, a uniformly bounded sequence $\gamma_N^{(k)} \in \cL^1_k$ converges to $\gamma^{(k)} \in \cL^1_k$ with respect to the weak* topology, if and only if $\eta_k (\gamma^{(k)}_N , \gamma^{(k)}) \to 0$ as $N \to \infty$. \medskip For a fixed $T > 0$, let $C ([0,T], \cL^1_k)$ be the space of functions of $t \in [0,T]$ with values in $\cL^1_k$ which are continuous with respect to the metric $\eta_k$. On $C ([0,T], \cL^1_k)$ we define the metric \begin{equation}\label{eq:whetak} \widehat \eta_k (\gamma^{(k)} (\cdot ) , \bar \gamma^{(k)} (\cdot )) := \sup_{t \in [0,T]} \eta_k (\gamma^{(k)} (t) , \bar \gamma^{(k)} (t))\,. \end{equation} Finally, we denote by $\tau_{\text{prod}}$ the topology on the space $\bigoplus_{k \geq 1} C([0,T], \cL^1_k)$ given by the product of the topologies generated by the metrics $\wh \eta_k$ on $C([0,T], \cL^1_k)$. \medskip \begin{proof}[Proof of Theorem \ref{thm:main2}] The proof is divided in four steps. \medskip {\it Step 1. Compactness of $\Gamma_{N,t}=\{ \gamma^{(k)}_{N,t} \}_{k\geq 1}$.} We fix $T>0$ and work on the interval $t\in [0, T]$. Negative times can be handled analogously. \medskip In Theorem \ref{thm:compactness} we show that the sequence $\Gamma_{N,t}^{(k)} = \{ \gamma^{(k)}_{N,t} \}_{k \geq 1} \in \bigoplus_{k \geq 1} C([0,T], \cL^1_k)$ is compact with respect to the product topology $\tau_{\text{prod}}$ defined above (we use the convention that $\gamma^{(k)}_{N,t} =0 $ if $k >N$). It also follows from Theorem \ref{thm:compactness} that any limit point $\Gamma_{\infty,t} = \{\gamma_{\infty,t}^{(k)} \}_{k\geq 1} \in \bigoplus_{k \geq 1} C ( [0,T], \cL^1_k)$ is such that, for every $k \geq 1$, $\gamma_{\infty,t}^{(k)} \geq 0$, and $\gamma_{\infty,t}^{(k)}$ is symmetric w.r.t. permutations. \medskip Using higher order energy estimates from Proposition \ref{prop:hk}, we show in Theorem \ref{thm:aprik} that an arbitrary limit point $\Gamma_{\infty,t} = \{ \gamma^{(k)}_{\infty,t} \}_{k \geq 1}$ of the sequence $\Gamma_{N,t}^{(k)}$ (with respect to the topology $\tau_{\text{prod}}$) is such that \begin{equation}\label{eq:apri} \tr \; (1-\Delta_1) \dots (1-\Delta_k) \, \gamma^{(k)}_{\infty,t} \leq C^k \end{equation} for every $t \in [0,T]$ and every $k \geq 1$. \medskip {\it Step 2. Convergence to the infinite hierarchy.} In Theorem \ref{thm:conv} we prove that any limit point $\Gamma_{\infty,t} = \{ \gamma_{\infty,t}^{(k)} \}_{k\geq 1} \in \bigoplus_{k\geq 1} C([0,T], \cL^1_k)$ of $\Gamma_{N,t} = \{ \gamma_{N,t}^{(k)} \}_{k\ge1}$ with respect to the product topology $\tau_{\text{prod}}$ is a solution of the infinite hierarchy of integral equations ($k=1,2, \ldots$) \begin{equation}\label{eq:BBGKYinf} \gamma_{\infty,t}^{(k)} = \; \cU^{(k)} (t) \gamma_{\infty,0}^{(k)} - 8\pi i a_0 \sum_{j=1}^k \int_0^t \rd s \, \cU^{(k)} (t-s) \tr_{k+1} \left[ \delta (x_j -x_{k+1}), \gamma_{\infty,s}^{(k+1)}\right]\, \end{equation} with initial data $\gamma_{\infty,0}^{(k)} = |\ph\rangle \langle\ph|^{\otimes k}$ (where $\ph \in H^1 (\bR^3)$ has been introduced in (\ref{eq:init})). Here $\tr_{k+1}$ denotes the partial trace over the $(k+1)$-th particle, and $\cU^{(k)} (t)$ is the free evolution, whose action on $k$-particle density matrices is given by \begin{equation}\label{eq:Uk} \cU^{(k)} (t) \gamma^{(k)} := e^{it\sum_{j=1}^k \Delta_j} \gamma^{(k)} e^{-it\sum_{j=1}^k \Delta_j}\,. \end{equation} \medskip We remark next that the family of factorized densities, \begin{equation} \label{eq:factsol} \gamma^{(k)}_t = |\ph_t \rangle \langle \ph_t|^{\otimes k}, \end{equation} is a solution of the infinite hierarchy (\ref{eq:BBGKYinf}) if $\ph_t$ is the solution of the nonlinear Gross-Pitaevskii equation (\ref{eq:GPthm}) with initial data $\ph_{t=0}= \ph$. The nonlinear Schr\"odinger equation (\ref{eq:GPthm}) is well-posed in $H^1 (\bR^3)$ and it conserves the energy, $\cE (\ph) := \frac{1}{2} \int |\nabla \ph|^2 + 4 \pi a_0 \int |\ph|^4$. {F}rom $\ph \in H^1 (\bR^3)$, we thus obtain that $\ph_t \in H^1 (\bR^3)$ for every $t \in \bR$, with a uniformly bounded $H^1$-norm. Therefore \begin{equation}\label{eq:phik} \tr \; (1-\Delta_1) \dots (1-\Delta_k) |\ph_t \rangle \langle \ph_t|^{\otimes k} \leq \| \ph_t \|_{H^1}^k \leq C^k \end{equation} for all $t \in \bR$, and a constant $C$ only depending on the $H^1$-norm of $\ph$. For the well-posedness of the subcritical nonlinear Schr\"odinger equation (\ref{eq:GPthm}) in $H^1$, see e.g. \cite{K}. We remark that the well-posedness has been established even for the critical (quintic) nonlinear Schr\"odinger equation in \cite{GV3,GV4,Str} for small data and in \cite{Bour,CKSTT} for large data. \medskip {\it Step 3. Uniqueness of the solution to the infinite hierarchy.} In Section 9 of \cite{ESY2} we proved the following theorem, which states the uniqueness of solution to the infinite hierarchy (\ref{eq:BBGKYinf}) in the space of densities satisfying the a priori bound (\ref{eq:apri}). The proof of this theorem is based on a diagrammatic expansion of the solution of (\ref{eq:BBGKYinf}). Remark that the uniqueness of the infinite hierarchy in a different space of densities was proven in \cite{KM}. \begin{theorem}\label{thm:uniqueness}[Theorem 9.1 of \cite{ESY2}] Suppose $\Gamma = \{ \gamma^{(k)} \}_{k \geq 1} \in \bigoplus_{k \geq 1} \cL^1_k$ is such that \begin{equation} \tr\; (1-\Delta_1) \dots (1-\Delta_k) \gamma^{(k)} \leq C^k \, . \end{equation} Then, for any fixed $T >0$, there exists at most one solution $\Gamma_t = \{ \gamma^{(k)}_t \}_{k \geq 1} \in \bigoplus_{k \geq 1} C([0,T], \cL^1_k)$ of (\ref{eq:BBGKYinf}) such that \begin{equation}\label{eq:bougam} \tr\; (1-\Delta_1) \dots (1-\Delta_k) \gamma^{(k)}_t \leq C^k \end{equation} for all $t \in [0,T]$ and for all $k \geq 1$. \end{theorem} \medskip {\it Step 4. Conclusion of the proof.} {F}rom Step 2 and Step 3 it follows that the sequence $\Gamma_{N,t} = \{ \gamma^{(k)}_{N,t} \}_{k \geq 1} \in \bigoplus_{k \geq 1} C([0,T],\cL_k^1)$ is convergent with respect to the product topology $\tau_{\text{prod}}$; in fact a compact sequence with only one limit point is always convergent. Since the family of densities $\Gamma_t = \{ \gamma^{(k)}_t \}_{k \geq 1}$ defined in (\ref{eq:factsol}) satisfies (\ref{eq:phik}) and it is a solution of (\ref{eq:BBGKYinf}), it follows that $\Gamma_{N,t} \to \Gamma_t$ w.r.t. the topology $\tau_{\text{prod}}$. In particular this implies that, for every fixed $k \geq 1$, and $t \in [0,T]$, $\gamma_{N,t}^{(k)} \to |\ph_t \rangle \langle \ph_t|^{\otimes k}$ with respect to the weak* topology of $\cL^1_k$, and thus, by a standard argument, also in the trace-norm topology. This completes the proof of Theorem \ref{thm:main2}. \end{proof} \medskip Next we prove Theorem \ref{thm:main}; to this end we have to combine Theorem \ref{thm:main2} with an approximation argument for the initial $N$-particle wave function, which is needed to make sure that the energy condition (\ref{eq:enerk}) is satisfied. This argument was already used in \cite{ESY}; we present it here for completeness. \begin{proof}[Proof of Theorem \ref{thm:main}] We assume here that, as in Theorem \ref{thm:main2}, the interaction potential $V$ is such that $|\nabla^{\alpha} V (x)| \leq C$ for all multi-indices $\alpha$ with $|\alpha| \leq 2$. We show how to remove this condition in Appendix \ref{app:nablaV}. \medskip Fix $\kappa >0$ and $\chi \in C_0^{\infty} (\bR)$, with $0\leq \chi\leq 1$, $\chi (s) = 1$, for $0 \leq s \leq 1$, and $\chi (s) =0$ if $s \geq 2$. We define the regularized initial wave function \[ \wt \psi_N := \frac{ \chi (\kappa H_N /N) \psi_N }{ \| \chi (\kappa H_N /N) \psi_N \|} , \] and we denote by $\wt\psi_{N,t}$ the solution of the Schr\"odinger equation (\ref{eq:schr}) with initial data $\wt \psi_N$. Denote by $\wt \Gamma_{N,t} = \{ \wt \gamma_{N,t}^{(k)} \}_{k=1}^\infty$ the family of marginal densities associated with $\wt \psi_{N,t}$. By convention, we set $\wt\gamma^{(k)}_{N,t}:=0$ if $k >N$. The tilde in the notation indicates the dependence on the cutoff parameter $\kappa$. In Proposition \ref{prop:initialdata}, part i), we prove that \begin{equation}\label{eq:thm2-1} \langle \wt \psi_{N,t} , H_N^k \wt\psi_{N,t} \rangle \leq {\wt C}^k N^k \end{equation} if $\kappa >0$ is sufficiently small (the constant $\wt C$ depends on $\kappa$). Moreover, in part iii) of Proposition \ref{prop:initialdata}, we show that, for every $J^{(k)} \in \cK_k$, \begin{equation}\label{eq:thm2-2} \tr \; J^{(k)} \left( \wt \gamma_N^{(k)} - |\ph \rangle \langle \ph|^{\otimes k} \right) \to 0 \end{equation} as $N \to \infty$. {F}rom (\ref{eq:thm2-1}) and (\ref{eq:thm2-2}), the assumptions (\ref{eq:enerk}) and (\ref{eq:init}) of Theorem \ref{thm:main2} are satisfied by the regularized wave function $\wt \psi_N$ and by the regularized marginal densities $\wt \gamma^{(k)}_{N,t}$. Applying Theorem \ref{thm:main2}, we obtain that, for every $t \in \bR$ and $k \geq 1$, \begin{equation}\label{eq:conve} \wt \gamma_{N,t}^{(k)} \to |\ph_t \rangle \langle \ph_t|^{\otimes k} \, , \end{equation} where $\ph_t$ is the solution of (\ref{eq:GPthm}) with initial data $\ph_{t=0} = \ph$. \medskip It remains to prove that the densities $\gamma^{(k)}_{N,t}$ associated with the original wave function $\psi_{N,t}$ (without cutoff $\kappa$) converge and have the same limit as the regularized densities $\wt \gamma^{(k)}_{N,t}$. This follows from Proposition \ref{prop:initialdata}, part ii), where we prove that \[ \| \psi_{N,t} - \wt\psi_{N,t} \| = \| \psi_{N} - \wt\psi_{N} \| \leq C \kappa^{1/2} \; , \] for a constant $C$ independent of $N$ and $\kappa$. This implies that, for every $J^{(k)} \in \cK_k$, we have \begin{equation}\label{eq:remove} \Big| \tr \; J^{(k)} \left( \gamma^{(k)}_{N,t} - \wt \gamma^{(k)}_{N,t} \right) \Big| \leq C \| J^{(k)} \| \, \kappa^{1/2}\,. \end{equation} Therefore, for every fixed $k \geq 1$, $t \in \bR$, $J^{(k)} \in \cK_k$, we have \begin{equation}\label{eq:lastproof} \begin{split} \Big| \tr \; J^{(k)} \left( \gamma^{(k)}_{N,t} - |\ph_t \rangle \langle \ph_t |^{\otimes k} \right) \Big| \leq &\; \Big| \tr \; J^{(k)} \left( \gamma^{(k)}_{N,t} - \wt \gamma^{(k)}_{N,t} \right) \Big| + \Big| \tr \; J^{(k)} \left( \wt \gamma^{(k)}_{N,t} - |\ph_t \rangle \langle \ph_t |^{\otimes k} \right) \Big| \\ \leq & \; C\, \| J^{(k)} \| \kappa^{1/2} + \Big| \tr \; J^{(k)} \left( \wt \gamma^{(k)}_{N,t} - |\ph_t \rangle \langle \ph_t |^{\otimes k} \right) \Big| \, . \end{split} \end{equation} Since $\kappa >0$ was arbitrary, it follows from (\ref{eq:conve}) that the l.h.s. of (\ref{eq:lastproof}) converges to zero as $N \to \infty$. This implies that, for arbitrary $k \geq 1$ and $t \in \bR$, $\gamma^{(k)}_{N,t} \to |\ph_t \rangle \langle \ph_t |^{\otimes k}$ in the weak* topology of $\cL^1_k$, and thus also in the trace-norm topology. This completes the proof of Theorem \ref{thm:main2}. \end{proof} \section{The wave operator and a-priori bounds on $\gamma_{N,t}^{(k)}$.} \setcounter{equation}{0} In order to derive a-priori bounds for the marginal densities $\gamma^{(k)}_{N,t}$, we need to introduce wave operators. We denote by $W$ and $W_N$ the wave operators associated with the one-particle Hamiltonian $\fh = -\Delta + (1/2) V (x)$ and, respectively, $\fh_N = -\Delta + (1/2) V_N (x)$, with $V_N (x) = N^2 V(Nx)$. The existence of these wave operators and their most important properties are stated in the following proposition (we denote by $s-\lim$ the limit in the strong operator topology). \begin{proposition} \label{prop:waveop} Suppose $V \geq 0$, with $V \in L^1 (\bR^3)$. Then: \begin{itemize} \item[i)] ({\it Existence of the wave operator}). The limit \[ W = s-\lim_{t\to \infty} e^{i\fh t} e^{i\Delta t} \] exists. \item[ii)] ({\it Completeness of the wave operator}). $W$ is a unitary operator on $L^2 (\bR^3)$ with \[ W^* = W^{-1} = s-\lim_{t\to \infty} e^{-i\Delta t} e^{-i\fh t} \] \item[iii)] ({\it Intertwining relations}). On $D(\fh) = D (-\Delta)$, we have \begin{equation}\label{eq:intertw} W^* \fh W = - \Delta \end{equation} \item[iv)] ({\it Yajima's bounds}). Suppose moreover that $V(x)\leq C \langle x \rangle^{-\sigma}$, for some $\sigma >5$. Then, for every $1 \leq p \leq \infty$, $W$ and $W^*$ map $L^p (\bR^3)$ into $L^p (\bR^3)$, that is \[ \| W \|_{L^p \to L^p} < \infty \qquad \text{for all } \quad 1 \leq p \leq \infty \] \item[v)] ({\it Rescaled wave operator}). If $\fh_N = -\Delta + (1/2) V_N (x)$, with $V_N (x) = N^2 V (Nx)$, then the limit \[ W_N = s-\lim_{t \to \infty} e^{i\fh_N t} e^{i\Delta t} \] exists and it defines a unitary operator $W_N$ on $L^2 (\bR^3)$ with \[ W_N^* = W_N^{-1} = s-\lim_{t\to \infty} e^{-i\Delta t} e^{-i\fh_N t} . \] The wave operator $W_N$ satisfies the intertwining relations \[ W_N^* \fh_N W_N = -\Delta \, .\] Moreover, the kernel of $W_N$ is given by \[ W_N (x;y) = N^3 W(Nx;Ny) \qquad \text{and} \qquad W^*_N (x;y) = N^3 W^* (Nx; Ny) \] where $W (x;y)$ and $W^* (x;y)$ denote the kernels of $W$ and $W^*$. In particular, it follows that, if for every $1 \leq p \leq \infty$, the norms \[ \| W_N \|_{L^p \to L^p} = \| W \|_{L^p \to L^p} < \infty \qquad \text{and} \qquad \| W^*_N \|_{L^p \to L^p} = \| W^* \|_{L^p \to L^p} < \infty \] are finite and independent of $N$. \end{itemize} \end{proposition} \begin{proof} The proof of i), ii), and iii) can be found in \cite{RS3}. Part iv) is proven in \cite{Ya0,Ya}. Part~v) follows by simple scaling arguments. \end{proof} In the following we will denote by $W_{(i,j)}$ and, respectively, by $W_{N, (i,j)}$, the wave operators $W$ and $W_N$ acting only on the relative variable $x_j - x_i$. In other words, the action of $W_{(i,j)}$ on a $N$-particle wave function $\psi_N \in L^2 (\bR^{3N})$ is given by \begin{equation}\label{eq:Wij} \left( W_{(i,j)} \psi_N \right) (\bx) = \int \rd v \; W (x_j - x_i; v) \, \psi_N \left( x_1, \dots , \frac{x_i + x_j}{2} + \frac{v}{2}, \dots , \frac{x_i + x_j}{2} - \frac{v}{2}, \dots, x_N \right) \end{equation} if $j j$ is similar). Here $W(x;y)$ is the kernel of the wave operator $W$. An analogous formula holds for the rescaled wave operator $W_N$. Similarly, we define $W^*_{(i,j)}$ and $W^*_{N,(i,j)}$. Using the wave operators, we have the following energy estimate. \begin{proposition}\label{prop:energ2} Suppose $V \geq 0$, $V \in L^1 (\bR^3)$ and $V(x) = V(-x)$ for all $x\in \bR^3$. Then we have, for every $i \neq j$, \begin{equation}\label{eq:energ2} \langle \psi_N , H_N^2 \psi_N \rangle \geq C N^2 \int \rd \bx \; \left| \left(\nabla_i \cdot \nabla_j \right)^2 W^*_{N,(i,j)} \psi_N \right|^2\, , \end{equation} where $W_{N,(i,j)}$ denotes the wave operator $W_N$ defined in Proposition \ref{prop:waveop} acting on the variable $v = x_j -x_i$ (defined similarly to (\ref{eq:Wij})). \end{proposition} \begin{proof} We define, for $j=1,\dots,N$, \[ h_j = -\Delta_j + \frac{1}{2}\sum_{i \neq j} V_N (x_i -x_j). \] Then we have $H_N = \sum_{j=1}^N h_j$ and thus \begin{equation}\label{eq:h1} \begin{split} \langle \psi_N, H_N^2 \, \psi_N \rangle \geq \; &N(N-1) \langle \psi_N, h_1 h_2 \psi_N \rangle \\ = \; &N(N-1) \left\langle \psi_N , \left( -\Delta_1 + \frac{1}{2} \sum_{i \neq 1} V_N (x_i -x_1) \right)\left( -\Delta_2 + \frac{1}{2} \sum_{j\neq 2} V_N (x_j -x_2) \right) \psi_N \right\rangle \\ \geq \; &N(N-1) \left\langle \psi_N, \left( -\Delta_1 + \frac{1}{2} V_N (x_1-x_2)\right) \left( -\Delta_2 + \frac{1}{2} V_N (x_1 -x_2)\right) \psi_N \right\rangle\,. \end{split} \end{equation} Now we define the new variables \[ u= \frac{x_1+x_2}{2}, \quad \text{and } \quad v=x_1 -x_2 \,.