Content-Type: multipart/mixed; boundary="-------------0408151153692" This is a multi-part message in MIME format. ---------------0408151153692 Content-Type: text/plain; name="04-252.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-252.comments" Revised version which replaces an older one. ---------------0408151153692 Content-Type: text/plain; name="04-252.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-252.keywords" Random Schroedinger equations, hydrodynamic limits ---------------0408151153692 Content-Type: application/x-tex; name="lpc.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="lpc.tex" \documentclass[12pt]{amsart} %\usepackage{times} %\documentclass[12pt]{article} \usepackage{amssymb,amsfonts,latexsym,amscd,epsfig,psfrag} %\usepackage{graphicx} %\pagestyle{myheadings} \setlength\textwidth{6.5 in} \setlength\textheight{8.5 in} \voffset=-0.6in \hoffset = -0.6in \parindent = 0.4in \pagestyle{plain} \begin{document} \def\A{{\mathcal A}} \def\alg{{\mathcal A}} \def\amp{{\rm Amp}} \def\Bound{{\mathcal B}} \def\bra{\big\langle} \def\C{{\Bbb C}} \def\Dom{\mathfrak{Dom}} \def\dist{{\rm dist}} \def\Exp{{\Bbb E}} \def\Expnd {{\Bbb E}_{n-d}} \def\Exptc{{\Bbb E}_{2-conn}} \def\Expd{{\Bbb E}_{disc}} \def\e{\varepsilon} \def\H{{\mathcal H}} \def\Hpl{{\Bbb H}} \def\Ie{I} \def\Im{{ Im}} \def\ket{\big\rangle} \def\lb{\left[} \def\mes{{\rm mes}} \def\N{{\Bbb N}} \def\nm{{|\!|\!|\,}} \def\p{r} \def\pip{\tau} \def\qm{q^{(m+1)}} \def\R{{\Bbb R}} \def\rb{\right]} %%\def\Re{{\mathcal Re}} \def\rc{\frac{1}{2}} \def\Rem{ R} \def\Sc{{\mathcal S}} \def\Tor{\Bbb T} \def\up{\underline{\vp}} \def\uk{\underline{\vk}} \def\utk{\underline{\tilde\vk}} \def\uvw{\underline{\vw}} \def\Z{{\Bbb Z}} \def\vx{{ x}} \def\vy{{ y}} \def\ve{{ e}} \def\vk{{ k}} \def\vl{{ l}} \def\vm{{ m}} \def\vn{{ n}} \def\vp{{ p}} \def\vQ{{ Q}} \def\vq{{ q}} \def\vr{{ r}} \def\vv{{ v}} \def\vw{{ w}} \def\1{{\bf 1}} \def\eqnn{\begin{eqnarray*}} \def\eeqnn{\end{eqnarray*}} \def\eqn{\begin{eqnarray}} \def\eeqn{\end{eqnarray}} \def\bal{\begin{align}} \def\eal{\end{align}} %\def\prf{\noindent{\em Proof.}$\;$} %\def\endprf{\vrule height .6em width .6em depth 0pt\bigbreak} \def\prf{\begin{proof}} \def\endprf{\end{proof}} %%\numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \newtheorem{definition}{Definition}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{lemma}{Lemma}[section] \title{\sc ${\mathcal L}^r$-Convergence of a Random Schr\"odinger to a Linear Boltzmann Evolution } \author{Thomas Chen} \address{Courant Institute of Mathematical Sciences\\ New York University\\ 251 Mercer Street\\ New York, NY 10012-1185.} \email{chenthom@cims.nyu.edu} \address{New address as of September 2004: Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544.} \date{} \maketitle \begin{abstract} We study the macroscopic scaling and weak coupling limit for a random Schr\"odinger equation on $\Z^3$. We prove that the Wigner transforms of a large class of "macroscopic" solutions converge in $\p$-th mean to solutions of a linear Boltzmann equation, for any finite value of $\p\in\R_+$. This extends previous results where convergence in expectation was established. \end{abstract} \section{Introduction} We consider the quantum mechanical dynamics of an electron against a background lattice of impurity ions exhibiting randomly distributed interaction strengths. Models of this type (Anderson model) are widely used to investigate qualitative features of technically highly relevant classes of materials that comprise semiconductors. Questions of key mathematical interest, treated intensively in the literature, address the emergence of electric conduction and insulation. While there exist landmark mathematical results explaining disorder-induced insulation at strong coupling (Anderson localization, \cite{aimo, frsp}), the weak coupling regime is at present far less understood. In the latter context, we shall here analyze issues regarding the derivation of macroscopic transport equations. We study the macroscopic scaling and weak coupling limit of the quantum dynamics generated by the Hamiltonian $$ H_\omega=-\Delta+\lambda \sum_{y\in\Z^3} \omega_y\delta(x-y) $$ on $\ell^2(\Z^3)$. Here, $\Delta$ is the nearest neighbor discrete Laplacian, $0<\lambda\ll1$ is a small coupling constant that defines the disorder strength, and $\omega_y$ are independent, identically distributed Gaussian random variables. Let $\phi_t\in\ell^2(\Z^3)$ be the solution of the Schr\"odinger equation \eqn \left\{ \begin{array}{rcl} i\partial_t\phi_t&=&H_\omega\phi_t \\ \phi_0&\in&\ell^2(\Z^3) \;, \end{array} \right. \eeqn with a non-random initial condition $\phi_0$ which is supported on a region of diameter $O(\lambda^{-2})$. Let $W_{\phi_t}(x,v)$ denote its {\em Wigner transform}, where $x\in\Z^3$, and $v\in[0,1]^3$. We consider a scaling for small $\lambda$, in which $(T,X):=\lambda^{2}(t,x)$, $V:=v$, are the macroscopic time, position, and velocity variables (while $(t,x,v)$ are the microscopic ones). We then focus on a corresponding, appropriately rescaled version $W^{resc}_\lambda(T,X,V)$ of $W_{\phi_t}(x,v)$. It was proved by Erd\"os and Yau for the continuum case, \cite{erdyau, erd}, and by the author for the lattice case, \cite{ch}, that for all test functions $J(X,V)$, and globally in macroscopic time $T$, \eqnn &&\lim_{\lambda\rightarrow0}\Exp\Big[ \int dX dV J(X,V)W_\lambda^{resc}(T,X,V)\Big] \nonumber\\ &&\hspace{3cm}=\int dX dV J(X,V)F(T,X,V)\;, \eeqnn where $F(T,X,V)$ is the solution of a linear Boltzmann equation. The corresponding local in $T$ result was achieved much earlier by Spohn, \cite{sp}. Our main result in this paper improves the mode of convergence in that we establish convergence in $r$-th mean for every finite $r\in\R_+$, \eqnn &&\lim_{\lambda\rightarrow0}\Exp\Big[\, \Big|\int dX dV J(X,V)W_\lambda^{resc}(T,X,V) \nonumber\\ &&\hspace{3cm}-\int dX dV J(X,V)F(T,X,V)\Big|^r\,\Big]=0\;. \eeqnn Hence in particular, the variance of $\int dX dV J(X,V) W_\lambda^{resc}(T,X,V)$ vanishes in this limit. As an immediate corollary, one obtains convergence in probability, and other standard modes of convergence. The proof comprises generalizations and extensions to the graph expansion method introduced by Erd\"os and Yau in \cite{erdyau, erd}, and further elaborated on in \cite{ch}. The structure of graphs entering the problem at hand is significantly more complicated. But as it turns out, it is not very hard to show that the corresponding amplitudes are extremely small. The strategy then consists of balancing these small amplitudes against the very large number of graphs, which is similar to the approach in the previous works \cite{erdyau, erd, ch}. The present work, as well as \cite{ch} and \cite{erdyau, erd}, address a time scale of order $O(\lambda^{-2})$, in which ballistic behavior is prevalent, because the average number of collisions experienced by the electron is finite. For this reason, the macroscopic dynamics is governed by a Boltzmann equation. Beyond this time scale, the average number of collisions is {\em infinite}, and the problem becomes much harder. Very recently, Erd\"os, Salmhofer and Yau have established that in a time scale of order $O(\lambda^{-2-\kappa})$ for an explicit numerical value of $\kappa>0$, the macroscopic dynamics in $d=3$ derived from the quantum dynamics is determined by a {\em diffusion equation}, \cite{erdsalmyau}. This is a breakthrough result which is the first of its kind. We note that control of the macroscopic dynamics up to a time scale $O(\lambda^{-2})$ allows to obtain lower bounds of a comparable scale on the localization lengths of eigenvectors of $H_\omega$ in 3 dimensions, for small $\lambda$, \cite{ch}. This extends recent results of Shubin, Schlag and Wolff, \cite{shscwo}, who derived lower bounds of this type for the weakly disordered Anderson model in dimensions $d=1,2$ by use of techniques of harmonic analysis. This work comprises a partial joint result with Laszlo Erd\"os (Lemma {~\ref{conngrbd-1}}), to whom the author is deeply grateful for his support and generosity. \section{Definition of the model and statement of main results} We consider the discrete random Schr\"odinger operator \eqn H_\omega = -\Delta + \lambda V_\omega \; \label{Homega-def} \eeqn on $\ell^2(\Z^3)$, with potential function \eqn V_{\omega}(\vx) =\sum_{ \vy \in\Z^3 } \omega_\vy \delta(\vx-\vy) \;, \eeqn where $\omega_y$ are independent, identically distributed Gaussian random variables satisfying $\Exp[\omega_x]=0$, $\Exp[\omega_x^2]=1$, for all $x\in\Z^3$. Expectations of higher powers of $\omega_x$ satisfy Wick's theorem, cf. \cite{erdyau}, and our discussion below. We use the convention \eqn \hat f(\vk) = \sum_{\vx\in\Z^3} e^{-2\pi i\vk\cdot\vx} f(\vx) \; \; , \; \; \check g(\vx) = \int_{\Tor^3} dk\; g(\vk) e^{2 \pi i \vk\cdot\vx} \; \eeqn for the Fourier transform and its inverse. In frequency space, the nearest neighbor lattice Laplacian defines the multiplication operator \eqn (-\Delta f)\hat{\;}(\vk) = e(\vk) \hat f(\vk) \;, \eeqn where \eqn e(\vk) &=& 2\sum_{i=1}^3 \big( 1- \cos(2\pi k_i) \big) \nonumber\\ &=&4\sum_{i=1}^3 \sin^2(\pi k_i) \label{kinendef} \eeqn is the quantum mechanical kinetic energy of the electron at momentum $k$. Let $\phi_t\in \ell^2(\Z^3)$ denote the solution of the random Schr\"odinger equation \eqn \left\{ \begin{array}{rcl} i\partial_t\phi_t&=&H_\omega \phi_t \\ \phi_0&\in& \ell^2(\Z^3)\;, \end{array} \right. \eeqn for a fixed realization of the random potential. Its (real, but not necessarily positive) Wigner transform $W_{\phi_t}:\Z^3\times\Tor^3\rightarrow\R$ is defined by \eqn W_{\phi_t}(x,v)=\sum_{y }\overline{\phi_t(x+\frac y2)}\phi_t(x-\frac y2) e^{2\pi iyv} \;, \eeqn and we note that Fourier transformation with respect to the variable $x$ yields \eqn \hat W_{\phi_t}(\xi,v)=\overline{\hat\phi_t(v+\frac\xi2)}\hat\phi_t(v-\frac\xi2)\;, \label{FTWx} \eeqn for $\xi\in\Tor^3$. The Wigner transform will serve as our key tool in the derivation of the macroscopic limit for the quantum dynamics described by (~\ref{RSE}). For this purpose, we introduce macroscopic variables $T:=\epsilon t$, $X:= \epsilon x$, $V:=v$, and consider the rescaled Wigner transform \eqn W^\e_{\phi_t}(X,V):=\e^{-3}W_{\phi_t}(\e^{-1}X,V) \eeqn for $T\geq0$, $X\in(\e\Z)^3$, and $V\in\Tor^3$. For a Schwartz class function $J\in \Sc(\R^3\times \Tor^3)$, we write \eqn \langle J,W^\e_{\phi_t}\rangle := \sum_{X\in(\e\Z)^3}\int_{\Tor^3}dV \overline{J(X,V)}W^\e_{\phi_t}(X,V) \; , \label{eqn3-1-0} \eeqn where $J_\e(x,v):=J(\e x,v)$. Let $\hat W^\e_{\phi_t}$ be defined as in (~\ref{FTWx}). Then, \eqn (~\ref{eqn3-1-0})&=& \langle \hat J,\hat W^\e_{\phi_t}\rangle \nonumber\\ &=& \int_{\Tor^3\times\Tor^3 }d\xi dv \overline{\hat J_\e(\xi,v)} \hat W_{\phi_t}(\xi,v) \; , \label{eqn3-1-1} \eeqn where we have defined \eqn \hat J_\e(\xi,v)&:=&\e^{-3}\hat J(\xi/\e,v) \;, \eeqn which is a smooth delta peak of width $\e$ with respect to the $\xi$-variable, and uniformly bounded with respect to $\e$ in the $v$-variable. The macroscopic limit under this scaling is determined by the linear Boltzmann equations, as was proven in \cite{ch} for the 3-dimensional lattice model, and non-Gaussian distributed random potentials (the Gaussian case follows also from \cite{ch}). The corresponding result for the 2- and 3-dimensional continuum model was proven in \cite{erdyau}. \begin{theorem}\label{Boltzlimthm} For $\e>0$, we consider the initial data \eqn \phi_0^\e(x):=\e^{\frac32} h(\e x) e^{\frac{i S(\e x)}{\e}} \;, \label{phi0-def} \eeqn where $h$ and $S$ are Schwartz class functions on $\R^3$. Let $\phi_t^\e$ be the solution of the random Schr\"odinger equation \eqn \label{RSE} i\partial_t\phi_t^\e = H_\omega \phi_t^\e \eeqn with initial condition given by $\phi_0^\e$, and let \eqn W_T^{(\e)}(X,V):= W^{\e}_{\phi_{\e^{-1}T}^{\e} } (X,V) \eeqn denote its corresponding rescaled Wigner transform. If the scaling factor $\e$ is set equal to $\e=\lambda^2$, where $\lambda$ is the coupling constant in (~\ref{Homega-def}), it follows that \eqn \lim_{\lambda\rightarrow0}\Exp\big[ \langle J, W_T^{(\lambda^2)} \rangle\big] =\langle J, F_T \rangle\;, \eeqn where $F_T(X,V)$ solves the linear Boltzmann equation \eqn &&\partial_T F_T(X,V) +\sum_{j=1}^3 (\sin2\pi V_j) \partial_{X_j} F_T(X,V) \nonumber\\ &&\hspace{2cm}= \int_{\Tor^3} dU \sigma(U,V) \lb F_T(X,U) - F_T(X,V)\rb \;. \label{linB} \eeqn The collision kernel is given by $$ \sigma(U,V):=4\pi\delta(e(U)-e(V)) \;. $$ and the initial condition by \eqn\label{initcondweaklim} F_0(X,V) = |h(X)|^2\delta(V-\nabla S(X))\;, \eeqn which is the weak limit of $W_{\phi^\e_0}^\e$ as $\e\rightarrow0$. \end{theorem} The present work aims at an extension of that result by significantly improving the mode of convergence. Our main result is the following theorem. \begin{theorem}\label{mainthm} For any fixed, finite $\p\in2\N$, $T>0$, and for any Schwartz class function $J$, the estimate \eqn \Big(\Exp\Big[\Big|\bra J,W_T^{(\lambda^2)}\ket- \Exp\big[\bra J,W_T^{(\lambda^2)}\ket\big]\Big|^\p\Big] \Big)^{\frac{1}{\p}}\leq c(r,T) \lambda^{\frac{1}{300\p}} \; \eeqn is satisfied for $\lambda$ sufficiently small, and a constant $c(r,T)$ that does not depend on $\lambda$. Consequently, convergence in $\p$-th mean, \eqn \lim_{\lambda\rightarrow0} \Exp\Big[\Big|\bra J,W_T^{(\lambda^2)}\ket- \bra J,F_T\ket\Big|^\p\Big]=0 \;, \label{lpconvmainthm} \eeqn holds for any finite $\p,T\in\R_+$. \end{theorem} We remark that in particular, the variance of $\bra J,W_T^{(\lambda^2)}\ket$ vanishes in the limit $\lambda\rightarrow0$. We shall here not list further