Content-Type: multipart/mixed; boundary="-------------0001310906277" This is a multi-part message in MIME format. ---------------0001310906277 Content-Type: text/plain; name="00-47.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-47.keywords" semiclassical limit, periodic potential ---------------0001310906277 Content-Type: application/x-tex; name="SLPP.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="SLPP.tex" \documentclass{amsart} \usepackage{amssymb} \def\R{\mathbb{R}} \def\C{\mathbb{C}} \def\epsi{\varepsilon} \newtheorem{The}{Theorem}[section] \newtheorem{Def}[The]{Definition} \newtheorem{Cor}[The]{Corollary} \newtheorem{Lem}[The]{Lemma} \theoremstyle{plain} \newtheorem*{Con}{Condition}\newtheorem*{Rem}{Remark} \numberwithin{equation}{section} \begin{document} \title[Semiclassical Limit with a Short Scale Periodic Potential]{Semiclassical Limit for the Schr\"{o}dinger Equation with a Short Scale Periodic Potential} \author{Frank H\"{o}vermann, Herbert Spohn, Stefan Teufel} \address{Zentrum Mathematik\\ Technische Universit\"{a}t M\"{u}n\-chen\\ D-80290 M\"{u}n\-chen\\ Germany} \email{spohn@ma.tum.de, teufel@ma.tum.de} \begin{abstract} We consider the dynamics generated by the Schr\"{o}ding\-er operator $H=-\frac{1}{2}\Delta+V(x)+W(\epsi x)$, where $V$ is a lattice periodic potential and $W$ an external potential which varies slowly on the scale set by the lattice spacing. We prove that in the limit $\epsi\to 0$ the time dependent position operator and, more generally, semiclassical observables converge strongly to a limit which is determined by the semiclassical dynamics. \end{abstract} \maketitle \section{Introduction} A basic problem of solid state physics is to understand the motion of electrons in the periodic potential which is generated by the ionic cores. While this problem is quantum mechanical, many electronic properties of solids can be understood already in the semiclassical approximation \cite{ashcroft-mermin,kohn,zak}. One argues that if the wave packet spreads over many lattice spacings, the kinetic energy $(\hbar k)^2/2m$ is modified to the $n$-th band energy $E_n(k)$. Otherwise the electron responds to external fields, $E_\mathrm{ex}$, $B_\mathrm{ex}$, as in the case of vanishing periodic potential. Thus the semiclassical equations of motion are $$\label{semiclassical_dynamics} \begin{array}{l} \displaystyle{ \dot{r}=v_n(k)=\nabla_k E_n(k)}\\ \\\displaystyle{ \hbar\dot{k}=e(E_{\mathrm{ex}}(r)+v_n(k)\wedge B_{\mathrm{ex}}(r))\,,} \end{array}$$ where $r$ is the position and $k$ the quasimomentum of the electron. Note that there is a semiclassical evolution for each band separately. The goal of our paper is to understand on a mathematical level how these semiclassical equations arise from the underlying Schr\"{o}dinger equation. We consider only the case where $B_{\mathrm{ex}}=0$. The setup is rather obvious. We start from the Schr\"{o}dinger equation $$\label{basic_dynamics} i\frac{\partial}{\partial t}\psi=H\psi$$ with Hamiltonian $$\label{basic_generator} H=-\frac{1}{2}\Delta+V(x)+W(\epsi x).$$ The electron moves in $\mathbb{R}^d$ and the solution to \eqref{basic_dynamics} defines the unitary time evolution $U^\epsi(t)\psi(x)=\mathrm{e}^{-itH}\psi(x)=\psi(x,t)$ in $L^2(\mathbb{R}^d)$. We have chosen units such that $\hbar=1$ and the mass of the particle $m=1$. $V(x)$ is a periodic potential with average lattice spacing $a$. The precise conditions on $V$ will be spelled out in the following section, where we also describe the direct fiber integral decomposition for periodic Schr\"{o}dinger operators. The lattice spacing $a$ defines the microscopic spatial scale. $W(\epsi x)$ is an external electrostatic potential with dimensionless scale parameter $\epsi$, $\epsi\ll 1$, which means that $W$ is slowly varying on the scale of the lattice. For real metals the condition of slow variation is satisfied even for the strongest external electrostatic fields available, cf.