\] Then we have \[ \nabla_1 = \frac{1}{2} \nabla_u + \nabla_v \quad \text{and } \quad \nabla_2 = \frac{1}{2} \nabla_u -\nabla_v \] and thus \[ \Delta_1 = \frac{1}{4} \Delta_u + \Delta_v + \nabla_u \cdot \nabla_v, \quad \text{and } \quad \Delta_2 = \frac{1}{4} \Delta_u +\Delta_v - \nabla_u \cdot \nabla_v \,.\] We set \[ h_v = -\Delta_v + \frac{1}{2} V_N (v) \,. \] Then \begin{equation}\label{eq:H-2} \begin{split} \langle \psi_N, H_N^2 \, \psi_N \rangle \geq \; &N(N-1) \left\langle \psi_N, \left( - \frac{1}{4} \Delta_u + h_v + \nabla_u \cdot \nabla_v \right) \left( -\frac{1}{4} \Delta_u + h_v - \nabla_u \cdot \nabla_v \right) \psi_N \right\rangle \\ = \; &N(N-1) \left\langle \psi_N, \left[ \left( - \frac{1}{4} \Delta_u + h_v \right)^2 - \left(\nabla_u \cdot \nabla_v \right)^2 + \frac{1}{2} \, \nabla_u \cdot \left(\nabla V_N (v)\right) \right] \psi_N \right\rangle\,. \end{split} \end{equation} Next we note that \begin{equation}\label{eq:inte} \begin{split} \langle \psi_N, \nabla_u &\cdot \nabla V_N (v) \psi_N \rangle \\ &= \int \rd u \rd v \; \overline{\psi}_N (u+v/2,u-v/2, \bx_{N-2}) \nabla V_N (v) \cdot \nabla_u \psi_N (u+v/2,u-v/2,\bx_{N-2}) = 0\,. \end{split} \end{equation} In fact, by the permutation symmetry, $\psi_N (x_1,x_2,\bx_{N-2}) = \psi_N (x_2,x_1,\bx_{N-2})$. This implies, in the $u,v$-coordinates, that $\psi_N (u+v/2,u-v/2,\bx_{N-2}) = \psi_N (u-v/2,u+v/2,\bx_{N-2})$ and also that $\nabla_u \psi_N (u+v/2,u-v/2,\bx_{N-2}) = \nabla_u \psi_N (u-v/2,u+v/2,\bx_{N-2})$. On the other hand $\left(\nabla V_N \right) (-v)=-\left(\nabla V_N \right) (v)$. Therefore, the integrand in (\ref{eq:inte}) is antisymmetric w.r.t. the change of variables $v \to -v$, and the integral vanishes. \medskip Using also that \[(\nabla_u \cdot \nabla_v)^2 \leq \left(-\Delta_u\right) \left(-\Delta_v\right) \leq \left(-\Delta_u \right) \, h_v \, , \] it follows from (\ref{eq:H-2}) that \begin{equation}\label{eq:H-3} \langle \psi_N, H_N^2 \, \psi_N \rangle \geq N(N-1) \left\langle \psi_N, \left( -\frac{1}{4} \Delta_u - h_v \right)^2 \psi_N \right\rangle \,. \end{equation} Next we make use of the wave operator $W_N$ defined in Proposition \ref{prop:waveop}, acting on the variable $v=x_2 - x_1$. By the intertwining relations (\ref{eq:intertw}), we find \begin{equation}\label{eq:H-4} \langle \psi_N, H_N^2 \, \psi_N \rangle \geq N(N-1) \left\langle W_{N,(1,2)}^* \psi_N, \left( \frac{1}{4} \Delta_u - \Delta_v \right)^2 \, W_{N,(1,2)}^* \psi_N \right\rangle\,. \end{equation} In terms of the coordinates $x_1$ and $x_2$, we have $\nabla_1 \cdot \nabla_2 = (1/4) \Delta_u - \Delta_v$. Therefore, by the permutation symmetry, the bound (\ref{eq:H-4}) implies (\ref{eq:energ2}). \end{proof} Proposition \ref{prop:energ2} implies strong a-priori bounds on the solution of the $N$-particle Schr\"odinger equation. \begin{proposition}\label{prop:apri2} Suppose that $V \geq 0$, $V \in L^1 (\bR^3)$, and $V(-x) = V(x)$ for all $x\in \bR^3$. Let $\psi_{N,t}$ be the solution of the Schr\"odinger equation (\ref{eq:schr}), with initial data satisfying the assumption (\ref{eq:enerk}) (with $k=2$) of Theorem \ref{thm:main2}, and let $\{ \gamma^{(k)}_{N,t} \}_{k=1}^N$ be the marginals associated with $\psi_{N,t}$. Then, for every $1\leq j \leq N$, we have \[ \langle \psi_{N,t}, (1-\Delta_j) \psi_{N,t} \rangle \leq C \qquad \text{and thus} \qquad \tr \; (1-\Delta_j) \gamma_{N,t}^{(k)} \leq C \] for every $1\leq j\leq k \leq N$ (and for a constant $C$ which only depends on the initial data $\psi_N$ through the constant on the r.h.s. of (\ref{eq:enerk})). Moreover, for any $i \neq j$, \[ \left\langle W_{N,(i,j)}^* \psi_{N,t} , \left((\nabla_i \cdot \nabla_j)^2 -\Delta_i - \Delta_j + 1 \right) W_{N,(i,j)}^* \psi_{N,t} \right\rangle \leq C \] uniformly in $N \geq 1$ and in $t \in \bR$. Here $W_{N,(i,j)}$ denotes the wave operator $W_N$ defined in Proposition~\ref{prop:waveop} acting on the variable $x_j - x_i$. In terms of density matrices, we obtain the a-priori bounds \[ \tr \; \left((\nabla_i \cdot \nabla_j)^2 -\Delta_i - \Delta_j + 1 \right) W^*_{N,(i,j)} \gamma^{(k)}_{N,t} W_{N,(i,j)} \leq C \] uniformly in $N \geq 1$ and in $t \in \bR$ and for all $1 \leq i < j \leq k$ (with a slight abuse of notation, we denote here by $W_{N,(i,j)}$ and $W^*_{N,(i,j)}$ the operators acting on the $k$-particle space $L^2 (\bR^{3k})$). \end{proposition} \begin{proof} The first bound follows simply by the symmetry of the wave function, by energy conservation, and by the condition $V \geq 0$. To prove the second bound, we compute \begin{equation}\label{eq:apri2-1} \begin{split} \langle W_{N,(i,j)}^* \psi_{N,t} , \left((\nabla_i \cdot \nabla_j)^2 -\Delta_i - \Delta_j + 1 \right) &W_{N,(i,j)}^* \psi_{N,t} \rangle \\ = \; & \langle W_{N, (1,2)}^* \psi_{N,t} ,(\nabla_1 \cdot \nabla_2)^2 W_{N,(1,2)}^* \psi_{N,t} \rangle \\ &+\langle W_{N,(1,2)}^* \psi_{N,t} , \left(-\Delta_1 - \Delta_2 \right) W_{N,(1,2)}^* \psi_{N,t} \rangle +1 \,. \end{split} \end{equation} The first term on the r.h.s. of the last equation can be bounded by \begin{equation} \begin{split} \langle W_{N,(1,2)}^* \psi_{N,t} , (\nabla_1 \cdot \nabla_2)^2 W_{N,(1,2)}^* \psi_{N,t} \rangle \leq \; & CN^{-2} \langle \psi_{N,t}, H_N^2 \psi_{N,t} \rangle \\ = \; & CN^{-2} \langle \psi_{N,0}, H_N^2 \psi_{N,0} \rangle \leq C \, \end{split} \end{equation} using Proposition \ref{prop:energ2} and (\ref{eq:enerk}). The second term on the r.h.s. of (\ref{eq:apri2-1}) is estimated by \begin{equation} \begin{split} \langle W_{N, (1,2)}^* \psi_{N,t} , & \left(-\Delta_1 - \Delta_2 \right)W_{N,(1,2)}^* \psi_{N,t} \rangle \\ & = 2 \, \left\langle W_{N,(1,2)}^* \psi_{N,t} , \left(-\Delta_{x_1-x_2} - \frac{1}{4} \Delta_{(x_1 + x_2)/2} \right) W_{N,(1,2)}^* \psi_{N,t} \right\rangle \\ & = 2\, \left\langle \psi_{N,t} , \left(-\Delta_{x_1-x_2} + \frac{1}{2} V_N (x_1 -x_2) - \frac{1}{4} \, \Delta_{(x_1 + x_2)/2} \right) \psi_{N,t} \right\rangle \\ & = \left\langle \psi_{N,t} , \left(-\Delta_{x_1} - \Delta_{x_2} + V_N (x_1 -x_2) \right) \psi_{N,t} \right\rangle \\ &\leq \frac{2}{N} \langle \psi_{N,t}, H_N \psi_{N,t} \rangle = \frac{2}{N} \langle \psi_{N,0} , H_N \psi_{N,0} \rangle \leq C\,. \end{split} \end{equation} \end{proof} \section{Compactness} \setcounter{equation}{0} In this section we prove the compactness of the sequence $ \Gamma_{N,t} = \{ \gamma^{(k)}_{N,t} \}_{k\ge1}$ w.r.t. the topology $\tau_{\text{prod}}$ (defined in Section \ref{sec:outline}). \begin{theorem}\label{thm:compactness} Let the assumptions of Theorem \ref{thm:main2} be satisfied and fix an arbitrary $T>0$. Then the sequence $\Gamma_{N,t} \in \bigoplus_{k \geq 1} C([0,T], \cL_k^1)$ is compact with respect to the product topology $\tau_{\text{prod}}$ generated by the metrics $\wh \eta_k$ (defined in Section \ref{sec:outline}). For any limit point $ \Gamma_{\infty,t} = \{ \gamma_{\infty,t}^{(k)} \}_{k \geq 1}$, $ \gamma^{(k)}_{\infty,t}$ is symmetric w.r.t. permutations, $ \gamma^{(k)}_{\infty,t} \geq 0$, and \begin{equation}\label{eq:bou} \tr \; \gamma^{(k)}_{\infty,t} \leq 1 \,\end{equation} for every $k \geq 1$. \end{theorem} \begin{proof} By a standard argument it is enough to prove the compactness of $\gamma_{N,t}^{(k)}$ for fixed $k \geq 1$ with respect to the metric $\wh \eta_k$. To this end, it is enough to show the equicontinuity of $\gamma_{N,t}^{(k)}$ with respect to the metric $\eta_k$. A useful criterium for equicontinuity is given by the following lemma, whose proof can be found in \cite[Proposition 9.2]{ESY2}. \begin{lemma}\label{lm:equi} Fix $k \in \bN$ and $T > 0$. A sequence $\gamma_{N, t}^{(k)} \in \cL^1_k$, $N=k,k+1, \ldots$, with $\gamma_{N,t}^{(k)} \geq 0$ and $\tr \; \gamma_{N,t}^{(k)} = 1$ for all $t \in [0,T]$ and $N \geq k$, is equicontinuous in $C([0,T], \cL^1_k)$ with respect to the metric $\eta_k$, if and only if there exists a dense subset $\cJ_k$ of $\cK_k$ such that for any $J^{(k)}\in \cJ_k$ and for every $\eps >0$ there exists a $\delta > 0$ such that \begin{equation}\label{eq:equi02} \sup_{N\ge 1}\Big| \tr \; J^{(k)} \left( \gamma_{N,t}^{(k)} - \gamma_{N,s}^{(k)} \right) \Big| \leq \eps \end{equation} for all $t,s \in [0,T]$ with $|t -s| \leq \delta$. \end{lemma} We prove (\ref{eq:equi02}) for all $J^{(k)} \in \cK_k$ such that $\tri J^{(k)} \tri < \infty$, where we introduced the norm \begin{equation}\label{eq:tri}\begin{split} \tri J^{(k)} \tri = \sup_{\bx_k , \bx'_k} &\la x_1 \ra^4 \dots \la x_k \ra^4 \la x'_1 \ra^4 \dots \la x'_k \ra^4 \, \\ &\times \left( |J^{(k)} (\bx_k; \bx'_k)| + \sum_{j=1}^k \left( |\nabla_{x_j} J^{(k)} (\bx_k; \bx'_k)| + |\nabla_{x'_j} J^{(k)} (\bx_k; \bx'_k)| \right) \right). \end{split}\end{equation} It is simple to check that the set of $J^{(k)} \in \cK_k$ for which $\tri J^{(k)} \tri < \infty$ is dense in $\cK_k$. \medskip Rewriting the BBGKY hierarchy (\ref{eq:BBGKY}) in integral form and multiplying with an arbitrary observable $J^{(k)} \in \cK_k$ with $\tri J^{(k)} \tri < \infty$, we obtain that, for any $r \leq t$, \begin{equation}\label{eq:equi-1} \begin{split} \Big| \tr \, J^{(k)} \left( \gamma_{N,t}^{(k)} - \gamma_{N,r}^{(k)} \right) \Big| \leq \; &\sum_{j=1}^k \int_r^t \rd s \, \Big| \tr \; J^{(k)} [ -\Delta_j , \gamma_{N,s}^{(k)}] \Big| + \sum_{i0$, and $\sum_{\ell} \lambda^{(k)}_{\ell} =1$ (here we omitted the dependence of $\xi^{(k)}_{\ell}$ and $\lambda^{(k)}_{\ell}$ on $N,s$ from the notation). Then we find, for example for the term with $i=1, j=2$, \begin{equation}\label{eq:equi-2ndterm} \begin{split} \tr \; J^{(k)} \, & V_N (x_1 -x_2) \gamma_{N,s}^{(k)} = \sum_{\ell} \lambda^{(k)}_{\ell} \, \int \rd \bx_k \, \rd \bx'_k \; J^{(k)} (\bx'_k ; \bx_k) V_N (x_1 -x_2) \xi^{(k)}_{\ell} ( \bx_k) \overline{\xi}^{(k)}_{\ell} (\bx'_k). \end{split} \end{equation} Denoting by $W_N$ the wave operator associated with the Hamiltonian $\fh_N = -\Delta + (1/2) V_N (x)$, and by $W_{N,(i,j)}$ the wave operator $W_N$ acting on the variable $x_j -x_i$ (as defined in (\ref{eq:Wij})), we can estimate (introducing the new variables $u= (x_1+ x_2)/2$ and $v = x_1-x_2$) \begin{equation*} \begin{split} \Big| \int \rd \bx_k \, \rd \bx'_k & J^{(k)} (\bx'_k ; \bx_k) V_N (x_1 -x_2) \xi^{(k)}_{\ell} ( \bx_k) \overline{\xi}^{(k)}_{\ell} (\bx'_k) \Big| \\ &= \Big| \int \rd u \rd v \rd x_3 \dots \rd x_k \rd \bx'_k \; J^{(k)} (\bx'_k ; u+ v/2, u-v/2, x_3, \dots ,x_k) \\ &\hspace{1.5cm} \times V_N (v) \xi^{(k)}_{\ell} (u+v/2,u-v/2,x_3, \dots ,x_k) \, \overline{\xi}^{(k)}_{\ell} (\bx'_k) \Big| \\ &= \Big| \int \rd u \rd v \rd v' \rd x_3 \dots \rd x_k \rd \bx'_k J^{(k)} (\bx'_k; u+v/2, u-v/2, x_3, \dots ,x_k)\\ &\hspace{1.5cm} \times V_N (v) W_N (v;v') \left(W_{N,(1,2)}^* \xi^{(k)}_{\ell} \right) (u+v'/2,u-v'/2,x_3, \dots ,x_k) \, \overline{\xi}^{(k)}_{\ell} (\bx'_k) \Big| \\ &\leq \int \rd u \rd v \rd v' \rd x_3 \dots \rd x_k \rd \bx'_k \Big| J^{(k)}(\bx'_k ; u+v/2, u-v/2, x_3, \dots ,x_k) \Big| \, |\xi_{\ell}^{(k)} (\bx'_k)| \\ &\hspace{1.5cm} \times V_N (v) |W_N (v;v')| \, \Big|\left(W_{N,(1,2)}^* \xi^{(k)}_{\ell} \right) (u+v'/2,u-v'/2,x_3, \dots ,x_k)\Big|. \end{split} \end{equation*} Applying a Schwarz inequality, we obtain \begin{equation}\label{eq:equi-est12a} \begin{split} \Big| \int &\rd \bx_k \, \rd \bx'_k \; J^{(k)} (\bx'_k ; \bx_k) V_N (x_1 -x_2) \xi^{(k)}_{\ell} ( \bx_k) \overline{\xi}^{(k)}_{\ell} (\bx'_k) \Big| \\ &\leq \| \xi_{\ell}^{(k)} \|^2 \; \left( \sup_v \int \rd v' \, |W_N (v;v')|\right) \left(\int \rd v \, V_N (v)\right) \\ &\hspace{1cm} \times \; \sup_{\bx'_k, v} \left( \int \rd u \rd x_3 \dots \rd x_k \; \Big| J^{(k)} (\bx'_k; u+v/2, u-v/2, x_3, \dots ,x_k) \Big| \right) \\ &\hspace{.5cm}+\sup_{\bx_k} \left( \int \rd \bx'_k \; \Big| J^{(k)} (\bx'_k ; \bx_k) \Big|\right) \, \left(\int \rd v \, V_N (v) \right) \\ &\hspace{1cm} \times \sup_v \left(\int \rd u \rd v' \rd x_3 \dots \rd x_k \; |W_N (v;v')| \Big|\left(W_{N,(1,2)}^* \xi^{(k)}_{\ell} \right) (u+v'/2,u-v'/2,x_3, \dots ,x_k)\Big|^2 \right) \\ &\leq \frac{C_k \, \tri J^{(k)} \tri}{N} \left( \| \xi_{\ell}^{(k)} \|^2 + \left\langle W_{N,(1,2)}^* \xi_{\ell}^{(k)}, \left( (\nabla_1 \cdot \nabla_2)^2 -\Delta_1 - \Delta_2 + 1\right) W_{N,(1,2)}^* \xi^{(k)}_{\ell} \right\rangle \right) \end{split} \end{equation} for a constant $C_k$ depending on $k$ (the norm $\tri \, . \, \tri$ is defined in (\ref{eq:tri})). In the last inequality we used Lemma \ref{lm:VL1} for the difference variable $v'$, and the second inequality from the bounds \begin{equation} \begin{split} \sup_{v' \in \bR^3} &\int \rd v \, |W(v;v')| \leq \| W \|_{L^1 \to L^1} < \infty \\ \sup_{v \in \bR^3} &\int \rd v' \, |W(v;v')| \leq \| W^* \|_{L^1 \to L^1} < \infty \end{split} \end{equation} which follow from Proposition \ref{prop:waveop}, part iv). {F}rom (\ref{eq:equi-2ndterm}) and from Proposition \ref{prop:apri2} we obtain that \begin{equation}\label{eq:equi-est12} \begin{split} \Big| \tr \; J^{(k)} \, V_N (x_1 -x_2) \gamma_{N,s}^{(k)} \Big| & \leq \frac{C_k \tri J^{(k)} \tri}{N} \, \tr \; \left( (\nabla_1 \cdot \nabla_2)^2 -\Delta_1 - \Delta_2 + 1\right) W_{N,(1,2)}^* \gamma_{N,s}^{(k)} W_{N,(1,2)} \\ &\leq \frac{C_k \tri J^{(k)} \tri}{N} \end{split} \end{equation} for all $s\in \bR$ and a constant $C_k$ only depending on $k$ (and on the constant appearing on the r.h.s. of (\ref{eq:enerk})). Similarly to (\ref{eq:equi-est12}), we can also show that \begin{equation}\label{eq:equi-est12b} \Big| \tr \; J^{(k)} \, \gamma_{N,s}^{(k)} V_N (x_1 -x_2) \Big| \leq \frac{C_k \, \tri J^{(k)} \tri}{N}\, . \end{equation} Since (\ref{eq:equi-est12}) and (\ref{eq:equi-est12b}) remain valid for all summands in the second term on the r.h.s. of (\ref{eq:equi-1}), we obtain that, for all $k \in \bN$, for all $t \in [0,T]$ and for all $J^{(k)} \in \cK_k$ with $\tri J^{(k)} \tri < \infty$, \begin{equation}\label{eq:equi-3} \sum_{i0$ (in our analysis, we will need that $N^{-1/2} \ll \ell \ll N^{-1/3}$), we set \begin{equation}\label{eq:hdef} h (x): = e^{-\frac{\sqrt{x^2 + \ell^2}}{\ell}}. \end{equation} Note that $h \simeq 0$ if $|x| \gg \ell$, and $h \simeq e^{-1}$ if $|x| \ll \ell$. For $i=1,\dots ,N$ we define the cutoff function \begin{equation}\label{eq:thetajdef} \theta_i (\bx) := \exp \left(-\frac{1}{\ell^{\eps}} \sum_{j\neq i} h (x_i -x_j)\right) \end{equation} for some $\eps >0$. Note that $\theta_i (\bx)$ is exponentially small if there is at least one other particle at distance of order $\ell$ from $x_i$, while $\theta_i(\bx)$ is exponentially close to 1 if there is no other particle near $x_i$ (on the length scale $\ell$). Next we define \begin{equation} \theta_i^{(n)}(\bx) := \theta_i (\bx)^{2^n} = \exp \left( -\frac{2^n}{\ell^{\eps}} \sum_{j \neq i} h (x_i -x_j) \right) \, \end{equation} and their cumulative versions, for $n, k\in \bN$, \be\label{eq:thetan} \Theta_k^{(n)} (\bx) := \theta_1^{(n)} (\bx) \dots \theta_k^{(n)} (\bx) = \exp \left( -\frac{2^n}{\ell^{\eps}} \sum_{i\leq k}\sum_{j \neq i} h (x_i -x_j) \right) \; . \ee To cover all cases in one formula, we introduce the notation $\Theta_k^{(n)}=1$ for any $k\leq 0$, $n\in \bN$. Some important properties of the function $\Theta_k^{(n)}$, used throughout the proof of Proposition \ref{prop:hk}, are collected, for completeness, in Lemma \ref{lm:theta}. \bigskip \begin{proposition}\label{prop:hk} Suppose that $V \geq 0$, with $|\nabla^{\alpha} V(x)| \leq C$ for all $|\alpha| \leq 2$. Let $\psi \in L^2_s (\bR^{3N})$ be a function symmetric in all its variables. Suppose that $\ell\gg N^{-1/2}$ (in the sense that there exists $\delta >0$ with $N^{1/2} \ell \geq N^{\delta}$). There exists $C_0 >0$ such that for every integer $k\ge 1$ there exists $N_0 = N_0 (k)$ such that \be \langle\psi, (H_N+N)^k\psi\rangle \ge C_0^k N^k \int \rd \bx \; \Theta_{k-1}^{(k)} (\bx) \, |\nabla_1\ldots \nabla_k\psi (\bx)|^2 \label{eq:hk} \ee for all $N\ge N_0$. \end{proposition} \begin{proof} We use induction over $k$. For $k=1$ the statement follows directly from $V_N\ge0$, since on the symmetric subspace \be H_N+N \ge \sum_{i=1}^N \nabla_i^*\nabla_i = N\nabla_j^*\nabla_j \label{eq:k=1} \ee for any fixed $j=1,2, \ldots, N$. We will present the $k=2$ case in details and then comment on the general case. Set $T=H_N+N\ge 0$ for brevity and use the induction hypothesis \be\label{eq:for} \begin{split} T^2 & \ge C_0N T^{1/2} \nabla_1^*\nabla_1 T^{1/2} \\ &\ge C_0N T^{1/2} \nabla_1^*\theta_1^4\nabla_1 T^{1/2} \\ & \ge \frac{1}{2}C_0N \nabla_1^* T^{1/2}\theta_1^4 T^{1/2}\nabla_1 - C_0N [T^{1/2}, \nabla_1]^*\theta_1^4 [ T^{1/2}, \nabla_1 ] \\ & \ge \frac{1}{4}C_0N \nabla_1^* \theta_1^2 T\theta_1^2\nabla_1 - C_0N \nabla_1^* [T^{1/2},\theta_1^2]^*[ T^{1/2},\theta_1^2]\nabla_1 - C_0N [T^{1/2}, \nabla_1]^*\theta_1^4 [ T^{1/2}, \nabla_1 ]. \end{split} \ee In the first term we use that $H_N + N \geq \sum_{j=2}^N \nabla^*_j \nabla_j$ to obtain \be\label{eq:back} \begin{split} \frac{1}{4}C_0 N \nabla_1^* \theta_1^2 T\theta_1^2\nabla_1 &\ge \frac{1}{4}C_0 N (N-1) \nabla_1^* \theta_1^2\nabla_2^*\nabla_2\theta_1^2\nabla_1 \\ & \ge \frac{1}{8}C_0 N^2 \nabla_1^* \nabla_2^*\theta_1^4\nabla_2\nabla_1 - C_0N^2 \nabla_1^*[ \nabla_2, \theta_1^2]^* [ \nabla_2, \theta_1^2]\nabla_1 \end{split} \ee for all $N$ large enough. Since $\Theta_1^{(2)} = \theta_1^4$, we would obtain \eqref{eq:hk} for $k=2$ with $C_0 < 1/8$ once we show that the commutator terms in \eqref{eq:for} and \eqref{eq:back} are negligible. The commutator in \eqref{eq:back} on symmetric functions can be estimated by \be\label{eq:symme} \begin{split} C_0N^2 \nabla_1^*[ \nabla_2, \theta_1^2]^* [ \nabla_2, \theta_1^2]\nabla_1 & = \frac{C_0N^2}{N-1}\sum_{j=2}^N \nabla_1^* (\nabla_j\theta_1^2)^2\nabla_1\\ &\leq O(\ell^{-2}N^{-1}) N^2 \nabla_1^* \nabla_1 \leq O(\ell^{-2}N^{-1})T^2 = o(1) T^2 \end{split} \ee where we used (\ref{eq:lmthetaiv}), recalling that $\theta_1^2 = \Theta_1^{(1)}$, and we also used $T\ge N$ and \eqref{eq:k=1}. To estimate the two commutators in \eqref{eq:for}, we express \be [T^{1/2}, A] = \frac{1}{\pi} \int_0^\infty \frac{1}{T+s} \, \left[A,T \right] \, \frac{1}{T+s} \, s^{1/2}\rd s \label{eq:com} \ee for any operator $A$. To estimate the first commutator term in \eqref{eq:for} we note that, by Schwarz inequality, \be \begin{split} [T^{1/2}, \theta_1^2]^*[T^{1/2}, \theta_1^2] \leq & \; C \, (\log K)\int_0^K \frac{1}{T+s} \, [\theta_1^2,T]^* \, \frac{1}{(T+s)^2} \, [\theta_1^2,T] \, \frac{1}{T+s} \, \langle s\rangle^2 \, \rd s \; \\ &+ C \int_K^\infty \frac{1}{T+s} \, [\theta_1^2,T]^* \, \frac{1}{(T+s)^2} \, [\theta_1^2,T] \, \frac{1}{T+s} \, \langle s\rangle^{5/2} \, \rd s, \end{split} \label{eq:comm} \ee where $K=\exp(N^\e)$ for some $\e>0$. Estimating $(T+s)^{-2}\leq \langle s\rangle ^{-2}$ (using $T\ge N$), we have \be \begin{split} N \nabla_1^* [T^{1/2},\theta_1^2]^*[ T^{1/2},\theta_1^2]\nabla_1 \leq & \; c N^{1+\e} \int_0^K \nabla_1^* \, \frac{1}{T+s} \, [T,\theta_1^2]^*[T,\theta_1^2] \, \frac{1}{T+s} \, \nabla_1 \, \rd s \\ &+ c N \int_K^\infty \nabla_1^* \frac{1}{T+s} \, [T,\theta_1^2]^*[T,\theta_1^2] \, \frac{1}{T+s} \, \nabla_1 \, \langle s\rangle^{1/2} \, \rd s\, , \end{split} \label{eq:comm1} \ee and we can estimate \be \begin{split} [T,\theta_1^2]^*[T,\theta_1^2] & = \sum_{i,j} \big(2\nabla_j^* \cdot (\nabla_j\theta_1^2) + (\Delta_j\theta_1^2)\big) \big(2 (\nabla_i\theta_1^2)\cdot\nabla_i + (\Delta_i\theta_1^2) \big)\\ &\leq c \sum_{i,j} \Big[ \nabla_j^* \cdot (\nabla_j\theta_1^2)(\nabla_i\theta_1^2)\cdot\nabla_i + \nabla^*_i \cdot |\Delta_j\theta_1^2| \nabla_i + |\nabla_i\theta_1^2| |\Delta_j\theta_1^2| |\nabla_i\theta_1^2| + |\Delta_i\theta_1^2||\Delta_j\theta_1^2|\Big]\\ &\leq c \sum_{i,j} \nabla^*_i \cdot\big( |\nabla_j\theta_1^2|^2 + |\Delta_j\theta_1^2|\big) \nabla_i + c\Big(\sum_i |\Delta_i\theta_1^2|\Big)^2 \label{eq:key} \\ &\leq c \ell^{-2} \sum_i \nabla_i^*\cdot\nabla_i + c\ell^{-4} \leq c\ell^{-2} T \end{split} \ee by using (\ref{eq:lmthetaiv}) and (\ref{eq:lmthetavi}) and the fact that $\theta_1^2 =\Theta_1^{(1)}$. Thus, the first commutator term in \eqref{eq:for} is estimated as \be \begin{split} N \nabla_1^* [T^{1/2},\theta_1^2]^*[ T^{1/2},\theta_1^2]\nabla_1 \leq \; & O(N^{1+\e}\ell^{-2}) \, \int_0^K \nabla_1^* \, \frac{1}{T+s} \, T \, % \Big[ \sum_i (-\Delta_i) + N\Big] \frac{1}{T+s} \, \nabla_1 \rd s \\ &+ O(N\ell^{-2}) \, \int_K^\infty \nabla_1^* \, \frac{1}{T+s} \, T \, \frac{1}{T+s} \, \nabla_1 \, \langle s\rangle^{1/2} \, \rd s\\ \leq \; &O(N^{1+\e}\ell^{-2}) \, \nabla_1^*\nabla_1 + O(N\ell^{-2}K^{-1/2}) \nabla_1^* \, T \, \nabla_1\\ \leq \; &O(N^{\e}\ell^{-2})T + o(1) T^2 \\ \leq \; &o(1) T^2 \end{split} \ee if we choose $\e > 0$ so small that $\ell^{-2}\ll N^{1-\e}$. When estimating the term $\nabla_1^* T\nabla_1$ in the last step, we could afford estimating any commutators, since $K^{-1/2}$ is exponentially small: \be\nonumber \begin{split} \nabla_1^*T\nabla_1 & = \frac{1}{N}\sum_j \nabla_j^* T\nabla_j = \frac{1}{N}\sum_j\Big[ (-\Delta_j) T - \nabla_j^*\cdot \sum_i(\nabla V_N)(x_j-x_i)\Big] \\ & = N^{-1}T^2 - N^{-1} \sum_{ij} \Big[ V_N(x_i-x_j)T -\nabla_j^*\cdot(\nabla V_N)(x_j-x_i)\Big] \\ &\leq 2N^{-1}T^2 + O(N^8) \end{split} \ee using that $|V_N (x)| \leq C N^2$, and $|\nabla V_N (x)|\leq C N^3$, for all $x \in \bR^3$. Finally, we estimate the second commutator term in \eqref{eq:for} by using again Schwarz inequality in \eqref{eq:com} but this time we do not split the integration: \be \begin{split} [T^{1/2}, &\nabla_1]^* \theta_1^4 [ T^{1/2}, \nabla_1 ] \\ & \leq c \int_0^\infty \frac{1}{T+s} [\nabla_1, T]^* \frac{1}{T+s}\theta_1^4 \frac{1}{T+s} [\nabla_1, T]\frac{1}{T+s} \langle s\rangle^{5/2}\rd s \\ & \leq cN\sum_{i\neq 1} \int_0^\infty \frac{1}{T+s} (\nabla V_N)(x_1-x_i) \frac{1}{T+s}\theta_1^4 \frac{1}{T+s} (\nabla V_N)(x_1-x_i)\frac{1}{T+s} \langle s\rangle^{5/2}\rd s \end{split} \label{eq:seccomm} \ee where we used $[\nabla_1, T] =\sum_{i\neq 1} (\nabla V_N)(x_1-x_i)$. Since $T_0= \sum_j -\Delta_j + N$ is a positivity preserving operator and $V \geq 0$, $T$ is also positivity preserving and its resolvent kernel satisfies $$ \frac{1}{T+s} (\bx ; \by) \leq \frac{1}{ T_0 + s}(\bx ; \by), $$ and thus \be \label{eq:TT} \begin{split} I:= \Big\| (\nabla V_N)(x_1-x_i) &\frac{1}{T+s}\, \theta_1^4 \, \frac{1}{T+s} (\nabla V_N)(x_1-x_i) \Big\| \\ & \leq \Big\| \left|(\nabla V_N)(x_1-x_i)\right| \frac{1}{T_0+s} \, e^{-4\ell^{-\e} h(x_1-x_i)} \, \frac{1}{T_0+s} \, \left|(\nabla V_N)(x_1-x_i)\right| \Big\| \, , \end{split} \ee where we also estimated $\theta_1$ by keeping only one summand in its definition (\ref{eq:thetajdef}). Introducing the variable $y=x_1 -x_i$, and observing that $L^2 (\bR^{3N}, \rd \bx) \simeq L^2 (\bR^3, \rd y ; L^2 (\bR^{3(N-1)} , \rd z \rd x_2 \dots \widehat{\rd x_i} \dots \rd x_N))$ (where the hat means that the variable $x_i$ is omitted), we obtain that \begin{equation*}\begin{split} I \leq \; &\sup_{M \geq 0} \Big\| |(\nabla V_N)(y)| \, \frac{1}{-\Delta_y + M +N+s} e^{-4\ell^{-\e} h(y)} \frac{1}{\Delta_y + M +N+s} \, | (\nabla V_N)(y)| \Big\| \\ \leq \; & \sup_{M \geq 0} \Big\| |(\nabla V_N)(y)| \frac{1}{-\Delta_y + M +N+s} e^{-4\ell^{-\e} h(y)} \frac{1}{\Delta_y + M +N+s} |(\nabla V_N)(y)| \Big\|_{\text{HS}} \end{split} \end{equation*} where the norms on the last two lines are, respectively, the operator norm and the Hilbert-Schmidt norm of an operator over $L^2 (\bR^3, \rd y)$. The last equation implies that \be\label{eq:I} \begin{split} I\leq \; & \int \rd y \rd y' e^{-4\ell^{-\e}h(y)} \Big|\frac{1}{\Delta + M +N+s}(y,y')\Big|^2 |\nabla V_N(y')|^2 \\ \leq \; &\int \rd y \rd y' e^{-4\ell^{-\e}h(y)} \frac{e^{-2\sqrt{N+s}|y-y'|}}{|y-y'|^2} |\nabla V_N(y')|^2 \\ \le \; & O (e^{-\ell^{-\e}}) \end{split} \ee since $h\approx e^{-1}$ on the support of $V_N$. We will use this bound for $s\leq K:= \exp(c\ell^{-\e})$ with a sufficiently small $c>0$. From \eqref{eq:TT}, we also have the trivial bound \be I \leq \frac{N^6}{\langle s \rangle^2} \label{eq:trivi} \ee that will be used for large $s$. Inserting these estimates into \eqref{eq:seccomm}, we have $$ [T^{1/2}, \nabla_1]^*\theta_1^4 [ T^{1/2}, \nabla_1 ] \leq O(N^2e^{-\ell^{-\e}} ) \int_0^K \frac{\langle s\rangle^{5/2}\rd s}{(T+s)^2} + O(N^8) \int_K^\infty \frac{\langle s\rangle^{1/2}\rd s}{(T+s)^2} = O(e^{-c\ell^{-\e}} ), $$ i.e. this commutator term is subexponentially small in $N$ and this completes the proof of \eqref{eq:hk} for $k=2$. \bigskip The proof for general $k>2$ follows the same pattern as for $k=2$. Introduce the notation $$ D_k = \nabla_1\nabla_2 \ldots \nabla_k\, . $$ We recall the summation convention: for any operator $A$, we denote $$ D_k^*AD_k := \sum_{\alpha_1=1}^3\ldots \sum_{\alpha_k=1}^3 \nabla_{x_{1,\alpha_1}}^* \ldots \nabla_{x_{k,\alpha_k}}^* A \nabla_{x_{k,\alpha_k}}\ldots\nabla_{x_{1,\alpha_1}} $$ where $x_j = (x_{j,1}, x_{j,2}, x_{j,3})$ are the three coordinates of $x_j\in \bR^3$. Using the induction hypothesis, $\Theta_{k-1}^{(k)} \geq [\Theta_{k}^{(k)}]^2=\Theta_k^{(k+1)}$ and \eqref{eq:k=1} we obtain, similarly to \eqref{eq:for} and \eqref{eq:back}, \be \begin{split} T^{k+1} &\ge C_0^{k}N^{k} T^{1/2} D_{k}^* \Theta_{k-1}^{(k)} D_k T^{1/2} \\ &\ge C_0^{k}N^{k} T^{1/2} D_{k}^* [\Theta_{k}^{(k)}]^2 D_k T^{1/2}\\ &\ge \frac{1}{8} C_0^k N^{k+1} D_{k+1}^* \Theta_{k}^{(k+1)} D_{k+1} - C_0^kN^k D_k^* [T^{1/2}, \Theta_{k}^{(k)}]^*[T^{1/2}, \Theta_{k}^{(k)}] D_k \\ &\;\;\;\; - C_0^kN^k [T^{1/2}, D_k]^*\Theta_{k}^{(k+1)} [T^{1/2}, D_k] - C_0^kN^{k+1} D_k^* [\nabla_{k+1}, \Theta_{k}^{(k)}]^*[\nabla_{k+1}, \Theta_{k}^{(k)}]D_k , \end{split} \label{eq:Tk+1} \ee for all $N$ sufficiently large (depending on $k$). The first term gives the desired result if $C_0<1/8$; in the sequel we show that all three commutator terms are negligible. The first commutator in \eqref{eq:Tk+1} is estimated exactly as the first commutator in \eqref{eq:for}, after replacing $\theta_1^2 = \Theta_1^{(1)}$ with $\Theta_k^{(k)}$. The estimates \eqref{eq:comm} are \eqref{eq:comm1} are identical for $k>1$ as well. In the key estimate \eqref{eq:key}, the only properties we used about $\theta_1^2= \Theta_1^{(1)}$ from Lemma \ref{lm:theta} were those that hold for $\Theta_k^{(k)}$ as well. The last commutator in \eqref{eq:Tk+1} can be estimated similarly to \eqref{eq:symme} by using (\ref{eq:lmthetaiv}) \be\label{eq:symmek} \begin{split} C_0^kN^{k+1} D_k^*[ \nabla_{k+1}, \Theta_k^{(k)}]^* [ \nabla_{k+1}, \Theta_k^{(k)}]D_k & = \frac{C_0^kN^{k+1}}{N-k}\sum_{j=k+1}^N D_k^* (\nabla_j\Theta_k^{(k)})^2D_k \\ &\leq O(\ell^{-2}N^{-1}) N^{k+1} D_k^* \Theta_k^{(k-1)} D_k \\ & \leq O_k(\ell^{-2}N^{-1}) N T^k = o_k(1) T^{k+1} \end{split} \ee by the induction hypothesis and $T\ge N$ (here we use the notation $f = o_k (g)$ if $f/ g \to 0$ as $N \to \infty$ for fixed $k$; analogously for $f=O_k (g)$). Finally, the estimate of the second commutator in \eqref{eq:Tk+1} is similar to that of the second commutator in \eqref{eq:for}, but more commutators need to be computed. Similarly to \eqref{eq:seccomm} and taking the permutation symmetry into account, we have \be \begin{split} &[T^{1/2}, D_k]^* \Theta_{k}^{(k+1)} [T^{1/2}, D_k] \\ & \leq C_kN \sum_{i\neq k} \int_0^\infty \frac{1}{T+s} D_{k-1}^*(\nabla V_N)(x_k-x_i) \frac{1}{T+s}\Theta_{k}^{(k+1)} \frac{1}{T+s} (\nabla V_N)(x_k-x_i)D_{k-1} \frac{1}{T+s} \langle s\rangle^{5/2}\rd s \\ & \quad + C_k \int_0^\infty \frac{1}{T+s} D_{k-2}^*(\nabla^2 V_N)(x_k-x_{k-1}) \frac{1}{T+s}\Theta_{k}^{(k+1)}\\ &\hspace{7cm} \times \frac{1}{T+s} (\nabla^2 V_N)(x_k-x_{k-1})D_{k-2} \frac{1}{T+s} \langle s\rangle^{5/2}\rd s \end{split} \label{eq:seccommk} \ee We will need the following lemma whose proof is postponed. \begin{lemma}\label{lemma:exp} Let $\psi \in L^2_s (\bR^{3N})$ be a function symmetric in all its variables and let $\delta>0$. Choose a strictly increasing sequence of positive constants $\{ c_k \}_{k \geq 1}$. Then for every integer $k\ge 1$ there exists $N_0 = N_0 (k,\delta)$ such that \be \langle\psi, (H_N+N)^k\psi\rangle \ge e^{-c_k N^\delta} \int \rd \bx \, |\nabla_1\ldots \nabla_k\psi (\bx)|^2 \label{eq:hk1} \ee for all $N\ge N_0$. \end{lemma} We demonstrate the estimate of the first term in (\ref{eq:seccommk}), the second one is similar. Using $\Theta_k^{(k+1)} \leq e^{-\ell^{-\e}h(x_k-x_i)}$, we obtain, similarly to \eqref{eq:TT}--\eqref{eq:trivi} that $$ I: = \Bigg\| (\nabla V_N)(x_k-x_i) \frac{1}{T+s}\Theta_{k}^{(k+1)} \frac{1}{T+s} (\nabla V_N)(x_k-x_i)\Bigg\| \leq O(e^{-\ell^{-\e}}) $$ and also $$ I\leq \frac{N^6}{\langle s\rangle^2}. $$ Let $K:= \exp(c\ell^{-\e})$ with a sufficiently small $c>0$. Choosing a sufficiently small $\delta$, so that $N^\delta \ll \ell^{-\e}$, by using \eqref{eq:hk1}, we have, \be\label{eq:nablaVterm} \begin{split} \int_0^\infty \frac{1}{T+s} & D_{k-1}^*(\nabla V_N)(x_k-x_i) \frac{1}{T+s}\Theta_{k}^{(k+1)} \frac{1}{T+s} (\nabla V_N)(x_k-x_i)D_{k-1} \frac{1}{T+s} \langle s\rangle^{5/2}\rd s \\ &\leq O(e^{-\ell^{-\e} +c_{k-1}N^\delta}) \int_0^K \frac{T^{k-1}}{(T+s)^2}\langle s\rangle^{5/2}\rd s + O(e^{c_{k-1}N^\delta})\int_K^\infty \frac{T^{k-1}}{(T+s)^2}\frac{N^6}{\langle s\rangle^2} \langle s\rangle^{5/2}\rd s \\ &\leq O(e^{-c'\ell^{-\e}})T^{k-1} \leq o(1) T^{k+1}. \end{split} \ee This completes the proof of Proposition \ref{prop:hk}. \end{proof} \begin{proof}[Proof of Lemma \ref{lemma:exp}] We proceed by a step two induction on $k$; for $k=1$ the claim follows from \eqref{eq:k=1}. We now consider the $k=2$ case. Similarly to \eqref{eq:h1}, but keeping also the $h_j^2$ terms in the expansion of $H_N^2$, we find \be\label{eq:lm62} \begin{split} T^2 & \ge N (N-1) \Big( -\Delta_1 + \frac{1}{2} V_N(x_1-x_2)\Big) \Big( -\Delta_2 + \frac{1}{2} V_N(x_1-x_2)\Big) + N \Big( -\Delta_1 + \frac{1}{2} V_N(x_1-x_2)\Big)^2 \\ & \ge (N^2/2) \Big( D_2^*D_2 - 2 \nabla_1^*\nabla_1 - 2 \nabla_2^*\nabla_2 - 4\| \nabla V_N\|_\infty^2 \Big) + N\Big( (\nabla_1^*\nabla_1)^2 - 2 \nabla_1^*\nabla_1- 4\| \nabla V_N\|_\infty^2 \Big)\\ &\ge (N^2/2) D_2^*D_2 + N (\nabla_1^*\nabla_1)^2 - CN^2 \nabla_1^*\nabla_1 - CN^8 \\ &\ge (N^2/2) D_2^*D_2 - O(N^8). \end{split} \ee Combining this bound with $T^2 \geq N^2$, it follows that \be T^2\ge cN^{-4}D_2^*D_2 \label{eq:T} \ee for a sufficiently small positive $c$. Now we show how to go from $k$ to $k+2$. By the induction hypothesis, we have \be T^{k+2} \ge e^{-c_kN^\delta}TD_k^* D_k T \ge e^{-c_kN^\delta}\Big( \frac{1}{2} D_k^* T^2 D_k - 2 [D_k, T]^*[D_k, T]\Big). \label{eq:Tk+2} \ee In the first term we can use \eqref{eq:T} in the form $T^2\ge cN^{-4} \nabla_{k+1}^*\nabla_{k+2}^*\nabla_{k+2}\nabla_{k+1}$, which holds for all $N$ large enough (because of the factors $D_k$, we only have symmetry on the last $N-k$ variables; this means that instead of (\ref{eq:lm62}), we are going to obtain $T^2 \geq (N-k) (N-k-1) \Delta_{k+1} \Delta_{k+2} \geq (N^2/2) \Delta_{k+1} \Delta_{k+2}$ for all $N$ large enough). The commutator term, after several Schwarz inequalities, can be estimated as \be\label{eq:commm} \begin{split} [D_k, T]^*[D_k, T] & \leq C_k \Big( N^2 D_{k-1}^* \|\nabla V_N \|^2 D_{k-1} + D_{k-2}^* \|\nabla^2 V_N \|^2 D_{k-2}\Big) \\ &\leq C_k N^8 \Big( D_{k-1}^* D_{k-1} + D_{k-2}^* D_{k-2}\Big) \\ &\leq C_k N^8 \, e^{c_{k-1} N^\delta}T^{k-1} \leq C_k N^8 \, e^{c_{k-1} N^\delta}T^{k+2} \, , \end{split} \ee where we used the induction hypothesis for $k-1$ and $k-2$ and, by convention, $D_m=1$ for $m\leq 0$. Inserting this estimate into \eqref{eq:Tk+2}, we obtain $$ T^{k+2} \ge cN^{-4} e^{-c_kN^\delta} D_{k+2}^*D_{k+2} - C_k N^8 e^{-(c_k-c_{k-1})N^\delta} T^{k+2} \; . $$ Since $c_k$ is strictly increasing, we obtain \eqref{eq:hk1} for $k+2$. Actually the proof shows that a sufficiently large $k$-dependent negative power, $N^{-\beta_k}$, would suffice on the r.h.s. of \eqref{eq:hk1} instead of the subexponentially small prefactor. \end{proof} \bigskip The higher order energy estimates proved in Proposition \ref{prop:hk} are used to show the following strong a-priori estimates on the limit points $\Gamma_{\infty,t} = \{ \gamma^{(k)}_{\infty,t} \}_{k \geq 1}$ of the sequence $\Gamma_{N,t}$. \begin{theorem}\label{thm:aprik} Suppose that the assumptions of Theorem \ref{thm:main2} are satisfied and fix $T>0$. Assume moreover that $\Gamma_{\infty,t}^{(k)} = \{ \gamma^{(k)}_{\infty,t} \}_{k \geq 1} \in \bigoplus_{k \geq 1} C([0,T], \cL^1_k)$ is a limit point of the sequence $\Gamma_{N,t} = \{ \gamma^{(k)}_{N,t} \}_{k=1}^N$ with respect to the topology $\tau_{\text{prod}}$. Then \begin{equation}\label{eq:aprik} \tr \; (1-\Delta_1) \dots (1-\Delta_k) \gamma^{(k)}_{\infty,t} \leq C^k \end{equation} for all $k \geq 1$ and $t \in [0,T]$. \end{theorem} \begin{proof} Theorem \ref{thm:aprik} follows from the higher order energy estimates of Theorem \ref{prop:hk}; the proof of this fact can be found in \cite[Proposition 6.3]{ESY}. \end{proof} \section{Convergence to the infinite hierarchy.} \setcounter{equation}{0} In order to prove Theorem \ref{thm:main2}, we need to prove the convergence of the BBGKY hierarchy towards a hierarchy of infinitely many equations. In the argument, we will make use of the apriori bounds from Propositions \ref{prop:apri2} and Theorem \ref{thm:aprik} for $k=2$. \begin{theorem}\label{thm:conv} Suppose that the assumptions of Theorem \ref{thm:main2} are satisfied and fix $T >0$. Suppose that $\Gamma_{\infty,t} = \{ \gamma^{(k)}_{\infty,t} \}_{k \geq 1} \in \bigoplus_{k \geq 1} C([0,T] , \cL_k^1)$ is a limit point of $\Gamma_{N,t} = \{ \gamma_{N,t}^{(k)} \}_{k =1}^N$ with respect to the topology $\tau_{\text{prod}}$. Then $\Gamma_{\infty,t}$ is a solution to the infinite hierarchy \begin{equation}\label{eq:conv} \gamma^{(k)}_{\infty,t} = \cU^{(k)} (t) \gamma^{(k)}_{\infty,0} - 8 \pi a_0 i \sum_{j=1}^k \int_0^t \rd s \, \cU^{(k)} (t-s) \tr_{k+1} \left[ \delta (x_j - x_{k+1}), \gamma_{\infty,s}^{(k+1)} \right] \end{equation} with initial data $\gamma_{\infty,0}^{(k)} = |\ph \rangle \langle \ph|^{\otimes k}$. Here $\cU^{(k)} (t)$ denotes the free evolution of $k$ particles defined in~(\ref{eq:Uk}). \end{theorem} \begin{proof} Fix $k \geq 1$. Passing to an appropriate subsequence, we can assume that, for every $J^{(k)} \in \cK_k$, \begin{equation}\label{eq:conv-0} \sup_{t \in [0,T]} \, \tr \; J^{(k)} \, \left( \gamma_{N,t}^{(k)} - \gamma_{\infty,t}^{(k)} \right) \to 0 \qquad \text{as } N \to \infty\,. \end{equation} We will prove (\ref{eq:conv}) by testing the limit point against a certain class of observables, that is dense in $\cK_k$. To characterize the class of observables we are going to consider, we define, for an arbitrary integer $k \geq 1$, $$ \Omega_k : = \prod_{j=1}^k \left( \la x_j \ra + \la i \nabla_j \ra \right) \; . $$ We will consider $J^{(k)} \in \cK_k$ such that \begin{equation}\label{eq:assJ} \Big\| \Omega_k^7 J^{(k)}\Omega_k^7 \Big\|_{\text{HS}} < \infty , \end{equation} where $\| A \|_{\text{HS}}$ denotes the Hilbert-Schmidt norm of the operator $A$. Note that the set of observables $J^{(k)}$ satisfying the condition (\ref{eq:assJ}) is a dense subset of $\cK_k$. Moreover, using the fact that $e^{i\Delta_j t} \la x_j \ra e^{-i\Delta_j t} = \la x_j - i t \nabla_j \ra$, it follows that \begin{equation}\label{eq:UkHS} \left\| \Omega_k^7 \, \cU^{(k)} (t) J^{(k)} \Omega_k^7 \right\|_{\text{HS}} \leq C \, (1 + |t|)^7 \left\| \Omega_k^7 \, J^{(k)} \Omega_k^7 \right\|_{\text{HS}} \,. \end{equation} Note also that, with the norm $\tri J^{(k)} \tri$ defined in (\ref{eq:tri}), we have \begin{equation}\label{eq:triJj} \tri J^{(k)} \tri \leq C_k \left\| \Omega_k^7 \, J^{(k)} \Omega_k^7 \right\|_{\text{HS}} \, \end{equation} for a constant $C_k$ only depending on $k$ (see \cite{ESY}, Eq. (7.8)). Combining (\ref{eq:UkHS}) with (\ref{eq:triJj}), we also have \begin{equation}\label{eq:triJjt} \tri \cU^{(k)} (t) J^{(k)} \tri \leq C_k \, (1 + |t|)^7 \, \left\| \Omega_k^7 \, J^{(k)} \Omega_k^7 \right\|_{\text{HS}}. \end{equation} \medskip In order to prove Theorem \ref{thm:conv} it is enough to show that, for every $J^{(k)} \in \cK_k$ satisfying (\ref{eq:assJ}), \begin{equation}\label{eq:conv-1} \tr \, J^{(k)} \gamma_{\infty,0}^{(k)} = \tr \, J^{(k)} |\ph \rangle \langle \ph|^{\otimes k} \end{equation} and \begin{equation}\label{eq:conv-2} \begin{split} \tr \; J^{(k)} \gamma_{\infty,t}^{(k)} = \tr \; J^{(k)} \cU^{(k)} (t) \gamma_{\infty,0}^{(k)} -8\pi a_0 i \sum_{j=1}^k \int_0^t \rd s \tr \, J^{(k)} \cU^{(k)} (t-s) \left[ \delta (x_j -x_{k+1}), \gamma^{(k+1)}_{\infty,s} \right]\, \end{split} \end{equation} for all $t \in [0,T]$. \medskip The relation (\ref{eq:conv-1}) follows from the assumption (\ref{eq:init}) and (\ref{eq:conv-0}). \medskip In order to prove (\ref{eq:conv-2}), we fix $t \in [0,T]$, we rewrite the BBGKY hierarchy (\ref{eq:BBGKY}) in integral form and we test it against the observable $J^{(k)}$. We obtain \begin{equation}\label{eq:conv-4} \begin{split} \tr \; J^{(k)} \, \gamma_{N,t}^{(k)} = \; & \tr \; J^{(k)} \, \cU^{(k)} (t) \gamma_{N,0}^{(k)} - i \sum_{i0$, we have \begin{equation}\label{eq:NVN-term} \begin{split} \Big| \tr \; J^{(k)} &\cU^{(k)} (t-s) N V_N (x_1 -x_{k+1}) \gamma^{(k+1)}_{N,s} - 8 \pi a_0 \tr \; J^{(k)} \cU^{(k)} (t-s) \delta (x_1 -x_{k+1}) \gamma^{(k+1)}_{\infty,s} \Big| \\ \leq \; & \Big| \tr \; J_{s-t}^{(k)} \, \left[ N V_N (x_1 -x_{k+1}) W_{N,(1,k+1)} - 8\pi a_0 \, \delta (x_1 - x_{k+1}) \right] W^*_{N,(1,k+1)} \gamma^{(k+1)}_{N,s} \Big| \\&+ 8 \pi a_0 \, \Big| \tr \; J_{s-t}^{(k)} \left[ \delta (x_1 -x_{k+1}) - \, h_{\alpha} (x_1 - x_{k+1}) \right] W^*_{N,(1,k+1)} \gamma^{(k+1)}_{N,s} \Big|\\ &+ 8 \pi a_0 \, \Big| \tr\; J_{s-t}^{(k)} \, h_{\alpha} (x_1 - x_{k+1}) \, (W_{N,(1,k+1)}^* -1) \gamma^{(k+1)}_{N,s} \Big| \\ &+ 8 \pi a_0 \, \Big| \tr\; J_{s-t}^{(k)} \, h_{\alpha} (x_1 - x_{k+1}) \, \left( \gamma^{(k+1)}_{N,s} - \gamma^{(k+1)}_{\infty,s} \right) \Big| \\ &+ 8 \pi a_0 \, \Big| \tr\; J_{s-t}^{(k)} \; \left[ h_{\alpha} (x_1 - x_{k+1}) -\delta (x_1 -x_{k+1}) \right] \, \gamma^{(k+1)}_{\infty,s} \Big|. \end{split} \end{equation} Here we insert the wave operator $W_{N, (1,k+1)}$, because we only have a-priori bounds on the quantity $W^*_{N,(1,k+1)} \gamma_{N,s}^{(k+1)}$ . Then we replace $NV_N (x_1 -x_{k+1}) W_{N,(1,k+1)}$ by $8 \pi a_0 \delta (x_1 -x_{k+1})$. Afterwards, in order to remove the inverse wave operator $W^*_{N, (1,k+1)}$ and to take the limit $\gamma_{N,s}^{(k+1)} \to \gamma_{\infty,s}^{(k+1)}$, we need to replace the delta-function by the bounded potential $h_{\alpha}$. At the end, $h_{\alpha}$ is changed back to the $\delta$-function. \medskip In Lemma \ref{lm:term1} and Lemma \ref{lm:term2} we prove that, for every $k \in \bN$, for every $0 \leq s \leq t \leq T$, and for every $J^{(k)} \in \cK_k$ with (\ref{eq:assJ}), \begin{equation}\label{eq:tm1to0} \Big| \tr \; J_{s-t}^{(k)} \, \left[ N V_N (x_1 -x_{k+1}) W_{N,(1,k+1)} - 8\pi a_0 \, \delta (x_1 - x_{k+1}) \right] W^*_{N,(1,k+1)} \gamma^{(k+1)}_{N,s} \Big| \to 0 \end{equation} as $N \to \infty$ and that \begin{equation}\label{eq:tm2to0} \Big| \tr \; J_{s-t}^{(k)} \, \left[ \delta (x_1 -x_{k+1}) - \, h_{\alpha} (x_1 - x_{k+1}) \right] W^*_{N,(1,k+1)} \gamma^{(k+1)}_{N,s} \Big| \to 0 \end{equation} as $\alpha \to 0$, uniformly in $N$. \medskip As for the third term on the r.h.s. of (\ref{eq:NVN-term}) we remark that, for fixed $k \in \bN$, $s \in [0,T]$, $J^{(k)} \in \cK_k$ and $\alpha >0$, \begin{equation}\label{eq:WNto1} \Big| \tr \; J_{s-t}^{(k)} \, h_{\alpha} (x_1 -x_{k+1}) (W_{N,(1,k+1)}^* -1 ) \gamma^{(k+1)}_{N,s} \Big| \to 0 \end{equation} as $N \to \infty$. In fact, for the bounded operator $A= J_{s-t}^{(k)}\, h_{\alpha} (x_1 -x_{k+1}) $, we can use the spectral decomposition $ \gamma^{(k+1)}_{N,s} = \sum_j \alpha_j |f_j\rangle \langle f_j|$ with $\sum_j \alpha_j =1$, $\alpha_j >0$, $\| f_j \|=1$, and estimate \begin{equation} \begin{split} \Big |\tr \; A \, (W_{N,(1,k+1)}^* -1 ) \, \gamma^{(k+1)}_{N,s} \Big| & \leq \| A \| \sum_j \alpha_j \| (W_{N,(1,k+1)}^* -1 ) f_j\|^2 \cr &\leq C\|A \| N^{-1/3} \; \tr (1-\Delta_1 - \Delta_k )\gamma^{(k+1)}_{N,s} \cr &\leq C\|A \|N^{2/3}\langle \psi_{N, s}, (H_N + N)\psi_{N, s}\rangle \leq C\| A\|N^{-1/3} \end{split} \end{equation} by the energy conservation and \eqref{eq:assH1}. From the first to the second line we used Lemma \ref{lm:WNto1}. Since the operator $A$ is bounded for any fixed $J^{(k)}$ and $\alpha>0$, we obtain \eqref{eq:WNto1}. \medskip To control the fourth term on the r.h.s. of (\ref{eq:NVN-term}) we observe that, for arbitrary $\delta >0$, \begin{equation} \begin{split} \tr\; J_{s-t}^{(k)} \, h_{\alpha} &(x_1 - x_{k+1}) \, \left( \gamma^{(k+1)}_{N,s} - \gamma^{(k+1)}_{\infty,s} \right) \\ = \; &\tr \; J_{s-t}^{(k)} \, h_{\alpha} (x_1 -x_{k+1}) \frac{1}{1+ \delta (1-\Delta_{k+1})^{1/2}} \left( \gamma^{(k+1)}_{N,s} - \gamma_{\infty,s}^{(k+1)}\right) \\ &+ \tr \; J_{s-t}^{(k)} \, h_{\alpha} (x_1 -x_{k+1}) \left(1-\frac{1}{1+ \delta ( 1-\Delta_{k+1})^{1/2}}\right) \left(\gamma^{(k+1)}_{N,s} -\gamma^{(k+1)}_{\infty,s}\right)\,. \end{split} \end{equation} The first term on the r.h.s. of the last equation converges to zero, as $N \to \infty$, for every fixed $\delta,\alpha >0$; this follows from the assumption (\ref{eq:conv-0}) and from the observation that $J_{s-t}^{(k)} h_{\alpha} (x_1 - x_{k+1}) (1+ \delta (1-\Delta_{k+1}))^{-1}$ is a compact operator on $L^2 (\bR^{3(k+1)})$. As for the second term, we notice that it can be bounded by \begin{equation} \begin{split} \Big| \tr \; J_{s-t}^{(k)} \, h_{\alpha} (x_1 -x_{k+1}) & \left(1-\frac{1}{1+ \delta \left( 1-\Delta_{k+1})^{1/2}\right)}\right) \left(\gamma^{(k+1)}_{N,s}-\gamma^{(k+1)}_{\infty,s}\right) \Big| \\ & \leq \delta \| J^{(k)} \| \, \| h_{\alpha} \|_{\infty} \tr \Big| (1-\Delta_{k+1})^{1/2} \left( \gamma^{(k+1)}_{N,s}-\gamma^{(k+1)}_{\infty,s}\right) \Big| \\ & \leq \delta \alpha^{-3} \| J^{(k)} \| \| h \|_{\infty} \left( \tr \; (1-\Delta_{k+1}) \gamma^{(k+1)}_{N,s} + \tr \; (1- \Delta_{k+1}) \gamma_{\infty,s}^{(k+1)} \right) \\ & \leq C \delta \alpha^{-3} \end{split}\end{equation} uniformly in $N$. Choosing, for example, $\delta = \alpha^4$, it follows that \begin{equation}\label{eq:eta} \Big| \tr\; J_{s-t}^{(k)} \, h_{\alpha} (x_1 - x_{k+1}) \, \left( \gamma^{(k+1)}_{N,s} - \gamma^{(k+1)}_{\infty,s} \right) \Big| \leq \eta (\alpha,N) + C \alpha \end{equation} where $\eta (\alpha,N) \to 0$ as $N \to \infty$, for every fixed $\alpha >0$, and where the constant $C$ only depends on $J^{(k)}$. \medskip Finally, using Lemma \ref{lm:sobsob} and Theorem \ref{thm:aprik}, the last term on the r.h.s. of (\ref{eq:NVN-term}) can be controlled by \begin{equation}\label{eq:term5} \begin{split} \Big| \tr\; J_{s-t}^{(k)} \; \left[ h_{\alpha} (x_1 - x_{k+1}) -\delta (x_1 -x_{k+1}) \right] \, \gamma^{(k+1)}_{\infty,s} \Big| \leq \; &C \alpha^{1/2} \, \tri J_{s-t}^{(k)} \tri \, \tr \; (1-\Delta_1) (1-\Delta_{k+1}) \gamma^{(k+1)}_{\infty,s} \\ \leq \; &C (k,T,J^{(k)}) \, \alpha^{1/2}\,. \end{split} \end{equation} \medskip {F}rom (\ref{eq:NVN-term}), (\ref{eq:tm1to0}), (\ref{eq:tm2to0}), (\ref{eq:WNto1}), (\ref{eq:eta}), and (\ref{eq:term5}) it follows that, for every $k \geq 1$, $0 \leq s \leq t \leq T$, and $J^{(k)} \in \cK_k$ with (\ref{eq:assJ}), \begin{equation}\label{eq:tmNto0} \Big| \tr \; J^{(k)} \cU^{(k)} (t-s) N V_N (x_1 -x_{k+1}) \gamma^{(k+1)}_{N,s} - 8 \pi a_0 \tr \; J^{(k)} \cU^{(k)} (t-s) \delta (x_1 -x_{k+1}) \gamma^{(k+1)}_{\infty,s} \Big| \to 0 \end{equation} as $N \to \infty$. %(in fact, for a fixed $\eps >0$, one can first choose $\alpha >0$ small enough %so that the %second and the fifth term on the r.h.s. of (\ref{eq:NVN-term}), as %well as the second term on %the r.h.s. of (\ref{eq:eta}), are smaller than %$\eps/4$, and then, choosing $N$ large enough, %one can make sure that also the % sum of the first and third term on the r.h.s. of %(\ref{eq:NVN-term}) and of %the term $\eta (\alpha,N)$ on the r.h.s. of (\ref{eq:eta}) is smaller %than $\eps/4$). Similarly to (\ref{eq:tmNto0}), we can also prove that \begin{equation}\label{eq:tmNto0b} \Big| \tr \; J^{(k)} \cU^{(k)} (t-s) \gamma^{(k+1)}_{N,s} N V_N (x_1 -x_{k+1}) - 8 \pi a_0 \tr \; J^{(k)} \cU^{(k)} (t-s) \gamma^{(k+1)}_{\infty,s} \delta (x_1 -x_{k+1}) \Big| \to 0 \end{equation} as $N \to \infty$. Since (\ref{eq:tmNto0}) and (\ref{eq:tmNto0b}) remain valid if we replace $x_1$ by any $x_j$, $j =2, \dots ,k$ in the potentials, it follows that, for every $k \geq 1$, $0\leq s \leq t \leq T$, and $J^{(k)} \in \cK_k$ with (\ref{eq:assJ}), \begin{equation}\label{eq:tmNto0c} \begin{split} \Big| \sum_{j=1}^k \Big( \tr \; J^{(k)} &\cU^{(k)} (t-s) \left[ N V_N (x_j -x_{k+1}), \gamma^{(k+1)}_{N,s}\right] \\ &- 8 \pi a_0 \tr \; J^{(k)} \cU^{(k)} (t-s) \left[\delta (x_j -x_{k+1}), \gamma^{(k+1)}_{\infty,s}\right] \Big) \Big| \to 