standard modes of convergence implied by (~\ref{lpconvmainthm}), which can be found in textbooks on probability theory, but only state the most important implication in the following corollary. \begin{corollary} The rescaled Wigner transform $W_T^{(\lambda^2) }$ convergences in probability weakly to a solution of the linear Boltzmann equations, globally in $T>0$, as $\lambda\rightarrow0$. That is, for any Schwartz class function $J$, and any finite $T>0$, \eqn \lim_{\nu\rightarrow0}{\Bbb P}\Big[\lim_{\lambda\rightarrow0} \Big|\bra J,W_T^{(\lambda^2) }\ket-\bra J,F_T\ket\Big|>\nu\Big] =0 \;, \label{convprobmainthm} \eeqn where $F_T$ solves (~\ref{linB}) with initial condition (~\ref{initcondweaklim}). \end{corollary} \section{Proof of Theorem {~\ref{mainthm}}} To begin with, we expand $\phi_t$ into the truncated Duhamel series \eqn \phi_t=\sum_{n=0}^N \phi_{n,t}+R_{N,t} \;, \eeqn where \eqn \phi_{n,t}&:=&(-i\lambda)^n \int_{\R_+^{n+1}} ds_0\cdots ds_n \delta(\sum_{j=0}^n s_j-t) \nonumber\\ &&\hspace{1cm}\times\,e^{i s_0 \Delta}V_\omega e^{is_1\Delta} \cdots V_\omega e^{is_n\Delta}\phi_0 \eeqn denotes the $n$-th Duhamel term, and where \eqn R_{N,t}=-i\lambda\int_0^t ds e^{-i(t-s)H_\omega}V_\omega \phi_{N,s} \; \eeqn is the remainder term. The number $N$ remains to be optimized. In frequency space, \eqn \hat \phi_{n,t}(k_0)&=&(-i\lambda)^n\int ds_0\cdots ds_n \delta(\sum_{j=0}^n s_j-t) \nonumber\\ &&\times\,\int_{(\Tor^3)^n}dk_1\cdots dk_n e^{-is_0 e(k_0)} \hat V_\omega(k_1-k_0)e^{-is_1 e(k_1)} \cdots \nonumber\\ &&\hspace{2.5cm} \cdots\hat V_\omega(k_{n}-k_{n-1}) e^{-is_n e(k_n)}\hat \phi_0(k_n) \;. \eeqn Representing the delta distribution by an oscillatory integral, we find \eqn \hat \phi_{n,t}(k_0)&=&(-i\lambda)^n e^{\e t}\int_{\R} d\alpha e^{-it\alpha} \nonumber\\ &&\times\,\int_{(\Tor^3)^n}dk_1\cdots dk_n \frac{1}{e(k_0)-\alpha-i\e}\hat V_\omega(k_1-k_0) \nonumber\\ &&\hspace{2 cm} \cdots \hat V_\omega(k_{n}-k_{n-1})\frac{1}{e(k_n)-\alpha-i\e} \hat \phi_0(k_n) \;. \label{hatphint-expans} \eeqn The function $\frac{1}{e(k)-\alpha-i\e}$ is referred to as a {\em particle propagator}, corresponding to the frequency space representation of the resolvent $\frac{1}{-\Delta-\alpha-i\e}$. Likewise, we note that (~\ref{hatphint-expans}) is equivalent to the $n$-th term in the resolvent expansion of \eqn \phi_t=\frac{1}{2\pi i}\int_{-i\e+\R} dz e^{-itz}\frac{1}{H_\omega-z}\phi_0 \;. \eeqn By analyticity of the integrand in (~\ref{hatphint-expans}) with respect to the variable $\alpha$, and due to its decay properties as $Im(z)\rightarrow-\infty$, the path of the $\alpha$-integration can, for any fixed $n\in\N$, be deformed away from $\R$ into the closed contour \eqn \Ie=I_0\cup I_1 \label{defIloop} \eeqn with \eqnn I_0 &:=& [-1, 13]\\ I_1&:=& ([-1, 13]-i)\cup (-1-i(0,1]) \cup (13-i(0,1]) \;, \eeqnn which encloses ${\rm spec}\big(- \Delta -i\e\big) = [0,12]-i\e $. The initial condition $\phi_0\equiv\phi_0^\e$ in the random Schr\"odinger equation (~\ref{RSE}), as characterized in (~\ref{phi0-def}), satisfies \eqn |\hat\phi_0(k)|=\sqrt{\delta_\e(q-k)}\;, \eeqn for some $q\in\Tor^3$, where $\delta_\e$ is a smooth bump function localized in a ball of radius $\e$, normalized by $\int_{\Tor^3} dk \, \delta_\e(k)=1$. In particular, \eqn \|f\hat\phi_0\|_{L^1(\Tor^3)}&<&c\e^{\frac32}\| f\|_{L^\infty(\Tor^3)}\;, \nonumber\\ \|\hat\phi_0\|_{L^2(\Tor^3)}&=&1\;, \label{initcond-ass} \eeqn for any $f\in L^\infty(\Tor^3)$. Applying the partial time integration method introduced in \cite{erdyau}, we choose $\kappa\in\N$ with $1\ll\kappa\ll\e^{-1}$, and subdivide $[0,t]$ into $\kappa$ subintervals bounded by the equidistant points $\theta_j=\frac{jt}{\kappa}$, where $j=1,\dots,\kappa$. Then, \eqn R_{N,t}=-i\lambda\sum_{j=0}^{\kappa-1}e^{-i(t-\theta_{j+1})H_\omega} \int_{\theta_j}^{\theta_{j+1}} ds \, e^{-isH_\omega}V_\omega \phi_{N,s} \;. \eeqn Let $\phi_{n,N,\theta}(s)$ denote the $n$-th Duhamel term conditioned on the requirement that the first $N$ collisions occur in the time interval $[0,\theta]$, and all remaining $n-N$ collisions take place in the time interval $(\theta,s]$. Further expanding $e^{-isH_\omega}$ into a truncated Duhamel series with $3N$ terms, we find \eqn R_{N,t}=R_{N,t}^{(<4N)}+R_{N,t}^{(4N)}\;, \eeqn where \eqn R_{N,t}^{(<4N)}&=&-i\lambda\sum_{n=N+1}^{4N-1}\tilde\phi_{n,N,t} \;, \\ \tilde\phi_{n,N,t}&:=&-i\lambda \sum_{j=1}^{\kappa} e^{-i(t-\theta_j)H_\omega}V_\omega\phi_{n,N,\theta_{j-1}}(\theta_{j}) \eeqn and \eqn R_{N,t}^{(4N)}=-i\lambda \sum_{j=1}^{\kappa}e^{-i(t-\theta_j)H_\omega} \int_{\theta_{j-1}}^{\theta_j}ds \; e^{-i(\theta_j-s)H_\omega} V_\omega \phi_{4N,N,\theta_{j-1}}(s) \;. \eeqn Clearly, the Schwarz inequality implies that \eqn \|R_{N,t}^{(<4N)}\|_2 \leq (3N\kappa) \sup_{N0$. This in turn implies that (~\ref{mainvarest-1}) holds for any fixed, finite $\p\in\R_+$, globally in $T$. This proves Theorem {~\ref{mainthm}}. \section{Graph expansions and main technical lemmata} The key technical lemmata required to establish (~\ref{mainvarest}) are formulated this section. The method of proof is based on graph expansions and estimation of singular integrals in momentum space, generalizing the analysis in \cite{erdyau} and \cite{ch} to cover the case of convergence in $\p$-th mean. \begin{lemma} \label{mainlm1} For any fixed $\p\geq2$ with $\p \in2\N$, and $\bar n:= n_1+n_2$, where $n_1,n_2\leq N$, \eqn \Big(\Exptc \Big[\big|\langle \hat J_\e,\hat W_{t;n_1,n_2}\rangle \big|^\p\Big]\Big)^{\frac1\p}\leq \e ( (\frac{\bar n \p}{2} ) !)^{\frac{1}{\p}} (\log\frac1\e)^{3} (c\lambda^2\e^{-1}\log\frac1\e)^{\frac{\bar n}{2}} \;. \label{keyest-1} \eeqn Furthermore, for any fixed $\p\geq2$, $\p\in2\N$, and $n\leq N$, \eqn \Big(\Exp\Big[ \|\phi_{n,t}\|_2^{2\p} \Big]\Big)^{\frac1\p}\leq ((n\p)!)^{\frac{1}{\p}}(\log\frac1\e)^{3} (c\lambda^2\e^{-1}\log\frac1\e)^{n} \;. \label{aprioribd-1} \eeqn \end{lemma} The estimate (~\ref{keyest-1}) is the key ingredient in our analysis. The central insight is that for every $\p\geq 2$, the expectation over {\em 2-connected graphs} (cf. Definition {~\ref{Expnd-def}} below) is a factor $\e^\gamma$ smaller than the a priori bound (~\ref{aprioribd-1}), for some $\gamma>0$. \begin{lemma} \label{mainlm2} For any fixed $\p\geq2$, $\p\in2\N$, and $Nn+1}\frac{1}{e(p_{\ell'}^{(j)})-\beta_j+i\e_j}\;. \eeqn Clearly, it follows that \eqn |\amp_{\hat J_\e}(\pi)|&\leq& \lambda^{2s\bar n} e^{2s\e t}\Big(\int_{\Tor^3} d\xi \sup_{p\in\Tor^3} |\hat J_\e(\xi,p)|\Big)^s \cdot (I)\cdot(II) \;, \eeqn where \eqn \int_{\Tor^3} d\xi \sup_{p\in\Tor^3}|\hat J_\e(\xi,p)|0$. \endprf \section{Proof of Lemma {~\ref{mainlm2}}} Based on the previous discussion, is straightforward to see that \eqn &&\Exp\big[\|\phi_{n,N,\theta_{j-1}}(\theta_j)\|_2^{2r}\big] \nonumber\\ &&\hspace{1cm} =e^{2r\e \theta_j}\int_{(\Ie\times \bar \Ie)^r} \prod_{j=1}^r d\alpha_jd\beta_j \,e^{-i\theta_j\sum_{j=1}^r(\alpha_j-\beta_j)} \nonumber\\ &&\hspace{1.5cm}\times\, \int_{(\Tor^3)^{(\bar n+2)r}}\prod_{j=1}^r d\up^{(j)} \delta(p_n^{(j)}-p_{n+1}^{(j)}) \Exp\Big[\prod_{j=1}^r U^{(j)}[\up^{(j)}]\Big]\, \nonumber\\ &&\hspace{1.5cm}\times \, \prod_{j=1}^r K^{(j)}_{n,N,\theta_{j-1},\kappa} [\up^{(j)},\alpha_j,\beta_j,\e ] \hat\phi_0(p_0^{(j)})\overline{\hat\phi_0(p_{\bar n+1}^{(j)})} \;. \label{ExpphinN-1} \eeqn The key differences between this expression and the integrals (~\ref{End-1}) considered above, are that now, we only study the special case $n=n_1=n_2$, that we replace the distinguished vertex $\hat J_\e(\xi,v)$ by $\delta(\xi)$, and that we are now considering the full instead of the non-disconnected expectation. We will refer to $\delta(\xi)$ as the "{\em $L^2$-delta}", since it is responsible for the $L^2$-inner product on the left hand side of (~\ref{ExpphinN-1}). Furthermore, \eqn &&K^{(j)}_{n,N,\theta_{j-1},\kappa} [\up^{(j)},\alpha_j,\beta_j,\e ] \nonumber\\ &&\hspace{1cm}:= \prod_{\ell_1=0}^{n-N} \frac{1}{e(p_{\ell_1}^{(j)})-\alpha_j-i\kappa\e} \prod_{\ell_2=n-N+1}^{n} \frac{1}{e(p_{\ell_2}^{(j)})-\alpha_j-\frac{i}{\theta_{j-1}}} \label{Kj-def-1}\\ &&\hspace{1.5cm}\times\, \prod_{\ell_3=n+1}^{n+N} \frac{1}{e(p_{\ell_3}^{(j)})-\beta_j+\frac{i}{\theta_{j-1}}} \prod_{\ell_4=n+N+1}^{2n+1} \frac{1}{e(p_{\ell_4}^{(j)})-\beta_j+i\kappa\e} \;, \nonumber \eeqn cf. the discussion following (~\ref{phinNtheta-2}). The expectation in (~\ref{ExpphinN-1}) again decomposes into a sum of products of delta distributions, and the corresponding contributions to (~\ref{ExpphinN-1}) can be represented by Feynman graphs. Referring to the notational conventions introduced after (~\ref{End-1}), we have $\bar n=2n$. Correspondingly, let $\Pi_{r;\bar n,n}$ denote the set of graphs on $r$ particle lines, each containing $\bar n$ vertices from copies of the random potential $\hat V_\omega$, and with the $L^2$-delta located between the $n$-th and the $n+1$-st $\hat V_\omega$-vertex. For $\pi\in\Pi_{r;n,\bar n}$, let $\amp_\delta(\pi)$ denote the amplitude corresponding to the graph $\pi$, given by the integral obtained from replacing $\Exp\big[\prod_{j=1}^r U^{(j)}[\up^{(j)}]\big]$ in (~\ref{ExpphinN-1}) by $\delta_\pi(\up^{(1)},\dots,\up^{(r)})$ (the product of delta distributions corresponding to the contraction graph $\pi$). The subscript in $\amp_\delta$ implies that instead of $\hat J_\e$ as before, we now have the $L^2$-delta at the distinguished vertex. Let $\Pi_{r;\bar n,n}^{conn}$ denote the subclass of $\Pi_{r;\bar n,n}$ comprising completely connected graphs. Then, the following estimate holds. \begin{lemma} \label{conngrbd-2} Let $s\geq1$, $s\in\N$, and let $\pi\in\Pi_{s;2n,n}^{conn}$ (that is, $\bar n=2n$) be a completely connected graph. Then, \eqn |\amp_\delta(\pi)|\leq \e^{2(s-1)}(\log\frac1\e)^{s+2}(c\lambda^2\e^{-1} \log\frac1\e)^{ sn} \;. \eeqn \end{lemma} \prf The proof is completely analogous to the one given for Lemma {~\ref{conngrbd-1}} (using $\theta_{j-1},\frac{1}{\kappa\e}\leq\frac1\e$), and shall not be reiterated here. All steps taken there can be adapted to the proof of Lemma {~\ref{conngrbd-2}} with minor modifications. \endprf In contrast to the situation in the context of Lemma {~\ref{conngrbd-1}}, the expectation in (~\ref{ExpphinN-1}) does not exclude completely disconnected graphs. We recall that the sum over all amplitudes of completely disconnected graphs on $r$ particle lines can be estimated by \eqn \sum_{\pi\in\Pi_{r;\bar n,n}^{disc}}|\amp_\delta(\pi)| \leq\Big(\sum_{\pi\in\Pi_{1;\bar n,n}^{conn}}|\amp_\delta(\pi)|\Big)^r\;. \eeqn The bound for $s=1$ in Lemma {~\ref{conngrbd-2}}, however, is not good enough, since there is no factor $\e^\gamma$ for any $\gamma>0$. Using Lemma {~\ref{conngrbd-2}} for $s=1$, one will not be able to compensate the large number $\sim 2^{nr}(n!)^r$ of disconnected graphs. We shall hence recall another result from \cite{ch} (the continuum version is proved in \cite{erdyau}), formulated in the following lemma. \begin{lemma} Let $\bar n=2n$. Then, \eqn \sum_{\pi\in\Pi_{1;\bar n,n}^{conn}}|\amp_\delta(\pi)|\leq \frac{(c\lambda^2\e^{-1})^n}{\sqrt{n!}} + (n!) \e^{\frac16}(\log\frac1\e)^{3}(c\lambda^2\e^{-1} \log\frac1\e)^{n} \;. \label{ch-mainbd-1} \eeqn \end{lemma} The term $\frac{(c\lambda^2\e^{-1})^n}{\sqrt{n!}}$ bounds the so-called {\em ladder contribution}, while the last term carries an additional $\e^{\frac16}$-factor relative to the bound provided by Lemma {~\ref{conngrbd-1}} for $s=1$. It is obtained from {\em crossing} and {\em nesting type subgraphs} that appear in all non-ladder graphs. This issue is discussed in much detail in \cite{ch}, and will not be recapitulated here. Since the number of non-ladder graphs is bounded by $n!2^n$, there is a factor $n!$. The sum over non-disconnected graphs can be estimated by the same bound as in Lemma {~\ref{Nondiscsum-est1}}. The result is formulated in the following lemma. \begin{lemma} \label{Nondiscsum-est2} Let $r\in2\N$, $\bar n=2n$, and $\Pi_{r;\bar n,n}^{n-d}\subset \Pi_{r;\bar n,n}$ denote the subclass of non-disconnected graphs. Then, \eqn \sum_{\pi\in\Pi_{r;\bar n,n}^{n-d}}|\amp_{\delta}(\pi)|\leq (r\bar n)!\e^{2}(\log\frac1\e)^{3r-2} (c\lambda^2\e^{-1} \log\frac1\e)^{\frac{r\bar n}{2}} \;. \eeqn \end{lemma} Combining (~\ref{ch-mainbd-1}) with Lemma {~\ref{Nondiscsum-est2}}, and applying the Minkowski inequality, the statement of Lemma {~\ref{mainlm2}} follows straightforwardly. \section{Proof of Lemma {~\ref{mainlm3}}} We have, for $r\in2\N$, $s\in[\theta_{j-1},\theta_j]$, $n=4N$, and $\bar n=8N$, \eqn &&\Exp\big[\|\phi_{4N,N,\theta_{j-1}}(s)\|_2^{2r}\big] \nonumber\\ &&\hspace{1cm} =e^{2r\e s}\int_{(\Ie\times \bar \Ie)^r} \prod_{j=1}^r d\alpha_jd\beta_j \,e^{-is\sum_{j=1}^r(\alpha_j-\beta_j)} \nonumber\\ &&\hspace{1.5cm}\times\, \int_{(\Tor^3)^{(\bar n+2)r}}\prod_{j=1}^r d\up^{(j)} \delta(p_{4N}^{(j)}-p_{4N+1}^{(j)}) \Exp\Big[\prod_{j=1}^r U^{(j)}[\up^{(j)}]\Big]\, \nonumber\\ &&\hspace{1.5cm}\times \, \prod_{j=1}^r K^{(j)}_{4N,N,\theta_{j-1},\kappa} [\up^{(j)},\alpha_j,\beta_j,\e ] \hat\phi_0(p_0^{(j)})\overline{\hat\phi_0(p_{8N+1}^{(j)})} \;, \label{ExpphinN-2} \eeqn where again, $\e=\frac1t$. All notations are the same as in the proof of Lemma {~\ref{mainlm2}}, in particlar, cf. (~\ref{Kj-def-1}) for the definition of $K^{(j)}_{4N,N,\theta_{j-1},\kappa}$. Let $\Pi_{s;8N,4N}^{conn}$ denote the subset of $\Pi_{s;8N,4N}$ of completely connected graphs. \begin{lemma} \label{conngrbd-3} Let $s\geq1$, $s\in\N$, and let $\pi\in\Pi_{s;8N,4N}^{conn}$ be a completely connected graph. Then, \eqn |\amp_\delta(\pi)|\leq \kappa^{-2rN} \e^{2(s-1)}(\log\frac1\e)^{s+2}(c\lambda^2\e^{-1} \log\frac1\e)^{\frac{s\bar n}{2}} \;. \eeqn \end{lemma} \prf We adapt the proof given for Lemma {~\ref{conngrbd-1}} in the following manner. First of all, we observe that (~\ref{ExpphinN-1}) contains $r(6N+2)$ propagators with imaginary part $\pm i\kappa\e$, and $2rN$ propagators with imaginary part $\pm \frac{i}{\theta_{j-1}}$. In the proof of Lemma {~\ref{conngrbd-1}}, $4rN$ out of all propagators were estimated in $L^\infty$-norm, while the remaining ones were estimated in $L^1$. Carrying out the same arguments line by line, we shall also estimate $4rN$ out of all propagators in (~\ref{ExpphinN-2}) in $L^\infty$. Since there are in total only $2rN$ propagators whose denominators carry an imaginary part $\pm\frac{i}{\theta_{j-1}}$, it follows that at least $2rN$ propagators bounded in $L^\infty$ exhibit a denominator with an imaginary part $\pm i\kappa\e$. Correspondingly, one obtains an improvement of the upper bound by a factor $\kappa^{-1}$ for each of the latter, in comparison to the $L^\infty$-bound of $\frac1\e$ used throughout the proof of Lemma {~\ref{conngrbd-1}}. Hence, there is in total a gain of a factor of at least $\kappa^{-2rN}$ above the estimate derived for Lemma {~\ref{conngrbd-1}}. \endprf \begin{lemma} \label{Nondiscsum-est3} Let $r\in2\N$. Then, \eqn \sum_{\pi\in\Pi_{r;8N,4N}}|\amp_{\delta}(\pi)|\leq (4rN)!\kappa^{-2rN}(\log\frac1\e)^{3r} (c\lambda^2\e^{-1} \log\frac1\e)^{4rN} \;. \eeqn \end{lemma} \prf Given a fixed $\pi\in\Pi_{r;8N,4N}$ with $m$ connectivity components, let us assume that $\pi$ comprises $s_1,\dots,s_m$ particle lines, where $\sum_{l=1}^m s_l=r$. Then, \eqn |\amp_{\delta}(\pi)|&\leq&\kappa^{-2N\sum_{l=1}^m s_l} \e^{2\sum_{l=1}^m (s_l-1)} \nonumber\\ &&\hspace{1cm}\times\, (\log\frac1\e)^{\sum_{l=1}^m (s_l+2)} (c\e^{-1}\lambda^2\log\frac1\e)^{\frac{\bar n\sum_{l=1}^m s_l}{2}} \nonumber\\ &\leq&\kappa^{-2rN} (\log\frac1\e)^{3r} (c\e^{-1}\lambda^2\log\frac1\e)^{4rN}\;. \eeqn Furthermore, $\Pi_{r;8N,4N}$ contains no more than $(4rN)!2^{4rN}$ elements. \endprf The corresponding sum over disconnected graphs can be estimated by the same bound as in Lemma {~\ref{Nondiscsum-est2}}. This proves Lemma {~\ref{mainlm3}}. \subsection*{Acknowledgements} The author is deeply grateful to H.-T. Yau and L. Erd\"os for their support, encouragement, advice, and generosity. He has benefitted immensely from numerous discussions with them, in later stages of this work especially from conversations with L. Erd\"os. He also thanks H.-T. Yau for his very generous hospitality during two visits at Stanford University. This work was supported by a Courant Instructorship, in part by a grant of the NYU Research Challenge Fund Program, and in part by NSF grant DMS-0407644. \begin{thebibliography}{99} \bibitem{aimo} Aizenman, M., Molchanov, S., {\em Localization at large disorder and at extreme energies: an elementary derivation}, Commun. Math. Phys. {\bf 157}, 245--278 (1993) \bibitem{ch} Chen, T., {\em Localization Lengths and Boltzmann Limit for the Anderson Model at Small Disorders in Dimension 3}, Submitted. \bibitem{erd} Erd\"os, L., {\em Linear Boltzmann equation as the scaling limit of the Schr\"odinger evolution coupled to a phonon bath}, J. Stat. Phys. 107(5), 1043-1127 (2002). \bibitem{erdyau} Erd\"os, L., Yau, H.-T., {\em Linear Boltzmann equation as the weak coupling limit of a random Schr\"odinger equation}, Comm. Pure Appl. Math., Vol. LIII, 667 - 753, (2000). \bibitem{erdsalmyau} Erd\"os, L., Salmhofer, M., Yau, H.-T., {\em Quantum diffusion of random Schr\"odinger evolution in the scaling limit}, announced. \bibitem{frsp} Fr\"ohlich, J., Spencer, T., {\em Absence of diffusion in the Anderson tight binding model for large disorder or low energy}, Commun. Math. Phys. {\bf 88}, 151--184 (1983) \bibitem{shscwo} Shubin, C., Schlag, W., Wolff, T., {\em Frequency concentration and localization lengths for the Anderson model at small disorders}, J. Anal. Math., 88 (2002). \bibitem{sp} Spohn, H., {\em Derivation of the transport equation for electrons moving through random impurities}, J. Statist. Phys., 17, no. 6, 385-412 (1977). \end{thebibliography} \end{document} ---------------0408151153692 Content-Type: application/postscript; name="lpc-fig1.