0 \end{split}\end{equation} as $N \to \infty$. {F}rom (\ref{eq:equi-4}) (with $J^{(k)}$ replaced by $\cU^{(k)} (s-t) J^{(k)}$, using the fact that $\tri \cU^{(k)} (s-t) J^{(k)} \tri \leq C_T$ for all $0 \leq s \leq t \leq T$), and from an estimate similar to (\ref{eq:equi-4}) but with $\gamma_{N,s}^{(k+1)}$ replaced by $\gamma_{\infty,s}^{(k+1)}$ and $NV_N (x_j - x_{k+1})$ replaced by $\delta (x_j - x_{k+1})$, we can now apply the dominated convergence theorem to conclude (\ref{eq:conv-6}). \end{proof} The following lemmas are important ingredients in the proof of Theorem \ref{thm:conv}. \begin{lemma}\label{lm:term1} Under the same assumptions of Theorem \ref{thm:conv}, and using the notation $J_t^{(k)} = \cU^{(k)} (t) J^{(k)}$, we have, for every $k \geq 1$, $\ell =1, \dots, k$, $0 \leq s \leq t \leq T$, $J^{(k)} \in \cK_k$ such that (\ref{eq:assJ}) is satisfied, \begin{equation*}\Big| \tr \; J_{s-t}^{(k)} \, \left[ N V_N (x_{\ell} -x_{k+1}) W_{N,(\ell,k+1)} - 8\pi a_0 \, \delta (x_{\ell} - x_{k+1}) \right] W^*_{N,(\ell,k+1)} \gamma^{(k+1)}_{N,s} \Big| \to 0 \end{equation*} as $N \to \infty$. \end{lemma} \begin{proof} We fix $\ell =1$. Decomposing $\gamma_{N,s}^{(k+1)} = \sum_j \lambda_j \, |\xi^{(k+1)}_j \rangle \langle \xi^{(k+1)}_j |$, and introducing the variables $u= (x_1 + x_{k+1})/2$ and $v= x_1 -x_{k+1}$, we find \begin{equation} \begin{split} \tr \; &J_{s-t}^{(k)} \, N V_N (x_1 -x_{k+1}) W_{N,(1,k+1)} W^*_{N,(1,k+1)} \gamma^{(k+1)}_{N,s} \\ =\; & \sum_j \lambda_j \int \rd u \rd v \rd v'\rd x_2 \dots \rd x_k \rd \bx'_k J_{s-t}^{(k)} (\bx'_k; u+v/2, x_2 , \dots ,x_k) N V_N (v) W_N (v;v') \\ &\hspace{.2cm} \times (W_{N,(1,k+1)}^* \, \xi^{(k+1)}_j) (u+v'/2, x_2 , \dots ,x_k , u-v'/2) \, \overline{\xi}^{(k+1)}_j (\bx'_k, u-v/2). \end{split}\end{equation} The potential $V_N (v)$ forces $v$ to be of order $1/N$. The wave operator $W_N (v;v')$ also forces $v'$ to be of order $1/N$; for this reason, we can first replace $v$ with $0$ in the observable $J_{s-t}^{(k)}$ and in $\overline{\xi}^{(k+1)}_j$ and then we replace $v'$ with $0$ in $(W_{N,(1,k+1)}^* \xi^{(k+1)}_j)$. \begin{equation}\label{eq:termI1} \begin{split} \tr \; &J_{s-t}^{(k)} \, N V_N (x_1 -x_{k+1}) W_{N,(1,k+1)} W^*_{N,(1,k+1)} \gamma^{(k+1)}_{N,s} \\ =\; &\sum_j \lambda_j \int \rd u \rd v \rd v' \rd x_2 .. \rd x_k \rd \bx'_k N V_N (v) W_N (v;v') (W_{N,(1,k+1)}^* \xi^{(k+1)}_j) (u+v'/2, x_2 , .. ,x_k , u-v'/2) \\ &\hspace{.2cm} \times \left[ J_{s-t}^{(k)} (\bx'_k; u+v/2, x_2 , \dots ,x_k) \overline{\xi}^{(k+1)}_j (\bx'_k, u-v/2) \right. \\ &\hspace{4cm} \left. - J_{s-t}^{(k)} (\bx'_k; u, x_2 , \dots ,x_k) \overline{\xi}^{(k+1)}_j (\bx'_k, u) \right] \\ &+\sum_j \lambda_j \int \rd u \rd v \rd v' \rd x_2 .. \rd x_k \rd \bx'_k \, J_{s-t}^{(k)} (\bx'_k; u, x_2 , .. ,x_k) \, \overline{\xi}^{(k+1)}_j (\bx'_k, u) N V_N (v)W_N (v;v') \\ &\hspace{.2cm} \times \left[ (W_{N,(1,k+1)}^* \xi^{(k+1)}_j) (u+v'/2, x_2 , .. ,x_k , u-v'/2) - (W_{N,(1,k+1)}^* \xi^{(k+1)}_j) (u, x_2 , .. ,x_k , u) \right] \\ &+\sum_j \lambda_j \left( \int \rd v \rd v' N V_N (v) W_N (v;v') \right) \int \rd u \rd x_2 \dots \rd x_k \rd \bx'_k \, J_{s-t}^{(k)} (\bx'_k; u, x_2 , \dots ,x_k) \\ &\hspace{.2cm} \times \overline{\xi}^{(k+1)}_j (\bx'_k, u) (W_{N,(1,k+1)}^* \xi^{(k+1)}_j) (u, x_2 , \dots ,x_k , u). \end{split} \end{equation} {F}rom Lemma \ref{lm:VW=a_0}, we know that \[ \int \rd v \rd v' N V_N (v) W_N (v;v') = 8 \pi a_0 \, .\] Therefore, from (\ref{eq:termI1}), we obtain that \begin{equation}\label{eq:Ij+IIj} \begin{split} \Big| \tr \; &J^{(k)}_{s-t} \, \left[ N V_N (x_1 -x_{k+1}) W_{N,(1,k+1)} - 8\pi a_0 \, \delta (x_1 - x_{k+1}) \right] W^*_{N,(1,k+1)} \gamma^{(k+1)}_{N,s} \Big| \\ \leq \; & \sum_j \lambda_j \, \Big| \int \rd u \rd v \rd v' \rd x_2 \dots \rd x_k \rd \bx'_k (W_{N,(1,k+1)}^* \xi^{(k+1)}_j) (u+v'/2, x_2 , \dots ,x_k , u-v'/2) \\ &\hspace{.3cm} \times N V_N (v) W_N (v;v') \left[ J_{s-t}^{(k)} (\bx'_k; u+v/2, x_2 , \dots ,x_k) \overline{\xi}^{(k+1)}_j (\bx'_k, u-v/2) \right. \\ &\hspace{6cm} \left. - J^{(k)}_{s-t} (\bx'_k; u, x_2 , \dots ,x_k) \overline{\xi}^{(k+1)}_j (\bx'_k, u) \right] \Big| \\ &+ \sum_j \lambda_j \Big| \int \rd u \rd v \rd v' \rd x_2 \dots \rd x_k \rd \bx'_k \, J_{s-t}^{(k)} (\bx'_k; u, x_2 , \dots ,x_k) \overline{\xi}^{(k+1)}_j (\bx'_k, u) \\ &\hspace{.3cm} \times N V_N (v) W_N (v;v') \left[ (W_{N,(1,k+1)}^* \xi^{(k+1)}_j) (u+v'/2, x_2 , \dots ,x_k , u-v'/2) \right. \\ &\hspace{6cm} \left. - (W_{N,(1,k+1)}^* \xi^{(k+1)}_j) (u, x_2 , \dots ,x_k , u) \right] \Big| \\ = : \; &\sum_j \lambda_j \left(\text{I}_j + \text{II}_j \right) \, . \end{split} \end{equation} Expanding \begin{equation} \begin{split} \Big[ J_{s-t}^{(k)} & (\bx'_k; u+v/2, x_2 , \dots ,x_k) \overline{\xi}^{(k+1)}_j (\bx'_k, u-v/2) \\ & \hspace{4cm} - J_{s-t}^{(k)} (\bx'_k; u, x_2 , \dots ,x_k) \overline{\xi}^{(k+1)}_j (\bx'_k, u) \Big] \\ &= \int_0^{1/2} \rd r \; \frac{\rd}{\rd r} J_{s-t}^{(k)} (\bx'_k; u+rv, x_2 , \dots ,x_k) \overline{\xi}^{(k+1)}_j (\bx'_k, u-rv) \\ &= \int_0^{1/2} \rd r \; \Big[ v \cdot \left( \nabla_1 J_{s-t}^{(k)} \right) (\bx'_k; u+rv, x_2 , \dots ,x_k) \overline{\xi}^{(k+1)}_j (\bx'_k, u-rv) \\ & \hspace{2cm} - J_{s-t}^{(k)} (\bx'_k; u+rv, x_2 , \dots ,x_k) v \cdot \left( \nabla_{k+1} \xi_j^{(k+1)} \right) (\bx'_k , u - r v) \Big], \end{split} \end{equation} we can bound the term $\text{I}_j$ on the r.h.s. of (\ref{eq:Ij+IIj}) by \begin{equation}\label{eq:|W|1} \begin{split} \text{I}_j \leq \; & \int \rd u \rd v \rd v' \rd x_2 \dots \rd x_k \rd \bx'_k \int_0^{1/2} \rd r \; \Big|(W_{N,(1,k+1)}^* \xi^{(k+1)}_j) (u+v'/2, x_2 , \dots ,x_k , u-v'/2) \Big| \\ &\hspace{.3cm} \times N V_N (v) |W_N (v;v')| \, |v| \, \Big|\left( \nabla_1 J_{s-t}^{(k)} \right) (\bx'_k; u+rv, x_2 , \dots ,x_k) \Big| \Big| \xi^{(k+1)}_j (\bx'_k, u-rv)\Big| \\ &+ \int \rd u \rd v \rd v' \rd x_2 \dots \rd x_k \rd \bx'_k \int_0^{1/2} \rd r \, \Big|(W_{N,(1,k+1)}^* \xi^{(k+1)}_j) (u+v'/2, x_2 , \dots ,x_k , u-v'/2)\Big| \\ &\hspace{.3cm} \times N V_N (v) |W_N (v;v')|\, |v| \, \Big| J^{(k)}_{s-t} (\bx'_k; u+rv, x_2 , \dots ,x_k) \Big| \Big| \nabla_{k+1} \xi^{(k+1)}_j (\bx'_k, u-rv)\Big| \\ =: \; & \text{A}_j + \text{B}_j \, . \end{split} \end{equation} With a Schwarz inequality, we obtain \begin{equation*} \begin{split} \text{B}_j \leq \; & \int \rd u \rd v \rd v' \rd x_2 \dots \rd x_k \rd \bx'_k \int_0^{1/2} \rd r \; N V_N (v) |v| |W_N (v;v')| \\ &\hspace{.2cm} \times \Big| J_{s-t}^{(k)} (\bx'_k; u+rv, x_2 , \dots ,x_k)\Big| \, \\ &\hspace{.2cm} \times \left( \Big| \nabla_{k+1} \xi^{(k+1)}_j (\bx'_k, u-rv)\Big|^2 + \Big|(W_{N,(1,k+1)}^* \xi^{(k+1)}_j) (u+v'/2, x_2 , \dots ,x_k , u-v'/2)\Big|^2 \right). \end{split} \end{equation*} In the first term we integrate out the observable $J^{(k)}_{s-t} (\bx'_k; u+rv, x_2 \dots, x_k)$ over the variables $x_2, \dots ,x_k$ and take the sup over the others. In the second term on the other hand, we integrate $J^{(k)}_{s-t} (\bx'_k; u+rv, x_2 \dots, x_k)$ over $\bx'_k$ and we take the sup over the other variables. We obtain \begin{equation*}\begin{split} \text{B}_j \leq \; & (1/2)\sup_{\bx'_k,x_1} \left( \int \rd x_2 \dots \rd x_k \,\Big| J_{s-t}^{(k)} (\bx'_k; \bx_k)\Big| \right) \, \left(\sup_v \int \rd v' |W_N (v;v')|\right) \\ &\hspace{0cm} \times \Big\| \nabla_{k+1} \xi_j^{(k+1)} \Big\|^2 \; \int \rd v \; N V_N (v) |v| \\ & + (1/2) \sup_{\bx_k}\left( \int \rd \bx'_k \Big|J_{s-t}^{(k)} (\bx'_k; \bx_k)\Big| \right) \, \left(\sup_v \int \rd v' \; |W_N (v;v')|\right) \\ &\hspace{.1cm} \times \int \rd x_1 \rd x_2 \dots \rd x_{k+1} \; N V_N (x_1- x_{k+1}) |x_1 -x_{k+1}| \Big|(W_{N,(1,k+1)}^* \xi^{(k+1)}_j) (x_1, x_2 , \dots ,x_k , x_{k+1})\Big|^2 \\ \leq & \; \frac{C (k,J^{(k)},T)}{N} \\ &\hspace{0cm} \times \left( \| \nabla_{k+1} \xi^{(k+1)}_j \|^2 + \left\langle W_{N,(1,k+1)}^* \xi_j^{(k+1)} , \left( (\nabla_1 \cdot \nabla_{k+1})^2 -\Delta_1 - \Delta_{k+1} +1 \right) W_{N,(1,k+1)}^* \xi_j^{(k+1)} \right\rangle \right), \end{split} \end{equation*} where we used (\ref{eq:triJjt}), Lemma \ref{lm:VL1}, the bound $\int \rd v \, N V_N (v) |v| \leq C/N$, and the fact that \begin{equation}\label{eq:L1W} \sup_v \int \rd v' \; |W_N (v;v')| \leq \| W_N \|_{L^1 \to L^1} \leq C \end{equation} uniformly in $N$ (by Proposition \ref{prop:waveop}). The term $A_j$, on the r.h.s. of (\ref{eq:|W|1}), can be bounded in a similar, but even simpler way, since $\nabla_{k+1} \xi_j^{(k+1)}$ is replaced here by $\xi_j^{(k+1)}$. Thus, the first term on the r.h.s. of (\ref{eq:Ij+IIj}) is controlled by \begin{equation}\label{eq:Ijto0} \begin{split} \sum_j \lambda_j \, \text{I}_j \leq &\; \frac{C}{N} \left( \tr \; (-\Delta_{k+1}) \gamma^{(k+1)}_{N,s} \right. \\ &\hspace{1cm} +\left. \tr \; \left( (\nabla_1 \cdot \nabla_{k+1})^2 -\Delta_1 - \Delta_{k+1} +1 \right) W_{N, (1,k+1)}^* \gamma^{(k+1)}_{N,s} W_{N, (1,k+1)} \right) \end{split} \end{equation} which converges to zero as $N \to \infty$ by Proposition \ref{prop:apri2}. \bigskip Next, we consider the second term on the r.h.s. of (\ref{eq:Ij+IIj}): \begin{equation}\label{eq:IIJ1} \begin{split} \text{II}_j = \; & \Big| \int \rd u \rd v' \rd x_2 \dots \rd x_k \rd \bx'_k \, J^{(k)}_{s-t} (\bx'_k; u, x_2 , \dots ,x_k) \overline{\xi}^{(k+1)}_j (\bx'_k, u) \\ &\hspace{1cm} \times \left( N^3 (W^* V)(Nv') - \delta (v') \right) (W_{N,(1,k+1)}^* \xi^{(k+1)}_j) (u+v'/2, x_2 , \dots ,x_k , u-v'/2)\Big| \,. \end{split} \end{equation} To control this contribution, we first insert a cutoff $\chi (v')$; this will allow us to apply Lemma \ref{lm:VL12} to bound the integral over $u$ and $v'$. To this end, we choose a function $\chi \in C^{\infty}_0 (\bR^3)$ such that $0 \leq \chi (x) \leq 1$, $\chi (x) = 1$ for $|x| \leq 1$ and $\chi (x) =0$ for $|x| \geq 2$, and we put $\bar{\chi} = 1-\chi$. Using $\chi$, we decompose the r.h.s. of (\ref{eq:IIJ1}) in two parts \begin{equation}\label{eq:IIJ2} \begin{split} \text{II}_j \leq \; &\Big|\int \rd u \rd v' \rd x_2 \dots \rd x_k \rd \bx'_k \, J_{s-t}^{(k)} (\bx'_k; u, x_2 , \dots ,x_k) \chi ( v' ) \overline{\xi}^{(k+1)}_j (\bx'_k, u) \\ &\hspace{1cm} \times \left[ N^3 (W^* V)(Nv') - \delta (v') \right] (W_{N,(1,k+1)}^* \xi^{(k+1)}_j ) (u+v'/2, x_2 , \dots ,x_k , u-v'/2) \Big| \\ &+ \Big| \int \rd u \rd v' \rd x_2 \dots \rd x_k \rd \bx'_k \, J_{s-t}^{(k)} (\bx'_k; u, x_2 , \dots ,x_k) \bar{\chi} (v') \overline{\xi}^{(k+1)}_j (\bx'_k, u) \\ &\hspace{1cm} \times N^3 (W^* V)(Nv') (W_{N,(1,k+1)}^* \xi^{(k+1)}_j) (u+v'/2, x_2 , \dots ,x_k , u-v'/2)\Big| \\ =: \; & \text{E}_j + \text{F}_j\, . \end{split} \end{equation} The term $\text{F}_j$ can be bounded by \begin{equation*} \begin{split} \text{F}_j \leq \; &\int \rd u \rd v' \rd x_2 \dots \rd x_k \rd \bx'_k \, \Big| J_{s-t}^{(k)} (\bx'_k; u, x_2 , \dots ,x_k) \Big| \, \bar{\chi} (v') N^3 (W^* V)(Nv') \\ &\hspace{.5cm} \times \left( |\xi^{(k+1)}_j (\bx'_k, u)|^2 + \Big|(W_{N,(1,k+1)}^* \xi^{(k+1)}_j) (u+v'/2, x_2 , \dots ,x_k , u-v'/2)\Big|^2 \right) \\ \leq \; & \, \| \xi^{(k+1)}_j \|^2 \, \left( \sup_{u,\bx'_k} \int \rd x_2 \dots \rd x_k \; \Big| J_{s-t}^{(k)} (\bx'_k; u, x_2 , \dots ,x_k) \Big| \right) \; \int_{|v'| \geq N} |(W^* V) (v')| \rd v'\\ &+ \left( \sup_{\bx_k} \int \rd \bx'_k \, \Big| J_{s-t}^{(k)} (\bx'_k; \bx_k)\Big| \right) \\ & \hspace{.5cm} \times \int \rd x_1 \dots \rd x_{k+1} \bar{\chi} (x_1 -x_{k+1}) N^3 (W^* V)(N (x_1 -x_{k+1})) \Big|(W_{N,(1,k+1)}^* \xi^{(k+1)}_j) (\bx_k, x_{k+1})\Big|^2. \end{split} \end{equation*} {F}rom Lemma \ref{lm:VL1}, we obtain \begin{equation}\label{eq:Fjto0} \begin{split} \sum_j \lambda_j \, \text{F}_j \leq \; &C (k,T, J^{(k)}) \, \left( \int_{|v'| \geq N} \rd v' \; |(W^* V) (v')| \right) \, \\ &\hspace{1cm} \times \left( \tr \; \left( (\nabla_1 \cdot \nabla_2)^2 - \Delta_1 -\Delta_2 + 1 \right) W_{N,(1,k+1)} \gamma_{N,s}^{(k+1)} W_{N,(1,k+1)}^* \right) \to 0 \end{split} \end{equation} as $N \to \infty$. Here we used Proposition \ref{prop:apri2} and the fact that, since $W^* V \in L^1 (\bR^3)$, \[ \int_{|x| >N} |W^* V (x)| \rd x \to 0 \qquad \text{ as} \quad N \to \infty. \] As for the term $\text{E}_j$ on the r.h.s. of (\ref{eq:IIJ2}), Lemma \ref{lm:VL12} implies that there exists a sequence $\delta_{N} \to 0$ as $N \to \infty$ ($\delta_N$ corresponds to the sequence $\beta_{1/N}$ defined in Lemma \ref{lm:VL12}, with $V$ replaced by $W^*V$) such that \begin{equation} \begin{split} \text{E}_j \leq \; &\delta_{N} \int \rd x_2 \dots \rd x_k \rd \bx'_k \, \\ & \hspace{.5cm} \times \Big( \int \rd u \rd v \; \Big| \left( (\Delta_u - \Delta_v)^2 -\Delta_u - \Delta_v + 1 \right)^{1/2} \\ &\hspace{5cm} \times \left( W_{N,(1,k+1)}^* \xi^{(k+1)}_j\right) (u+v/2, x_2, \dots , x_k, u-v/2) \Big|^{2} \Big)^{1/2} \\ &\hspace{.5cm} \times \left( \int \rd u \rd v \, \Big| (1-\Delta_u+\Delta_v^2)^{1/2} \chi (v) \, J^{(k)}_{s-t} (\bx'_k; u, x_2 , \dots ,x_k) \overline{\xi}^{(k+1)}_j (\bx'_k, u) \Big|^2 \right)^{1/2} \\ \leq \; & \delta_{N} \| \chi \|_{H^2} \int \rd x_2 \dots \rd x_k \rd \bx'_k \, \left( \int \rd u \Big| (1-\Delta_u)^{1/2} \overline{\xi}^{(k+1)}_j (\bx'_k, u) \Big|^2 \right)^{1/2} \\ &\hspace{.5cm} \times \sup_u \left[ \Big| J^{(k)}_{s-t} (\bx'_k; u, x_2 , \dots ,x_k)\Big| + \Big| \nabla_u \, J_{s-t}^{(k)} (\bx'_k; u, x_2 , \dots ,x_k)\Big| \right] \\ & \hspace{.5cm} \times \Big( \int \rd u \rd v \; \Big| \left( (\Delta_u - \Delta_v)^2 -\Delta_u - \Delta_v + 1 \right)^{1/2} \\ &\hspace{5cm} \times \left( W_{N,(1,k+1)}^* \xi^{(k+1)}_j\right) (u+v/2, x_2, \dots , x_k, u-v/2) \Big|^{2} \Big)^{1/2}. \end{split} \end{equation} With a Schwarz inequality, we find \begin{equation} \begin{split} \text{E}_j \leq &\; \delta_{N} \| \chi \|_{H^2} \int \rd x_2 \dots \rd x_k \rd \bx'_k \, \\ &\hspace{.05cm} \times \sup_u \left[ \Big| J_{s-t}^{(k)} (\bx'_k; u, x_2 , \dots ,x_k)\Big| + \Big| \nabla_u \, J^{(k)}_{s-t} (\bx'_k; u, x_2 , \dots ,x_k)\Big| \right] \\ &\hspace{.05cm} \times \left(\int \rd u \Big| (1-\Delta_u)^{1/2} \overline{\xi}^{(k+1)}_j (\bx'_k, u) \Big|^2 \right. \\ &\hspace{.1cm} \left.+ \int \rd u \rd v \Big| \left( (\Delta_u - \Delta_v)^2 -\Delta_u - \Delta_v + 1 \right)^{1/2} \left( W_{N,(1,k+1)}^* \xi^{(k+1)}_j\right) (u+v/2, x_2,.., x_k, u-v/2) \Big|^{2} \right) \\ \leq \; & C (k,T,J^{(k)}) \, \delta_{N} \, \left( \langle \xi_j^{(k+1)}, (1-\Delta_u) \xi^{(k+1)}_j \rangle \right. \\ &\left. \hspace{2cm} + \langle W_{N,(1,k+1)}^* \xi_j^{(k+1)}, \left( (\nabla_1 \cdot \nabla_{k+1})^2 -\Delta_1 -\Delta_{k+1} + 1 \right) W_{N,(1,k+1)}^* \xi_j^{(k+1)} \rangle \right)\,. \end{split} \end{equation} {F}rom Lemma \ref{lm:VL12} and Proposition \ref{prop:apri2}, we find \begin{equation}\label{eq:Ejto0} \sum_j \lambda_j \, \text{E}_j \leq C \delta_{N} \to 0 \qquad \text{as } N \to \infty \end{equation} and this, with (\ref{eq:Fjto0}), implies that \begin{equation*} \sum_j \lambda_j \text{II}_j \to 0 \qquad \text{as } N \to \infty \, . \end{equation*} Together with (\ref{eq:Ijto0}) and (\ref{eq:Ij+IIj}), this concludes the proof of the lemma. \end{proof} \begin{lemma}\label{lm:term2} Under the same conditions as in Theorem \ref{thm:conv}, we have, for every $k \geq 1$, $\ell=1,\dots,k$, $0\leq s \leq T$, and $J^{(k)} \in \cK_k$ satisfying (\ref{eq:assJ}), \begin{equation*} \Big| \tr \; J_{s-t}^{(k)} \, \left[ \delta (x_{\ell} -x_{k+1}) - \, h_{\alpha} (x_{\ell} - x_{k+1}) \right] W^*_{N,(\ell,k+1)} \gamma^{(k+1)}_{N,s} \Big| \to 0 \end{equation*} as $\alpha \to 0$, uniformly in $N$. Here we use notation $J^{(k)}_t = \cU^{(k)} (t) J^{(k)}$. \end{lemma} \begin{proof} We fix $\ell =1$. Using the decomposition $\gamma_{N,s}^{(k+1)} = \sum_j \lambda_j |\xi^{(k+1)}_j \rangle \langle \xi^{(k+1)}_j |$, we find that \begin{equation*} \begin{split} \Big| \tr \; &J_{s-t}^{(k)} \, \left[ \delta (x_1 - x_{k+1}) - \, h_{\alpha} (x_1 - x_{k+1}) \right] W^*_{N,(1,k+1)} \gamma^{(k+1)}_{N,s} \Big| \\ \leq \; &\sum_j \lambda_j \, \Big| \int \rd u \rd v \rd x_2 \dots \rd x_k \rd \bx'_k \, J_{s-t}^{(k)} (\bx'_k; u+v/2, x_2 , \dots ,x_k) \, \left[ \delta (v) -h_{\alpha} (v) \right]\\ &\hspace{.5cm} \times \; \overline{\xi}^{(k+1)}_j (\bx'_k, u-v/2) (W_{N,(1,k+1)}^* \xi^{(k+1)}_j) (u+v/2, x_2 , \dots ,x_k , u-v/2) \Big| \\ \leq \; &\sum_j \lambda_j \, \Big| \int \rd u \rd v \rd x_2 \dots \rd x_k \rd \bx'_k \, h_{\alpha} (v) \\ &\hspace{.5cm}\times \left[ J_{s-t}^{(k)} (\bx'_k; u+v/2, x_2 , \dots ,x_k) \overline{\xi}^{(k+1)}_j (\bx'_k, u-v/2) \right. \\ &\hspace{4cm} \times (W_{N,(1,k+1)}^* \xi^{(k+1)}_j) (u+v/2, x_2 , \dots ,x_k , u-v/2) \\ &\hspace{1cm} - \left. J_{s-t}^{(k)} (\bx'_k; u, x_2 , \dots ,x_k) \overline{\xi}^{(k+1)}_j (\bx'_k, u) (W_{N,(1,k+1)}^* \xi^{(k+1)}_j) (u, x_2 , \dots ,x_k , u) \right] \Big| \, . \end{split} \end{equation*} Similarly to (\ref{eq:termI1}), we first replace $v$ by $0$ in $J_{s-t}^{(k)} (\bx'_k; u+v/2, x_2 , \dots ,x_k) \overline{\xi}^{(k+1)}_j (\bx'_k, u-v/2)$ and then in $(W_{N,(1,k+1)}^* \xi^{(k+1)}_j) (u+v/2, x_2 , \dots ,x_k , u-v/2)$. We obtain \begin{equation}\label{eq:alpha1} \begin{split} \Big| \tr \; &J_{s-t}^{(k)} \, \left[ \delta (x_1 - x_{k+1}) - \, h_{\alpha} (x_1 - x_{k+1}) \right] W^*_{N,(1,k+1)} \gamma^{(k+1)}_{N,s} \Big| \\ \leq \; & \sum_j \lambda_j \, \Big| \int \rd u \rd v \rd x_2 \dots \rd x_k \rd \bx'_k \, h_{\alpha} (v) \, (W_{N,(1,k+1)}^* \xi^{(k+1)}_j) (u+v/2, x_2 , \dots ,x_k , u-v/2) \, \\ &\hspace{.5cm} \times \left[ J_{s-t}^{(k)} (\bx'_k; u+v/2, x_2 , \dots ,x_k) \overline{\xi}^{(k+1)}_j (\bx'_k, u-v/2) \right. \\ &\hspace{4cm}\left. - J_{s-t}^{(k)} (\bx'_k; u, x_2 , \dots ,x_k) \overline{\xi}^{(k+1)}_j (\bx'_k, u) \right] \Big| \\ & + \sum_j \lambda_j \, \Big| \int \rd u \rd v \rd x_2 \dots \rd x_k \rd \bx'_k \, h_{\alpha} (v) \, J_{s-t}^{(k)} (\bx'_k; u, x_2 , \dots ,x_k)\overline{\xi}^{(k+1)}_j (\bx'_k, u) \\ &\hspace{.5cm} \times \left[(W_{N,(1,k+1)}^* \xi^{(k+1)}_j) (u+v/2, x_2 , \dots ,x_k , u-v/2) - (W_{N,(1,k+1)}^* \xi^{(k+1)}_j) (u, x_2 , \dots ,x_k , u) \right] \Big| \\ =: \; & \sum_j \lambda_j \; \left( \text{III}_j + \text{IV}_j \right). \end{split} \end{equation} To bound the first term, we expand the difference in an integral \begin{equation} \begin{split} \Big[ J_{s-t}^{(k)} (\bx'_k; u+v/2, &x_2 , \dots ,x_k) \overline{\xi}^{(k+1)}_j (\bx'_k, u-v/2) %\\ &\hspace{4cm}\left. - J_{s-t}^{(k)} (\bx'_k; u, x_2 , \dots ,x_k) \overline{\xi}^{(k+1)}_j (\bx'_k, u) \Big] \\ &\hspace{1cm} = \int_0^{1/2} \rd r \, v \cdot \nabla_1 J_{s-t}^{(k)} (\bx'_k; u+rv, x_2 , \dots ,x_k) \overline{\xi}^{(k+1)}_j (\bx'_k, u-rv) \\ &\hspace{1.5cm} - \int_0^{1/2} \rd r \, J_{s-t}^{(k)} (\bx'_k; u+rv, x_2 , \dots ,x_k) v \cdot \nabla_{k+1} \overline{\xi}^{(k+1)}_j (\bx'_k, u-rv) \end{split} \end{equation} and we obtain that \begin{equation} \begin{split} \text{III}_j \leq \; & \int \rd u \rd v \rd x_2 \dots \rd x_k \rd \bx'_k \, \int_0^{1/2} \rd r \, h_{\alpha} (v) |v| \Big|(W_N^* \xi^{(k+1)}_j) (u+v/2, x_2 , \dots ,x_k , u-v/2) \Big| \\ &\hspace{.5cm} \times \left( \Big| \nabla_1 J_{s-t}^{(k)} (\bx'_k; u+rv, x_2 , \dots ,x_k)\Big| | \xi^{(k+1)}_j (\bx'_k, u-rv)| \right. \\ &\hspace{2cm} \left. + \Big| J_{s-t}^{(k)} (\bx'_k; u+rv, x_2 , \dots ,x_k)\Big| \Big| \nabla_{k+1} \xi^{(k+1)}_j (\bx'_k, u-rv) \Big| \right) \end{split} \end{equation} which implies that \begin{equation*} \begin{split} \sum_j \lambda_j \; \text{III}_j \leq \; & \alpha \; C (k,T,J^{(k)}) \, \left( \tr \; (1-\Delta_{k+1}) \gamma^{(k+1)}_{N,s} + \tr\; \left( (\nabla_1 \cdot \nabla_{k+1})^2 - \Delta_1 -\Delta_{k+1} + 1\right) \gamma^{(k+1)}_{N,s} \right) \\ \leq \; &C \; \alpha. \end{split} \end{equation*} The terms $\text{IV}_j$ can be estimated similarly to the terms $\text{II}_j$ considered in (\ref{eq:IIJ1}); in particular, analogously to (\ref{eq:Fjto0}) and (\ref{eq:Ejto0}), we also find \begin{equation*} \begin{split} \sum_j \lambda_j \; \text{IV}_j \leq \; & C (k,T,J^{(k)}) \, \beta_{\alpha} \, \left( \tr \; (1-\Delta_{k+1}) \gamma^{(k+1)}_{N,s} + \tr \; \left( (\nabla_1 \cdot \nabla_{k+1})^2 - \Delta_1 -\Delta_{k+1} + 1\right) \gamma^{(k+1)}_{N,s} \right) \\ \leq \; &C \beta_{\alpha} \end{split} \end{equation*} where $\beta_{\alpha} \to 0$ as $ \alpha \to 0$ uniformly in $N$ (the sequence $\beta_{\alpha}$ comes from Lemma \ref{lm:VL12}, with $V$ replaced by $h$). This concludes the proof of the lemma. \end{proof} \bigskip \begin{lemma}\label{lm:VW=a_0} Suppose that $V \geq 0$, with $V (x) \leq C \langle x \rangle^{-\sigma}$ for some $\sigma >5$ (this implies, in particular that $V \in L^1 (\bR^3) \cap L^2 (\bR^3)$ and thus that $V$ is in the Rollnik class of potentials). Let $W$ denote the wave operator (as defined in Proposition \ref{prop:waveop}) associated with the Hamiltonian $\fh= -\Delta + (1/2) V (x)$. Then \[ \int \rd x \; (W^* V) (x) = 8 \pi a_0 \] where $a_0$ is the scattering length of the potential $V$. \end{lemma} \begin{proof} First of all, we observe that, under the assumption that $V \geq 0$ and $V (x) \leq C \langle x \rangle^{-\sigma}$, for some $\sigma >5$, the operator $\fh = -\Delta + (1/2) V$ cannot have a zero energy resonance (recall that a zero-energy resonance of $\fh$ is a solution $\ph$ of $(-\Delta + (1/2) V) \ph = 0$ such that $|\ph (x)| \leq C/|x|$ for all $x \in \bR^3$); this can be proven using the maximum principle. We will make use of this observation in the proof of this lemma. \medskip Next, we note that, since $W^*$ maps $L^1 (\bR^3)$ into $L^1 (\bR^3)$ (see Proposition \ref{prop:waveop}), we have that $(W^* V) \in L^1 (\bR^3)$ and thus \begin{equation}\label{eq:WV0} \begin{split} \int \rd x \; (W^* V) (x) &= \lim_{\eps \to 0} \int \rd x \; (W^* V) (x) \; \chi_{\eps} (x) = \lim_{\eps \to 0} \int \rd x \, V(x) (W \chi_{\eps}) (x) \end{split} \end{equation} with \[ \chi_{\eps} (x) = \frac{1}{1+\eps x^2} \,. \] \medskip We expand $W \chi_{\eps}$ in terms of solutions $\ph (x,k)$ of the Lippman-Schwinger equation \begin{equation}\label{eq:LSE} \ph (x,k) = e^{i k \cdot x} - \frac{1}{8\pi} \int \rd y \frac{e^{i|k||x-y|}}{|x-y|} \, V(y) \ph (y,k) \, .\end{equation} It follows from \cite[Theorem XI.41, a)]{RS3} that Eq. (\ref{eq:LSE}) has a unique solution $\ph (x,k)$, such that $\ph (x,k) V^{1/2} (x) \in L^2 (\bR^3)$, for all $k \in \bR^3$ such that $k^2 \not \in \cE$, for an exceptional set $\cE$ with Lebesgue measure zero. The set $\cE$ consists of all values of $k^2$ for which zero is an eigenvalue of the operator \begin{equation}\label{eq:Mk} M_{|k|} = 1 + \frac{1}{2} V^{1/2} \frac{1}{-\Delta -k^2} V^{1/2} \, . \end{equation} {F}rom the observation that the operator $\fh= -\Delta + (1/2) V$ does not have a zero energy resonance, it follows immediately that $0 \not \in \cE$; in fact, if $M_{0} \psi = 0$ for some $\psi \in L^2 (\bR^3)$, then \[ \psi (x) = -\frac{1}{2} V^{1/2} (x) \int \rd y \, \frac{1}{|x-y|} V^{1/2} (y) \psi (y) \, , \] which implies that $\psi (x)/ V^{1/2} (x) \leq C / |x|$ for $|x| \gg 1$ and thus that $\ph (x) := \psi (x) / V^{1/2} (x)$ is a zero-energy resonance solution of $\left(-\Delta + (1/2) V \right) \ph = 0$. Since $M_0$ is a non-negative Fredholm operator with no eigenvalue at zero, it follows that there exists $\lambda > 0$ with $\sigma (M_0) \subset (\lambda,\infty)$ (here $\sigma (M_0)$ indicates the spectrum of $M_0$). Since moreover $M_{|k|} - M_0$ is a compact operator with kernel \begin{equation} \left(M_{|k|} - M_0\right) (x;y) = \frac{1}{2} V^{1/2} (x) \frac{e^{i|k||x-y|} - 1}{|x-y|} V^{1/2} (y) \end{equation} we obtain that \begin{equation} \begin{split} \| M_{|k|} - M_0 \|^2_{\text{HS}} = \; &\frac{1}{4} \int \rd x \rd y \, V(x) V(y) \, \frac{\left|e^{i|k||x-y|} - 1\right|^2}{|x-y|^2} \leq \;\frac{|k|^2 \| V \|_{L^1}^2}{4} \end{split} \end{equation} and thus that there exists $\kappa >0$ such that $\sigma (M_{|k|}) \subset (\lambda/2, \infty)$ for all $|k| \leq \kappa$. In particular it follows that \begin{equation}\label{eq:Mk-1} \| M_{|k|}^{-1} \| \leq 2/\lambda \qquad \text{for all $k \in \bR^3$ with $|k| \leq \kappa$.} \end{equation} \medskip {F}rom \cite[Theorem XI.41, e)]{RS3} we also find \begin{equation}\label{eq:RS1} \begin{split} (W \chi_{\eps}) (x) = \text{L.I.M.} \, (2\pi)^{-3/2} \int \rd k\; \ph (x,k) \, \widehat{\chi}_{\eps} (k) = \text{L.I.M.} \int \rd k\; \ph (x,k) \frac{e^{-|k|/\sqrt{\eps}}}{4\pi |k|\eps} \, , \end{split}\end{equation} where L.I.M. denotes the $L^2$-limit as $M \to \infty$ and $\delta \to 0$ of the integral over $\{ k \in \bR^3: |k| \leq M \text{ and } \text{dist } (k^2, \cE) > \delta \}$. Inserting (\ref{eq:RS1}) in the r.h.s. of (\ref{eq:WV0}), we find (recalling that $\kappa >0$ is chosen such that (\ref{eq:Mk-1}) holds true) \begin{equation}\label{eq:WV1} \begin{split} \int \rd x \; (W^* V) (x) \; \chi_{\eps} (x) = \; & \int_{|k| > \kappa} \rd x \rd k \, V(x) \ph (x,k) \frac{e^{-|k|/\sqrt{\eps}}}{4\pi |k|\eps}+ \int_{|k| \leq \kappa} \rd x \rd k \, V(x) \ph (x,k) \frac{e^{-|k|/\sqrt{\eps}}}{4\pi |k|\eps}\,. \end{split} \end{equation} The first term on the r.h.s. of (\ref{eq:WV1}) can be controlled by \begin{equation}\label{eq:WV2} \begin{split} \left| \int_{|k| > \kappa} \rd x \rd k \, V(x) \ph (x,k) \frac{e^{-|k|/\sqrt{\eps}}}{4\pi |k|\eps} \right| & = \left| \int_{|k| > \kappa} \rd k \, V^{\sharp} (k) \frac{e^{-|k|/\sqrt{\eps}}}{4\pi |k|\eps} \right| \\ &\leq C \, \| V^{\sharp} \|_{L^2} \left( \int_{|k| \geq \kappa} \rd k \, \frac{e^{-2|k|/\sqrt{\eps}}}{|k|^2 \eps^2} \right)^{1/2} \\ &\leq C \| V \|_{L^2} \frac{e^{-\kappa /(2 \sqrt{\eps})}}{\eps^{3/4}} \to 0 \end{split} \end{equation} as $\eps \to 0$. Here we introduced the function \[ V^{\sharp} (k) = \text{l.i.m.} (2\pi)^{-3/2} \int \rd x \, V(x) \ph (x,k) \] where l.i.m. denotes the $L^2$-limit of the integral over $|x| \leq M$ as $M \to \infty$; the existence of $V^{\sharp}$ for $V \in L^2 (\bR^3)$ and the fact that $\| V^{\sharp}\|_{L^2} \leq \| V \|_{L^2}$ (actually, in our case, $\| V^{\sharp}\|_{L^2} = \| V \|_{L^2}$) are proven in \cite[Theorem IX.41]{RS3}. As for the second term on the r.h.s. of (\ref{eq:WV1}), we have \begin{equation}\label{eq:WV3} \begin{split} \int_{|k| \leq \kappa} \rd k \rd x \, V(x) \ph (x,k) \frac{e^{-|k|/\sqrt{\eps}}}{4\pi |k|\eps} = \; & \int_{|k| \leq \kappa} \rd k \rd x \, V(x) \ph (x,0) \frac{e^{-|k|/\sqrt{\eps}}}{4 \pi |k|\eps} \\ &+ \int_{|k| \leq \kappa} \rd k \rd x \, V(x) \left( \ph (x,k) -\ph (x,0) \right) \frac{e^{-|k|/\sqrt{\eps}}}{4\pi |k|\eps} \\ = \; & \left(1 - (1+ \kappa \eps^{-1/2}) e^{-\kappa \eps^{-1/2}} \right) \int \rd x \, V(x) \ph (x,0) \\ &+ \int_{|k| \leq \kappa} \rd k \rd x \, V(x) \left( \ph (x,k) -\ph (x,0) \right) \frac{e^{-|k|/\sqrt{\eps}}}{4\pi |k|\eps}\, . \end{split} \end{equation} Using that $\ph (x,0)$ is the solution of the zero energy scattering equation \[ \left( -\Delta + (1/2) V(x) \right) \ph (x,0) = 0 \] with the boundary condition $\ph (x,0) \to 1$ as $|x| \to \infty$, it follows that (see (\ref{eq:a0})) \begin{equation}\label{eq:WV4} \int \rd x \, V(x) \ph (x,0) = 8 \pi a_0 \, . \end{equation} To bound the second term on the r.h.s. of (\ref{eq:WV3}), we define \[ \psi_k (x) = V^{1/2} (x) \ph (x,k) \] and we observe that, from the Lippman-Schwinger equation (\ref{eq:LSE}), \begin{equation} \begin{split} \psi_k (x) - \psi_0 (x) = \; & V^{1/2} (x) \left( e^{ik \cdot x} - 1 \right) -\frac{1}{8\pi} \int \rd y V^{1/2} (x) \frac{e^{i|k||x-y|}}{|x-y|} V^{1/2} (y) \left( \psi_k (y) - \psi_0 (y) \right) \\ &-\frac{1}{8\pi} \int \rd y V^{1/2} (x) \frac{e^{i|k||x-y|}-1}{|x-y|} V^{1/2} (y) \psi_0 (y) \, , \end{split} \end{equation} which implies that (with $M_{|k|}$ defined in (\ref{eq:Mk})) \begin{equation} \begin{split} \left[ M_{|k|} \left( \psi_k - \psi_0 \right) \right] (x) = \; & V^{1/2} (x) \left( e^{ik\cdot x} - 1 \right) - \frac{1}{4\pi} \int \rd y V^{1/2} (x) \frac{e^{i|k||x-y|}-1}{|x-y|} V^{1/2} (y) \psi_0 (y)\, . \end{split} \end{equation} By (\ref{eq:Mk-1}), we have \begin{equation} \begin{split} \Big\| \psi_k - \psi_0 \Big\|_{L^2} \leq \; &C \left( \Big\| V^{1/2} (x) \left(e^{i k \cdot x} -1\right) \Big\|_{L^2} + \| V^{1/2} \|_{L^2} |k| \int V^{1/2} (y) |\psi_0 (y)| \right) \\ \leq \; & C |k| \left( \| |x|^2 V \|^{1/2}_{L^1} + \| V \|_{L^1}^{3/2} \right) \\ \leq \; & C |k| \, . \end{split} \end{equation} Therefore, the second term on the r.h.s. of (\ref{eq:WV3}) can be bounded by \begin{equation}\begin{split} \Big| \int_{|k| \leq \kappa} \rd k \rd x \, V(x) \left( \ph (x,k) -\ph (x,0) \right) \frac{e^{-|k|/\sqrt{\eps}}}{4\pi |k|\eps} \Big| \leq \; &\int_{|k| \leq \kappa} \rd k \rd x \, V^{1/2} (x) \left| \psi_k (x) -\psi_0 (x) \right| \frac{e^{-|k|/\sqrt{\eps}}}{4\pi |k|\eps} \\ \leq \; &\| V \|_{L^1}^{1/2} \int_{|k| \leq \kappa} \rd k \, \| \psi_k -\psi_0 \|_{L^2} \frac{e^{-|k|/\sqrt{\eps}}}{4\pi|k|\eps} \\ \leq \; & C \| V \|_{L^1}^{1/2} \int_{|k| \leq \kappa} \rd k \frac{e^{-|k|/\sqrt{\eps}}}{\eps} \\ \leq \; & C \eps^{1/2} \end{split} \end{equation} and thus it converges to zero as $\eps \to 0$. The last equation, together with (\ref{eq:WV0}), (\ref{eq:WV1}), (\ref{eq:WV2}), (\ref{eq:WV3}), and (\ref{eq:WV4}), concludes the proof of the lemma. \end{proof} \begin{lemma}\label{lm:WNto1} Suppose that $V \geq 0$ and $V(x) \leq C \langle x \rangle^{-\sigma}$, for some $\sigma \geq 5$. Then, for every $g \in L^2 (\bR^3, \rd x)$, we have \[ \Big\| \left(W_{N} - 1 \right) g \Big\| \leq CN^{-1/6} \| g\|_{H^1} \, .