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="lpc-fig1.eps" %!PS-Adobe-3.0 EPSF-3.0 %%HiResBoundingBox: 0.000000 0.000000 404.000000 210.000000 %APL_DSC_Encoding: UTF8 %%Title: (Unknown) %%Creator: (Unknown) %%CreationDate: (Unknown) %%For: (Unknown) %%DocumentData: Clean7Bit %%LanguageLevel: 2 %%Pages: 1 %%BoundingBox: 0 0 404 210 %%EndComments %%BeginProlog %%BeginFile: cg-pdf.ps %%Copyright: Copyright 2000-2002 Apple Computer Incorporated. %%Copyright: All Rights Reserved. currentpacking true setpacking /cg_md 140 dict def cg_md begin /L3? languagelevel 3 ge def /bd{bind def}bind def /ld{load def}bd /xs{exch store}bd /xd{exch def}bd /cmmtx matrix def mark /sc/setcolor /scs/setcolorspace /dr/defineresource /fr/findresource /T/true /F/false /d/setdash /w/setlinewidth /J/setlinecap /j/setlinejoin 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}ifelse store /srcDecode currentdict/Decode known {Decode}{Range}ifelse store /nRange Range length 2 idiv store end }bd /FunEvalN { begin nDomM1 -1 0 { 2 mul/mIndex xs Domain mIndex get max Domain mIndex 1 add get min Domain mIndex get Domain mIndex 1 add get srcEncode mIndex get srcEncode mIndex 1 add get inter round cvi 0 max sizem1 mIndex 2 idiv get min nDomM1 1 add 1 roll }for nDomM1 1 add array astore/val xs nDomM1 0 gt { 0 nDomM1 -1 0 { dup 0 gt { /mIndex xs val mIndex get 1 index add Size mIndex 1 sub get mul add }{ val exch get add }ifelse }for }{ val 0 get }ifelse nRange mul /ival xs 0 1 nRange 1 sub { dup 2 mul/mIndex xs ival add DataSource exch get 0 255 srcDecode mIndex 2 copy get 3 1 roll 1 add get inter Range mIndex get max Range mIndex 1 add get min }for end }bd /sh2 { /Coords load aload pop 3 index 3 index translate 3 -1 roll sub 3 1 roll exch sub 2 copy dup mul exch dup mul add sqrt dup scale atan rotate /Function load setupFunEvalN clippath {pathbbox}stopped {0 0 0 0}if newpath /ymax xs /xmax xs /ymin xs /xmin xs currentdict/Extend known { /Extend load 0 get { /Domain load 0 get /Function load FunEvalN sc xmin ymin xmin abs ymax ymin sub rectfill }if }if /dx/Function load/Size get 0 get 1 sub 1 exch div store gsave /di ymax ymin sub store /Function load dup /Domain get dup 0 get exch 1 get 2 copy exch sub dx mul exch { 1 index FunEvalN sc 0 ymin dx di rectfill dx 0 translate }for pop grestore currentdict/Extend known { /Extend load 1 get { /Domain load 1 get /Function load FunEvalN sc 1 ymin xmax 1 sub abs ymax ymin sub rectfill }if }if }bd /shp { 4 copy dup 0 gt{ 0 exch a1 a0 arc }{ pop 0 moveto }ifelse dup 0 gt{ 0 exch a0 a1 arcn }{ pop 0 lineto }ifelse fill dup 0 gt{ 0 exch a0 a1 arc }{ pop 0 moveto }ifelse dup 0 gt{ 0 exch a1 a0 arcn }{ pop 0 lineto }ifelse fill }bd /calcmaxs { xmin dup mul ymin dup mul add sqrt xmax dup mul ymin dup mul add sqrt xmin dup mul ymax dup mul add sqrt xmax dup mul ymax dup mul add sqrt max max max }bd /sh3 { /Coords load aload pop 5 index 5 index translate 3 -1 roll 6 -1 roll sub 3 -1 roll 5 -1 roll sub 2 copy dup mul exch dup mul add sqrt /dx xs 2 copy 0 ne exch 0 ne or { exch atan rotate }{ pop pop }ifelse /r2 xs /r1 xs /Function load dup/Size get 0 get 1 sub /Nsteps xs setupFunEvalN dx r2 add r1 lt{ 0 }{ dx r1 add r2 le { 1 }{ r1 r2 eq { 2 }{ 3 }ifelse }ifelse }ifelse /sh3tp xs clippath {pathbbox}stopped {0 0 0 0}if newpath /ymax xs /xmax xs /ymin xs /xmin xs dx dup mul r2 r1 sub dup mul sub dup 0 gt { sqrt r2 r1 sub atan /a0 exch 180 exch sub store /a1 a0 neg store }{ pop /a0 0 store /a1 360 store }ifelse currentdict/Extend known { /Extend load 0 get r1 0 gt and { /Domain load 0 get/Function load FunEvalN sc { { dx 0 r1 360 0 arcn xmin ymin moveto xmax ymin lineto xmax ymax lineto xmin ymax lineto xmin ymin lineto eofill } { r1 0 gt{0 0 r1 0 360 arc fill}if } { 0 r1 xmin abs r1 add neg r1 shp } { r2 r1 gt{ 0 r1 r1 neg r2 r1 sub div dx mul 0 shp }{ 0 r1 calcmaxs dup r2 add dx mul dx r1 r2 sub sub div neg exch 1 index abs exch sub shp }ifelse } }sh3tp get exec }if }if /d0 0 store /r0 r1 store /di dx Nsteps div store /ri r2 r1 sub Nsteps div store /Function load /Domain load dup 0 get exch 1 get 2 copy exch sub Nsteps div exch { 1 index FunEvalN sc d0 di add r0 ri add d0 r0 shp { d0 0 r0 a1 a0 arc d0 di add 0 r0 ri add a0 a1 arcn fill d0 0 r0 a0 a1 arc d0 di add 0 r0 ri add a1 a0 arcn fill }pop /d0 d0 di add store /r0 r0 ri add store }for pop currentdict/Extend known { /Extend load 1 get r2 0 gt and { /Domain load 1 get/Function load FunEvalN sc { { dx 0 r2 0 360 arc fill } { dx 0 r2 360 0 arcn xmin ymin moveto xmax ymin lineto xmax ymax lineto xmin ymax lineto xmin ymin lineto eofill } { xmax abs r1 add r1 dx r1 shp } { r2 r1 gt{ calcmaxs dup r1 add dx mul dx r2 r1 sub sub div exch 1 index exch sub dx r2 shp }{ r1 neg r2 r1 sub div dx mul 0 dx r2 shp }ifelse } } sh3tp get exec }if }if }bd /sh { begin /ShadingType load dup dup 2 eq exch 3 eq or { gsave newpath 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