\] \end{lemma} \begin{proof} Let $\frak{h}_N = -\Delta + (1/2) V_N (x)$. Since \[ W_N = s-\lim_{t \to \infty} e^{i \frak{h}_N t} e^{i\Delta t} \] it is enough to prove that \begin{equation}\label{eq:equiv} \sup_{t \in \, \bR} \left\| \left(e^{-i \frak{h}_N t} - e^{i\Delta t} \right) g \right\|\leq CN^{-1/6} \| g\|_{H^1}\,. \end{equation} Note that \begin{equation*} \begin{split} \frac{\rd}{\rd t} \; \left\| \left(e^{-i \frak{h}_N t} - e^{i\Delta t} \right) g \right\|^2 = 2 \text{Im } \langle e^{-i \fh_N t} g, V_N (x) e^{i\Delta t} g \rangle \end{split} \end{equation*} which implies that \begin{equation}\label{eq:ints} \left\| \left(e^{-i \fh_N t} - e^{i\Delta t} \right) g \right\|^2 \leq 2 \int_0^t \rd s \; \left|\langle e^{-i \fh_N s} g, V_N (x) e^{i\Delta s} g \rangle\right|\,. \end{equation} Next we observe that \begin{equation}\label{eq:es1} \left| \langle e^{-i \fh_N s} g, V_N (x) e^{i\Delta s} g \rangle \right| \leq \| e^{-i \fh_N s} g \|_{\infty} \, \| e^{i\Delta s} g \|_{\infty} \, \| V_N \|_1 \leq \frac{\| V \|_1 \, \| g \|^2_{1} \, \| W \|_{\infty \to \infty}\, \| W^* \|_{1 \to 1}}{Ns^3}\, , \end{equation} where we used the fact that \[ \| W_N \|_{p \to p} = \| W \|_{p \to p} \] for every $N$ and $1 \leq p \leq \infty$. For small $s$ we need a different estimate of the integrand on the r.h.s. of (\ref{eq:ints}). To this end we remark that \begin{equation}\label{eq:es2} \begin{split} \left|\langle e^{-i \fh_N s} g, V_N (x) e^{i\Delta s} g \rangle\right| \leq \; & \langle e^{i \fh_N s} g, V_N (x) e^{i \fh_N s} g \rangle^{1/2} \, \langle e^{i \Delta s} g, V_N (x) e^{i\Delta s} g \rangle \\ \leq \; & C \, \| V_N \|_{3/2} \| \nabla e^{i \fh_N s} g \| \| \nabla e^{i\Delta s} g \| \\ \leq \; & C\, \| V \|_{3/2} (1 + \| V \|_{3/2} )^{1/2} \, \| g \|^2_{H^1}\, , \end{split} \end{equation} where we used that $\| V_N \|_{3/2} = \| V \|_{3/2}$ and we estimated \begin{equation*} \begin{split} \| \nabla e^{i \fh_N s} g \|^2 = \langle e^{i \fh_N s} g, -\Delta e^{i \fh_N s} g \rangle \leq \langle g, \fh_N g \rangle \leq (1 + \| V \|_{3/2}) \| g \|^2_{H^1}\,. \end{split} \end{equation*} Combining (\ref{eq:es1}) and (\ref{eq:es2}), we obtain from (\ref{eq:ints}) \begin{equation} \begin{split} \left\| \left(e^{-i \fh_N t} - e^{i\Delta t} \right) g \right\|^2 \leq \; & 2 \int_0^{N^{-\alpha}} \rd s \; \| V \|_{3/2} (1 + \| V \|_{3/2} )^{1/2} \, \| g \|^2_{H^1} \\ &+ 2 \int_{N^{-\alpha}}^t \, \rd s \frac{\| V \|_1 \, \| g \|^2_{H^1} \, \| W \|_{\infty \to \infty}\, \| W^* \|_{1 \to 1}}{Ns^3} \\ \leq \; & \Big( C_1 N^{-\alpha} + C_2 N^{2\alpha-1}\Big)\| g \|^2_{H^1} \end{split} \end{equation} for every $t \in \bR$. Choosing $\alpha = 1/3$, we obtain (\ref{eq:equiv}). \end{proof} \section{Approximation of the initial data}\label{sec:appro} \setcounter{equation}{0} In this section we show how to regularize the initial wave function $\psi_N$ given in Theorem \ref{thm:main2}. \begin{proposition}\label{prop:initialdata} Suppose that $\psi_N \in L^2 (\bR^{3N})$ with $\| \psi_N \| =1$ is a family of $N$-particle wave functions with \begin{equation}\label{eq:initener} \langle \psi_N, H_N \psi_N \rangle \leq C N \end{equation} and with one-particle marginal density $\gamma^{(1)}_{N}$ such that \begin{equation} \label{eq:initcond} \gamma^{(1)}_{N} \to |\ph \rangle \langle \ph | \qquad \text{as } N \to \infty \end{equation} for a $\ph \in H^1 (\bR^3)$. For $\kappa >0$ we define \begin{equation}\label{eq:wtpsi} \wt \psi_N: = \frac{\chi ( \kappa H_N/N ) \psi_N}{\| \chi (\kappa H_N/N) \psi_N \|} \, . \end{equation} Here $\chi \in C^{\infty}_0 ( \bR )$ is a cutoff function such that $0\leq \chi\leq 1$, $\chi (s) =1$ for $0 \leq s \leq 1$ and $\chi (s) =0$ for $s \geq 2$. We denote by $\wt \gamma^{(k)}_{N}$, for $k =1, \dots, N$, the marginal densities associated with $\wt \psi_N$. \begin{enumerate} \item[i)] For every integer $k \geq 1$ we have \begin{equation} \langle \wt \psi_N , H_N^k \, \wt \psi_N \rangle \leq \frac{2^k N^k}{\kappa^k}\,. \end{equation} \item[ii)] We have \[ \sup_N \| \psi_N - \wt\psi_N \| \leq C \kappa^{1/2} \; . \] \item[iii)] For $\kappa >0$ small enough and for every fixed $k \geq 1$ we have \begin{equation}\label{eq:init-3} \lim_{N\to\infty} \tr \; \Big| \wt \gamma^{(k)}_{N} - |\ph \rangle \langle \ph|^{\otimes k} \Big| = 0\; . \end{equation} \end{enumerate} \end{proposition} \begin{proof} For the proof of part i) and ii) see \cite[Proposition 8.1]{ESY}. To prove iii), we begin by noticing (see (\ref{eq:1tok2})) that it is enough to show that \[ \lim_{N \to \infty} \tr \Big| \wt \gamma^{(1)}_{N} - |\ph \rangle \langle \ph| \Big| = 0 \,. \] Moreover, since the limiting density is an orthogonal projection, trace-norm convergence is equivalent to weak* convergence. In other words, it is enough to prove that, for every compact operator $J^{(1)} \in \cK_1$ and for every $\eps >0$ there exists $N_0 = N_0 (J^{(1)},\eps)$ such that \begin{equation}\label{eq:iii1} \Big| \tr \; J^{(1)} \left( \wt \gamma^{(1)}_{N} - |\ph \rangle \langle \ph| \right) \Big| \leq \eps \end{equation} for $N > N_0$. To show (\ref{eq:iii1}), we start by observing that, from (\ref{eq:initcond}), there exists a sequence $\xi_N^{(N-1)} \in L^2 (\bR^{3(N-1)})$, with $\| \xi_N^{(N-1)} \| = 1$ such that \begin{equation}\label{eq:ale} \| \psi_N - \ph \otimes \xi_N^{(N-1)} \| \to 0 \qquad \text{as } N \to \infty \, . \end{equation} This was proven by Alessandro Michelangeli in \cite{M} using the following argument. Choose an orthonormal basis $\{ f_i \}_{i \geq 1}$ of $L^2 (\bR^3)$ with $f_1 = \ph$. Choose also an orthonormal basis $\{ g_j \}_{j \geq 1}$ of $L^2 (\bR^{3(N-1)})$. Then one can write \[ \psi_N = \sum_{ij} \alpha^{(N)}_{ij} f_i \otimes g_j \] and \[ |\psi_N \rangle \langle \psi_N| = \sum_{i,j,i',j'} \overline{\alpha}^{(N)}_{i,j} \alpha^{(N)}_{i',j'} |f_i \rangle \langle f_{i'}| \otimes |g_j \rangle \langle g_{j'}| \, . \] This implies that \begin{equation*} \begin{split} \gamma^{(1)}_N = \; &\sum_{j} \left( |\alpha_{1,j}^{(N)}|^2 |\ph \rangle \langle \ph | + \alpha_{1,j}^{(N)} \sum_{i \neq 1} \overline{\alpha}_{i,j}^{(N)} |\ph \rangle \langle f_i| + \overline{\alpha}_{1,j}^{(N)} \sum_{i \neq 1} \alpha_{i,j}^{(N)} |f_i \rangle \langle \ph| + \sum_{i,i' \neq 1} \overline{\alpha}_{i,j}^{(N)} \alpha_{i',j}^{(N)} |f_i \rangle \langle f_{i'}|\right) \end{split} \end{equation*} and therefore, using (\ref{eq:initcond}), that \[ \sum_{j} |\alpha_{1,j}^{(N)}|^2 \to 1 \] as $N \to \infty$. Thus, putting $\wt\xi^{(N-1)}_N = \sum_j \alpha_{1,j}^{(N)} g_j$, we get \[ \| \psi_N - \ph \otimes \wt\xi_N^{(N-1)} \|^2 = \sum_j \sum_{i \neq 1} |\alpha_{i,j}^{(N)}|^2 = 1 - \sum_j |\alpha_{1,j}^{(N)}|^2 \to 0 \] as $N \to \infty$. It is then simple to check that $\xi_N^{(N-1)} = \wt \xi_N^{(N-1)} / \| \wt \xi_N^{(N-1)} \|$ satisfies (\ref{eq:ale}). \medskip On the other hand, there exists $\ph_* \in H^2 (\bR^3)$ with $\| \ph_* \|=1$ and such that \[ \| \ph - \ph_* \| \leq \frac{\eps}{32 \| J^{(1)}\|} \, . \] Let $\Xi = \chi (\kappa H_N/N)$. Then \[ \| (\Xi -1) \psi_N \|^2 \leq \frac{\kappa}{N} \langle \psi_N, H_N \psi_N \rangle \leq C \kappa \] independently of $N$. Therefore, choosing $\kappa >0$ so small that $\| \Xi \psi_N \| \geq 1/2$, we find \begin{equation} \begin{split} \left\| \frac{\Xi \psi_N}{\| \Xi \psi_N \|} - \frac{\Xi \left(\ph_* \otimes \xi_{N}^{(N-1)}\right)}{ \|\Xi \left(\ph_* \otimes \xi^{(N-1)}_N \right)\|} \right\| & \leq \frac{2}{\| \Xi \psi_N \|} \left\| \Xi \left( \psi_N - \ph_* \otimes \xi^{(N-1)}_{N} \right) \right\| \\ & \leq 4 \left\| \psi_N - \ph_* \otimes \xi^{(N-1)}_{N} \right\| \\ & \leq 4 \left\| \psi_N - \ph \otimes \xi^{(N-1)}_{N} \right\| + 4 \| \ph - \ph_* \| \\ &\leq \frac{\eps}{6 \| J^{(1)} \|} \end{split} \end{equation} for all $N$ sufficiently large. Next we define the Hamiltonian \begin{equation} \wh H_N := -\sum_{j=2}^N \Delta_j + \sum_{10$ is small enough, \begin{equation}\label{eq:hatXipsi-ph} \begin{split} \Big\| \frac{\Xi \psi_N}{\| \Xi \psi_N \|} - \frac{\wh\Xi \left( \ph_* \otimes \xi^{(N-1)}_{N} \right)}{\|\wh\Xi \left( \ph_* \otimes \xi^{(N-1)}_{N} \right)\|} \Big\| \leq \frac{\e}{3 \| J^{(1)}\|} \end{split} \end{equation} for $N$ sufficiently large. The proof of (\ref{eq:hatXipsi-ph}) can be found in \cite[Proposition 8.1]{ESY}. To get (\ref{eq:iii1}) we define \[ \wh\psi_{N} := \frac{\wh\Xi \left( \ph_* \otimes \xi^{(N-1)}_{N} \right)}{\|\wh\Xi \left( \ph_* \otimes \xi^{(N-1)}_{N} \right)\|} = \ph_* \otimes \frac{\wh\Xi \xi^{(N-1)}_{N}}{\|\wh\Xi \xi^{(N-1)}_{N} \|} \] where we used the fact that $\wh \Xi$ acts only on the last $N-1$ variables and the fact that $\| \ph_*\|=1$. Define \[ \wh \gamma_N^{(1)} (x_1;x'_1) := \int \rd \bx_{N-1} \, \wh \psi_{N} (x_1, \bx_{N-k}) \overline{\wh\psi}_{N} (x'_1,\bx_{N-k})\,. \] Note that $\wh \psi_N$ is not symmetric in all variables, but it is symmetric in the last $N-1$ variables. In particular, $\wh\gamma_N^{(1)}$ is a density matrix and clearly $\wh \gamma_N^{(1)} = |\ph_* \rangle \langle \ph_*|$. Therefore, since $\| \wt \psi_N - \wh \psi_{N} \| \leq \e /(3\| J^{(1)}\|)$ by (\ref{eq:hatXipsi-ph}) and since $\| \ph - \ph_*\| \leq \e/(32 \|J^{(1)}\|)$, we have \begin{equation} \begin{split} \Big|\tr \; J^{(1)} \left( \wt\gamma^{(1)}_{N} - |\ph \rangle \langle \ph| \right) \Big| \leq \; &\Big|\tr \; J^{(1)} \left( \wt\gamma^{(1)}_{N} - |\ph_* \rangle \langle \ph_*| \right) \Big| + \Big| \tr \; J^{(1)} \left( |\ph_* \rangle \langle \ph_*| - |\ph\rangle \langle \ph| \right)\Big| \\ \leq \; &2 \| J^{(1)} \| \, \| \wt \psi_N - \wh \psi_{N} \| + 2 \| J^{(1)}\| \, \| \ph - \ph_* \| \leq \e \end{split} \end{equation} for $N$ sufficiently large (for arbitrary $\kappa, \e >0$ small enough). This proves (\ref{eq:iii1}). \end{proof} \section{Poincar{\'e}-Sobolev type inequalities} \setcounter{equation}{0} In the proof of the convergence we need to estimate potentials converging to a delta functions, and their difference to a normalized delta-function. To this end we make use of the following three lemmas. \begin{lemma}\label{lm:VL1} Suppose $V \in L^1 (\bR^3)$. Then \begin{equation} \begin{split} \left|\langle \ph, V (x_1 -x_2) \psi \rangle \right| \leq C \| V \|_1 \, \langle &\psi, \left( (\nabla_1 \cdot \nabla_2)^2 -\Delta_1 - \Delta_2 + 1 \right) \psi \rangle^{1/2} \; \\ & \times \langle \ph, \left( (\nabla_1 \cdot \nabla_2)^2 -\Delta_1 - \Delta_2 + 1 \right) \ph \rangle^{1/2}\end{split} \end{equation} for every $\psi,\ph \in L^2 ( \bR^6, \rd x_1 , \rd x_2)$. \end{lemma} \begin{proof} Switching to Fourier space, we find \begin{equation} \begin{split} \langle \ph, V (x_1 -x_2) \psi \rangle =\; & \int \rd p_1 \rd p_2 \rd q_1 \rd q_2 \; \overline{\widehat\ph (p_1, p_2)} \widehat{\psi} (q_1, q_2) \, \widehat{V} (q_1 -p_1)\, \delta (p_1 + p_2 - q_1 -q_2) \, . \end{split} \end{equation} Therefore, by a weighted Schwarz inequality, \begin{equation} \begin{split} \Big| \langle \ph, V &(x_1 -x_2) \psi \rangle \Big| \\ \leq \; & \| \widehat{V} \|_{\infty} \left( \int \rd p_1 \rd p_2 \rd q_1 \rd q_2 \; \frac{(p_1 \cdot p_2)^2 + p_1^2 + p^2_2 +1}{(q_1 \cdot q_2)^2 + q_1^2 + q^2_2 +1} \, |\widehat\ph (p_1, p_2)|^2 \delta (p_1 + p_2 - q_1 -q_2) \right)^{1/2} \\ & \hspace{1cm} \times \left(\int \rd p_1 \rd p_2 \rd q_1 \rd q_2 \frac{(q_1 \cdot q_2)^2 + q_1^2 + q^2_2 +1}{(p_1 \cdot p_2)^2 + p_1^2 + p^2_2 +1} |\widehat{\psi} (q_1, q_2)|^2 \,\delta (p_1 + p_2 - q_1 -q_2)\right)^{1/2} \\ \leq \; & \| V \|_{1} \; \left( \sup_{p} \int \rd q \; \frac{1}{(q \cdot (p -q))^2 + q^2 + (p - q)^2 +1}\right) \\ & \hspace{1cm} \times \left\langle \psi, \left((\nabla_1 \cdot \nabla_2)^2 -\Delta_1 - \Delta_2 + 1 \right) \psi \right\rangle^{1/2} \, \left\langle \ph, \left((\nabla_1 \cdot \nabla_2)^2 -\Delta_1 - \Delta_2 + 1 \right) \ph \right\rangle^{1/2}\,. \end{split} \end{equation} The lemma will then follow from \begin{equation}\label{eq:int1} \sup_{p \in \bR^3} \int \rd q \; \frac{1}{(q \cdot (p -q))^2 + q^2 + (p - q)^2 +1} < \infty \, . \end{equation} To prove (\ref{eq:int1}), we proceed as follows. \begin{equation}\label{eq:int2} \begin{split} \int \rd q \; \frac{1}{(q \cdot (p -q))^2 + q^2 + (p - q)^2 +1} = \; & \int_{|q-\frac{p}{2}| > |p|} \rd q \; \frac{1}{\left( \left(q-\frac{p}{2}\right)^2 - \frac{p^2}{4}\right)^2 + q^2 + (p - q)^2 +1} \\ & + \int_{|q-\frac{p}{2}| < |p|} \rd q \; \frac{1}{\left( \left(q-\frac{p}{2}\right)^2 - \frac{p^2}{4}\right)^2 + q^2 + (p - q)^2 +1}\,. \end{split} \end{equation} The first term on the r.h.s. of the last equation is bounded by \begin{equation} \begin{split} \int_{|q-\frac{p}{2}| > |p|} \rd q \; \frac{1}{\left( \left(q-\frac{p}{2}\right)^2 - \frac{p^2}{4}\right)^2 + q^2 + (p - q)^2 +1} \leq \; & \int_{|q-\frac{p}{2}| > |p|} \rd q \; \frac{1}{\frac{9}{16} \left|q-\frac{p}{2}\right|^4 +1} \\ \leq \; & \frac{16}{9} \int_{\bR^3} \rd q \; \frac{1}{|q|^4 +1} < \infty\,, \end{split} \end{equation} uniformly in $p \in \bR^3$. As for the second term on the r.h.s. of (\ref{eq:int2}), we observe that \begin{equation} \begin{split} \int_{|q-\frac{p}{2}| < |p|} \rd q \; &\frac{1}{\left( \left(q-\frac{p}{2}\right)^2 - \frac{p^2}{4}\right)^2 + q^2 + (p - q)^2 +1} \\ = \; & \int_{|x| < |p|} \rd x \; \frac{1}{\left( x^2 - \frac{p^2}{4}\right)^2 + \left( x+\frac{p}{2} \right)^2 + \left(x-\frac{p}{2} \right)^2 +1} \\ = \; & 4\pi \int_{0}^{|p|} \rd r \frac{r^2}{\left( r^2 - \frac{|p|^2}{4}\right)^2 + 2r^2 + \frac{|p|^2}{2} +1} \\ \leq \; & C |p|^2 \int_{-|p|/2}^{|p|/2} \rd r \frac{1}{ r^2 \left( r + |p|\right)^2 + \left( r+\frac{|p|}{2} \right)^2 + \frac{|p|^2}{4} +1} \\ \leq\; & C \int_{-|p|/2}^{|p|/2} \rd r \, \frac{1}{ r^2 + 1} \leq C \int_{\bR} \rd r \, \frac{1}{r^2 + 1} < \infty , \end{split} \end{equation} uniformly in $p$. \end{proof} \begin{lemma}\label{lm:VL12} Suppose $V \in L^1 (\bR^3)$ with $\int \, V(x) \rd x = 1$. For $\alpha >0$ , let $V_{\alpha} (x) = \alpha^{-3} V (x/\alpha)$. Then there exists a sequence $\beta_{\alpha}$ with $\beta_{\alpha} \to 0$ as $\alpha \to 0$ such that \begin{equation}\label{eq:VL12} \begin{split} \left|\langle \ph, \left( V_{\alpha} (x_1 -x_2) - \delta (x_1 -x_2) \right) \psi \rangle \right| \leq C\beta_{\alpha} \, \langle &\psi, \left( (\nabla_1 \cdot \nabla_2)^2 -\Delta_1 - \Delta_2 + 1 \right) \psi \rangle^{1/2} \; \\ & \times \langle \ph, \left( (\nabla_1-\nabla_2)^4 + (\nabla_1 + \nabla_2)^2 + 1 \right) \ph \rangle^{1/2}, \end{split} \end{equation} for all $\ph,\psi \in L^2 (\bR^6)$. \end{lemma} \begin{proof} Switching to Fourier space we find \begin{equation*} \left\langle \ph, \Big( V_{\alpha} (x_1 -x_2) - \delta (x_1 -x_2) \Big) \psi \right\rangle = \text{I} + \text{II} \, , \end{equation*} where we defined \begin{equation}\label{eq:IandII} \begin{split} \text{I} = \; & \int_{| x \cdot (p_1 - q_1)| < \alpha^{-1/2}} \rd p_1 \rd p_2 \rd q_1 \rd q_2 \rd x \, V(x) \, \overline{\widehat{\ph}} (p_1, p_2) \left( e^{i\alpha x \cdot (p_1 -q_1)} - 1 \right) \widehat{\psi} (q_1, q_2) \delta (p_1 +p_2 - q_1 -q_2) , \\ \text{II} = \; & \int_{| x \cdot (p_1 - q_1)| \geq \alpha^{-1/2}} \rd p_1 \rd p_2 \rd q_1 \rd q_2 \rd x \, V(x) \, \overline{\widehat{\ph}} (p_1, p_2) \left( e^{i\alpha x \cdot (p_1 -q_1)} - 1 \right) \widehat{\psi} (q_1, q_2) \delta (p_1 +p_2 - q_1 -q_2)\,. \end{split} \end{equation} To bound the first term we use that $|e^{i\kappa} - 1| \leq |\kappa|$, for $\kappa \in \bR$, and we observe that \begin{equation}\label{eq:I1} \begin{split} |\text{I}| \leq \; &\alpha^{1/2} \| V \|_1 \int \rd p_1 \rd p_2 \rd q_1 \rd q_2 \frac{\sqrt{|p_1 - p_2|^4 + (p_1 + p_2)^2 + 1}}{\sqrt{(q_1 \cdot q_2)^2 + q_1^2 + q_2^2 +1}} \, |\widehat{\ph} (p_1, p_2)| \\ &\hspace{4cm} \times \frac{\sqrt{(q_1 \cdot q_2)^2 + q_1^2 + q_2^2 +1}}{\sqrt{|p_1 - p_2|^4 + (p_1 + p_2)^2 + 1}} \, |\widehat{\psi} (q_1, q_2)| \, \delta (p_1 +p_2 - q_1 -q_2) \end{split} \end{equation} With a Schwarz inequality, we obtain that \begin{equation*}\begin{split} |\text{I}| \leq \; &\alpha^{1/2} \| V \|_1 \left( \int \rd p_1 \rd p_2 \rd q_1 \rd q_2 \, \frac{|p_1 - p_2|^4 + (p_1 + p_2)^2 + 1}{(q_1 \cdot q_2)^2 + q_1^2 + q_2^2 +1} \, |\widehat{\ph} (p_1, p_2)|^2 \, \delta (p_1 + p_2 - q_1 - q_2) \right)^{1/2} \\ & \times \left( \int \rd p_1 \rd p_2 \rd q_1 \rd q_2 \, \frac{\sqrt{(q_1 \cdot q_2)^2 + q_1^2 + q_2^2 +1}}{\sqrt{|p_1 - p_2|^4 + (p_1 + p_2)^2 + 1}} \,|\widehat{\psi} (q_1, q_2)|^2 \, \delta (p_1 +p_2 - q_1 -q_2)\right)^{1/2} \\ \leq \; &\alpha^{1/2} \| V \|_1 \; \Big\langle \ph, \left( (\nabla_1 - \nabla_2)^4 + (\nabla_1 + \nabla_2)^2 + 1 \right) \ph \Big\rangle^{1/2} \, \Big\langle \psi, \left( (\nabla_1 \cdot \nabla_2)^2 -\Delta_1 - \Delta_2 + 1 \right) \psi \Big\rangle^{1/2} \\ & \times\left( \sup_{p \in \bR^3} \int \frac{\rd q}{|q-p|^4 + p^2 + 1} \right)^{1/2} \,\left( \sup_{q \in \bR^3} \int \frac{\rd p}{(p\cdot (q-p))^2 + p^2 + (q-p)^2+ 1} \right)^{1/2}\,. \end{split} \end{equation*} {F}rom \[ \sup_{p\in \bR^3} \int \frac{\rd q}{|q-p|^4 + p^2 + 1} \leq \int \frac{\rd q}{|q|^4 + 1} < \infty \] and (\ref{eq:int1}) it follows that \begin{equation}\label{eq:Ifin} |\text{I}| \leq C \alpha^{1/2} \Big\langle \ph, \left( (\nabla_1 - \nabla_2)^4 + (\nabla_1 + \nabla_2)^2 + 1 \right) \ph \Big\rangle^{1/2} \, \Big\langle \psi, \left( (\nabla_1 \cdot \nabla_2)^2 -\Delta_1 - \Delta_2 + 1 \right) \psi \Big\rangle^{1/2} \, .\end{equation} In order to control the second term in (\ref{eq:IandII}), we bound it by \begin{equation}\label{eq:II1} \begin{split} | \text{II}| \leq \; &2 \int_{| x \cdot (p_1 - q_1)| \geq \alpha^{-1/2}} \rd p_1 \rd p_2 \rd q_1 \rd q_2 \rd x \, |V(x)| \, |\widehat\ph (p_1, p_2)| \, | \widehat\psi (q_1, q_2)| \, \delta (p_1 +p_2 - q_1 -q_2) \\ \leq \; &2 \int_{|x| \geq \alpha^{-1/4}} \rd p_1 \rd p_2 \rd q_1 \rd q_2 \rd x \, |V(x)| \, |\widehat\ph (p_1, p_2)| \, |\widehat \psi (q_1, q_2)| \, \delta (p_1 +p_2 - q_1 -q_2) \\ &+ 2 \int_{|p_1-q_1| \geq \alpha^{-1/4}} \rd p_1 \rd p_2 \rd q_1 \rd q_2 \rd x \, |V(x)| \, |\ph (p_1, p_2)| \, | \psi (q_1, q_2)| \, \delta (p_1 +p_2 - q_1 -q_2) \\ \leq \; &\beta_{1,\alpha} \int \rd p_1 \rd p_2 \rd q_1 \rd q_2 \, |\widehat\ph (p_1, p_2)| \, |\widehat \psi (q_1, q_2)| \, \delta (p_1 +p_2 - q_1 -q_2) \\ &+2\| V \|_1 \int_{|p_1-q_1| \geq \alpha^{-1/4}} \rd p_1 \rd p_2 \rd q_1 \rd q_2 \, |\widehat\ph (p_1, p_2)| \, | \widehat\psi (q_1, q_2)| \, \delta (p_1 +p_2 - q_1 -q_2)\\ \leq \; &\beta_{1,\alpha} \left\langle \ph , \left((\nabla_1 - \nabla_2)^4 + (\nabla_1 + \nabla_2)^2 +1 \right) \ph \right\rangle^{1/2} \, \left\langle \psi , \left((\nabla_1 \cdot \nabla_2)^4 - \Delta_1 - \Delta_2 +1 \right) \ph \right\rangle^{1/2} \\ &+2\| V \|_1 \int_{|p_1-q_1| \geq \alpha^{-1/4}} \rd p_1 \rd p_2 \rd q_1 \rd q_2 \, |\widehat\ph (p_1, p_2)| \, | \widehat\psi (q_1, q_2)| \, \delta (p_1 +p_2 - q_1 -q_2)\, , \end{split} \end{equation} where we defined \[ \beta_{1,\alpha} = 2 \int_{|x| \geq \alpha^{-1/4}} |V (x)| \, , \] and we bounded the first integral analogously as we did with the integral in (\ref{eq:I1}). Note that $\beta_{1,\alpha} \to 0$ as $\alpha \to 0$, because $V \in L^1 (\bR^3)$. We still need to control the last integral, on the r.h.s. of the last equation. To this end, we observe that \begin{equation}\label{eq:II2} \begin{split} \int_{|p_1-q_1| \geq \alpha^{-1/4}} \rd p_1 &\rd p_2 \rd q_1 \rd q_2 \, |\widehat\ph (p_1, p_2)| \, |\widehat \psi (q_1, q_2)| \, \delta (p_1 +p_2 - q_1 -q_2) \\ \leq \; & 2 \int_{|q_1| \geq \alpha^{-1/4}/8} \rd p_1 \rd p_2 \rd q_1 \rd q_2 \, |\widehat\ph (p_1, p_2)| \, |\widehat \psi (q_1, q_2)| \, \delta (p_1 +p_2 - q_1 -q_2) \\ &+ \int_{|q_2| \geq \alpha^{-1/4}/8} \rd p_1 \rd p_2 \rd q_1 \rd q_2 \, |\widehat\ph (p_1, p_2)| \, |\widehat \psi (q_1, q_2)| \, \delta (p_1 +p_2 - q_1 -q_2) \\ &+\int_{\stackrel{|p_1| \geq \alpha^{-1/4}/2}{|q_1 + q_2| \leq \alpha^{-1/4}/4}} \rd p_1 \rd p_2 \rd q_1 \rd q_2 \, |\widehat\ph (p_1, p_2)| \, |\widehat \psi (q_1, q_2)| \, \delta (p_1 +p_2 - q_1 -q_2)\,. \end{split} \end{equation} The first two terms can be bounded by \begin{equation}\label{eq:bdq} \begin{split} \int_{|q_j| \geq \alpha^{-1/4}/8} &\rd p_1 \rd p_2 \rd q_1 \rd q_2 \, |\widehat\ph (p_1, p_2)| \, | \widehat\psi (q_1, q_2)| \, \delta (p_1 +p_2 - q_1 -q_2) \\ \leq \; &C \alpha^{1/12} \left\langle \ph , \left((\nabla_1 - \nabla_2)^4 + (\nabla_1 + \nabla_2)^2 +1 \right) \ph \right\rangle^{1/2} \, \left\langle \psi , \left((\nabla_1 \cdot \nabla_2)^4 - \Delta_1 - \Delta_2 +1 \right) \ph \right\rangle^{1/2}\, \end{split} \end{equation} which holds for both $j=1,2$, and for a universal constant $C$, independent of $\alpha, \ph,\psi$. To show (\ref{eq:bdq}) note that, proceeding as in (\ref{eq:I1}) (for example for $j=1$), we have \begin{equation} \begin{split} &\int_{|q_1| \geq \alpha^{-1/4}/8} \rd p_1 \rd p_2 \rd q_1 \rd q_2 \, |\widehat\ph (p_1, p_2)| \, |\widehat \psi (q_1, q_2)| \, \delta (p_1 +p_2 - q_1 -q_2) \\ &\hspace{.1cm} \leq \; \left\langle \ph , \left((\nabla_1 - \nabla_2)^4 + (\nabla_1 + \nabla_2)^2 +1 \right) \ph \right\rangle^{1/2} \, \left\langle \psi , \left((\nabla_1 \cdot \nabla_2)^4 - \Delta_1 - \Delta_2 +1 \right) \ph \right\rangle^{1/2} \\ &\hspace{.3cm} \times \left( \sup_{q \in \bR^3} \int \frac{\rd p}{(p\cdot (q-p))^2 + p^2 + (q-p)^2 + 1} \right)^{1/2} \, \left(\sup_{p \in \bR^3} \int_{|q| \geq \alpha^{-1/4}/8} \frac{\rd q}{|q-p|^4 + p^2 + 1}\right)^{1/2} \end{split} \end{equation} and thus (\ref{eq:bdq}) follows from (\ref{eq:int1}) and \begin{equation} \begin{split} \sup_{p\in \bR^3} \int_{|q| \geq \alpha^{-1/4}/8} \frac{\rd q}{|q-p|^4 + p^2 + 1} & \leq (8\alpha^{1/4})^{1/3} \sup_{p\in \bR^3} \int \rd q \frac{ |q|^{1/3}}{|q-p|^4 + p^2 + 1} \\ & \leq (8\alpha^{1/4})^{1/3} \left( \sup_{q,p \in \bR^3} \frac{|q+p|^{1/3}}{(|q|^4 + p^2 + 1)^{1/6}} \right) \, \int \frac{\rd q}{(|q|^4 + 1)^{5/6}} \\ & \leq C \alpha^{1/12}\,. \end{split} \end{equation} As for the last term on the r.h.s. of (\ref{eq:II2}), we note that \begin{equation} \begin{split} &\int_{\stackrel{|p_1| \geq \alpha^{-1/4}/2}{|q_1 + q_2| \leq \alpha^{-1/4}/4}} \rd p_1 \rd p_2 \rd q_1 \rd q_2 \, |\widehat\ph (p_1, p_2)| \, | \widehat\psi (q_1, q_2)| \, \delta (p_1 +p_2 - q_1 -q_2) \\& \leq \left\langle \ph , \left((\nabla_1 - \nabla_2)^4 + (\nabla_1 + \nabla_2)^2 +1 \right) \ph \right\rangle^{1/2} \, \left\langle \psi , \left((\nabla_1 \cdot \nabla_2)^4 - \Delta_1 - \Delta_2 +1 \right) \ph \right\rangle^{1/2} \\ & \hspace{.2cm} \times \left( \sup_{p\in \bR^3} \int \frac{\rd q}{|q-p|^4 + p^2 + 1} \right)^{1/2} \left( \sup_{q} \int_{\stackrel{|p| \geq 2 |q|}{|p| \geq \alpha^{-1/4}/2}} \frac{\rd p}{(p \cdot (q-p))^2 + p^2 + (q-p)^2 + 1} \right)^{1/2}\,. \end{split} \end{equation} Since \begin{equation} \begin{split} \int_{\stackrel{|p| \geq 2 |q|}{|p| \geq \alpha^{-1/4}/2}} & \frac{\rd p}{(p \cdot (q-p))^2 + p^2 + (q-p)^2 + 1} \\ & \leq (2\alpha^{1/4})^{1/3} \int_{|p| \geq 2 |q|} \rd p \, \frac{|p|^{1/3}}{\left(\left(p-\frac{q}{2} \right)^2 - \frac{q^2}{4} \right)^2 + p^2 + (q-p)^2 + 1} \\ & \leq C \alpha^{1/12} \int_{|p| \geq |q|} \rd p \, \frac{\left|p+ \frac{q}{2} \right|^{1/2}}{\left(p^2 - \frac{q^2}{4} \right)^2 + p^2 + \frac{q^2}{4} + 1} \\ & \leq C \alpha^{1/12} \int \rd p \frac{|p|^{1/2}}{\frac{9}{16} |p|^4 + 1} \\ & \leq C \alpha^{1/12}\, , \end{split} \end{equation} it follows that the last term on the r.h.s. of (\ref{eq:II2}) is bounded by \begin{equation} \begin{split} &\int_{\stackrel{|p_1| \geq \alpha^{-1/4}/2}{|q_1 + q_2| \leq \alpha^{-1/4}/2}} \rd p_1 \rd p_2 \rd q_1 \rd q_2 \, |\widehat\ph (p_1, p_2)| \, |\widehat \psi (q_1, q_2)| \, \delta (p_1 +p_2 - q_1 -q_2) \\& \leq C \alpha^{1/12} \, \left\langle \ph , \left((\nabla_1 - \nabla_2)^4 + (\nabla_1 + \nabla_2)^2 +1 \right) \ph \right\rangle^{1/2} \, \left\langle \psi , \left((\nabla_1 \cdot \nabla_2)^4 - \Delta_1 - \Delta_2 +1 \right) \ph \right\rangle^{1/2}\,. \end{split} \end{equation} {F}rom the last equation, (\ref{eq:bdq}), (\ref{eq:II2}), and (\ref{eq:II1}), it follows that \begin{equation*} \begin{split} |\text{II}| \leq \; &C (\beta_{1,\alpha} + \alpha^{1/12}) \\ &\hspace{.5cm} \times \left\langle \ph , \left((\nabla_1 - \nabla_2)^4 + (\nabla_1 + \nabla_2)^2 +1 \right) \ph \right\rangle^{1/2} \, \left\langle \psi , \left((\nabla_1 \cdot \nabla_2)^4 - \Delta_1 - \Delta_2 +1 \right) \ph \right\rangle^{1/2}\,. \end{split} \end{equation*} This together with (\ref{eq:Ifin}), implies (\ref{eq:VL12}) with $\beta_{\alpha} = C (\beta_{1,\alpha} + \alpha^{1/12} + \alpha^{1/2})$. \end{proof} When dealing with the limit points $\gamma^{(k)}_{\infty,t}$, for which we have stronger a-priori estimates, we will make use of the following lemma, whose proof can be found in \cite{ESY2} (Lemma 8.2). \begin{lemma}\label{lm:sobsob} Suppose that $\delta_\alpha(x)$ is a function satisfying $0 \leq \delta_{\alpha} (x) \leq C \alpha^{-3} {\bf 1} (|x| \leq \alpha)$ and $\int \delta_{\alpha} (x) \rd x=1$ (for example $\delta_{\alpha} (x) = \alpha^{-3} g (x/\alpha)$, for a bounded probability density $g(x)$ supported in $\{ x : |x| \leq 1\}$). Moreover, for $J^{(k)} \in \cK_k$, and for $j=1,\dots ,k$, we define the norm \be\label{eq:Jnorm} \tri J^{(k)} \tri_{j} := \sup_{\bx_k, \bx'_k} \la x_1 \ra^4 \dots \la x_k \ra^4 \la x'_1 \ra^4 \dots \la x'_k \ra^4 \left( |J^{(k)} (\bx_k ; \bx'_k)| + |\nabla_{x_j} J^{(k)} (\bx_k;\bx'_k)| + |\nabla_{x'_j} J^{(k)} (\bx_k;\bx'_k)| \right) \, \ee and $S_j = (1-\Delta_{x_j})$ (here $\la x \ra^2 := 1+ x^2$). Then if $\gamma^{(k+1)} (\bx_{k+1};\bx'_{k+1})$ is the kernel of a density matrix on $L^2 (\bR^{3(k+1)})$, we have, for any $j\leq k$, \begin{multline}\label{eq:gammaintbound} \Big| \int \rd \bx_{k+1} \rd \bx'_{k+1} \, J^{(k)} (\bx_k ; \bx'_k) \left(\delta_{\alpha_1} (x_{k+1} - x'_{k+1}) \delta_{\alpha_2} (x_j -x_{k+1}) - \delta (x_{k+1} -x'_{k+1}) \delta (x_j - x_{k+1})\right)\\ \times \gamma^{(k+1)} (\bx_{k+1} ; \bx'_{k+1}) \Big| \\ \leq C_k \, \tri J^{(k)} \tri_j \left( \alpha_1 + \sqrt{\alpha_2}\right) \, \tr \, | S_j S_{k+1} \gamma^{(k+1)} S_j S_{k+1}|\;. \end{multline} The same bound holds if $x_j$ is replaced with $x_j'$ in (\ref{eq:gammaintbound}) by symmetry. \end{lemma} \appendix \section{Properties of the cutoff function $\theta_i^{(n)}$} \setcounter{equation}{0} Recall that the cutoff functions $\Theta_k^{(n)}=\Theta_k^{(n)}(\bx)$ defined for $k=1,\dots,N$ and $n \in \bN$, in Eq. (\ref{eq:thetan}). In the following lemma, whose proof can be found in \cite[Appendix A]{ESY}, we collect some of their important properties which were used in the energy estimate, Proposition \ref{prop:hk}. \begin{lemma}\label{lm:theta} \begin{itemize} \item[i)] The functions $\Theta_k^{(n)}$ are monotonic in both indices, that is for any $n, k \in \bN$, \[ \Theta_{k+1}^{(n)} \leq \Theta_k^{(n)} \leq 1\; ,\qquad \Theta_k^{(n+1)} \leq \Theta_k^{(n)} \leq 1 \; . \] Moreover, $\Theta^{(n)}_k$ is permutation symmetric in the first $k$ and the last $N-k$ variables. %\item[ii)] We have, for any $n\in \bN$, $k=1,\dots,N$, %\begin{equation}\label{eq:lmthetaii} %\left( \frac{2^n}{\ell^{\eps}} \sum_{i=1}^k \sum_{j\neq i}^N h_{ij} %\right)^m \Theta_k^{(n)} \leq C_m \; \Theta_k^{(n-1)}\,. %\end{equation} %\item[iii)] For every $k = 1, \dots , N$, $n \in \bN$, we have %\begin{equation}\label{eq:lmthetaiiia} %\begin{split} %|\nabla_i & \Theta_k^{(n)}| \leq C \ell^{-1} \left( %\frac{2^{n}}{\ell^{\eps}} \sum_{r=1}^N h_{ri} \right) \Theta_k^{(n)} %\leq C \ell^{-1} %\Theta_k^{(n-1)} \qquad \text{if } i \leq k \\ %|\nabla_i &\Theta_k^{(n)}| \leq C \ell^{-1} \left( %\frac{2^{n}}{\ell^{\eps}} \sum_{r=1}^k h_{ri} \right) \Theta_k^{(n)} %\leq C \ell^{-1} \Theta_k^{(n-1)} \qquad \text{if } i %> k %\end{split} %\end{equation} \item[ii)] For every $k =1,\dots ,N$, $n \in\bN$ we have \begin{equation}\label{eq:lmthetaiv} \begin{split} \sum_{j=1}^N &\frac{\left| \nabla_j \Theta_k^{(n)} \right|^2}{\Theta_k^{(n)}} \leq C \ell^{-2} \Theta_k^{(n-1)} \end{split} \end{equation} \item[iii)] For every fixed $k =1,\dots,N$ and $n \in \bN$ we have \begin{equation}\label{eq:lmthetavi} \begin{split} \sum_{i,j} \left| \nabla_i \nabla_j \Theta_k^{(n)} \right| \leq C \ell^{-2} \Theta_k^{(n-1)}\,. \end{split} \end{equation} \end{itemize} \end{lemma} \section{Removal of the assumption on derivatives of $V$} \label{app:nablaV} The goal of this appendix is to explain how the assumption \begin{equation}\label{eq:assV2} |\nabla^{\alpha} V (x)| \leq C \qquad \text{for all } x \in \bR^3, \; |\alpha|\leq 2 \end{equation} in Theorem \ref{thm:main} can be removed. The main observation is that (\ref{eq:assV2}) is only used in the proof of the higher order energy estimate, Proposition \ref{prop:hk}, in the form $\|\nabla V_N\|_\infty\leq CN^3$, $\|\nabla^2 V_N\|_\infty \leq CN^4$. More precisely, the estimate on $\|\nabla V_N\|_\infty$ is first used in the study of the third term on the r.h.s. of (\ref{eq:Tk+1}) (the third term on the r.h.s. of (\ref{eq:for}) in the case $k=2$); namely the term containing the commutator $[T^{1/2}, D_k] = [(H_N+N)^{1/2}, \nabla_1 \dots \nabla_k ]$. Bounds on the first and second derivatives are also used in the proof of Lemma \ref{lemma:exp}. However, in both cases, the final estimates turn out to be subexponentially small in $N$ (see (\ref{eq:nablaVterm}) and (\ref{eq:commm})). For this reason, the proof of Proposition \ref{prop:hk} remains unchanged if, instead of (\ref{eq:assV2}), we allow $V= V^{(N)}$ to depend on $N$ and only assume \be \| \nabla^\alpha V^{(N)}\|_\infty \leq e^{cN^\kappa}, \quad |\alpha|\leq 2, \label{eq:sube} \ee for some sufficiently small $\kappa>0$. \medskip More precisely, suppose that the potential $V\geq 0$ satisfies $V(x) \leq C \langle x \rangle^{-\sigma}$ for some $\sigma >5$, with no assumptions on the derivatives $\nabla^{\alpha} V$, for $|\alpha| \geq 1$. Then consider the evolution $\psi_{N,t} = e^{-iH_N t} \psi_N$ of an initial $N$-body wave function $\psi_N$ satisfying the two assumptions (\ref{eq:assH1}) and (\ref{eq:asscond}), with respect to the evolution generated by the Hamiltonian \[ H_N = \sum_{j=1}^N -\Delta_j + \sum_{i0$ (here the constant is chosen so that $\int \rd x \, \nu^{(N)} (x) =1$), and we consider the evolution $\wt \psi_{N,t} = e^{-i\wt H_N t} \psi_N$ of the initial data $\psi_N$ with respect to the modified Hamiltonian \[ \wt H_N = -\sum_{j=1}^N \Delta_j + \sum^N_{i0$, applying a Schwarz inequality, and changing variables $q \to p$ in $\widehat{\wt \psi}_{N,t}$, we obtain \begin{equation} \begin{split} \sup_{y\in \bR^3}\Big| \langle \psi_{N,t}, (\delta_y - \nu_{N,y}) &(x_1 - x_2) \wt \psi_{N,t} \rangle \Big| \\ \leq \; C e^{-\frac{N^{\kappa}}{8}} \int &\rd \bp_{N-2} \rd p_1 \rd p_2 \rd q_1 \rd q_2 \; \delta (p_1 + p_2 - q_1 -q_2) \frac{|q_1|^{1/2}+1}{(q_1^2 + 1) (q_2^2 + 1)} \\ &\times (p_1^2 + 1)(p_2^2 + 1) \left(\beta |\widehat{\psi}_{N,t} (p_1,p_2, \bp_{N-2})|^2 + \beta^{-1} |\widehat{\wt \psi}_{N,t} (p_1,p_2,\bp_{N-2})|^2 \right) \end{split} \end{equation} for every $\beta >0$. 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