\ \cite{ashcroft-mermin}, Chapter 13. The external forces due to $W$ are of order $\epsi$ and therefore have to act over a time of order $\epsi^{-1}$ to produce finite changes, which defines the macroscopic time scale. We will mostly work in the microscopic coordinates $(x,t)$ of (\ref{basic_dynamics}). For sake of comparison we note that the macroscopic space-time scale $(x',t')$ is defined through $x=\epsi^{-1}x'$ and $t=\epsi^{-1}t'$. With this scale change Eqs.\ \eqref{basic_dynamics}, \eqref{basic_generator} read $$\label{scaled_dynamics} \begin{array}{l} \displaystyle{ i\epsi\frac{\partial}{\partial t^\prime}\psi= H\psi\,,}\\ \\ \displaystyle{ H=\left(-\epsi^2\frac{1}{2}\Delta^\prime +V(x^\prime/\epsi)+W(x^\prime)\right)} \end{array}$$ with initial conditions $\psi^\epsi(x^\prime)=\epsi^{-d/2}\psi(x^\prime/\epsi)$. If $V=0$, Eq.\ \eqref{scaled_dynamics} is the usual semiclassical limit with $\epsi$ set equal to $\hbar$. Thus our problem is to understand how an additional periodic, but rapidly oscillating potential modifies the standard picture. The two scale problem \eqref{basic_dynamics}, \eqref{basic_generator} can be attacked along several routes. A first choice would be time dependent WKB \cite{buslaev,buslaev-grigis,gerard-martinez-sjostrand,guillot-ralston-trubowitz}. In the limit $\epsi\to 0$, for each energy band separately, one obtains a Hamilton-Jacobi equation for the phase and a transport equation for the amplitude of the wave function $\psi(x,t)$. As a main draw-back of this method, generically, the solution to the Hamilton-Jacobi equation develops singularities after some finite macroscopic time. If $V=0$, it is well understood how to go beyond such caustics by introducing new coordinates on the Lagrangian manifold. For \eqref{basic_dynamics}, \eqref{basic_generator} a corresponding program has not yet been attempted. The results \cite{buslaev,buslaev-grigis,gerard-martinez-sjostrand,guillot-ralston-trubowitz} are valid only over a finite macroscopic time span with a duration depending on the initial wave function. Another variant is to establish the semiclassical limit through the convergence of Wigner functions. In our context one defines a band Wigner function $W^\epsi_n(r,k,t)$ depending on the band index $n$ and as a function of the position and quasimomentum. One then wants to prove that in the limit $\epsi\to 0$ $W^\epsi_n(t)$ converges to $\overline{W}_n(t)$, which is the initial band Wigner function $\overline{W}_n(0)$ evolved according to the semiclassical flow \eqref{semiclassical_dynamics}. Such a result is established in \cite{gerard-markowich-mauser-poupaud,markowich-mauser-poupaud} for the case of zero external potential, the general case being left open as a challenging problem. A third approach to the semiclassical limit for $V=0$ is the strong convergence of Heisenberg operators \cite{asch-knauf,avron-seiler-yaffe,nenciu,spohn}. We briefly recall its main features. We define, as unbounded operators on $L^2(\mathbb{R}^d)$, \begin{eqnarray*} &x(t):=\mathrm{e}^{itH}x\mathrm{e}^{-itH},\\ \\ &p(t):=\mathrm{e}^{itH}p\mathrm{e}^{-itH},\quad p=-i\nabla_x, \end{eqnarray*} where $H$ is the Hamiltonian in \eqref{basic_generator} with $V=0$. The goal is to establish the strong limit of $$x^\epsi(t)\psi=\epsi x(\epsi^{-1}t)\psi,\quad p^\epsi(t)\psi=p(\epsi^{-1}t)\psi$$ as $\epsi\to 0$ with $\psi$ in a suitable domain. In the trivial case of free motion, $W=0$, this amounts to the strong convergence of $x^\epsi(t)\psi=(\epsi x+pt)\psi$, $p^\epsi(t)\psi=p\psi$, which yields $\lim_{\epsi\to 0}x^\epsi(t)=pt$, $\lim_{\epsi\to 0}p^\epsi(t)=p$. The general case requires more work \cite{robert}. One obtains the strong limits $$\label{semiclassical_limit} \begin{array}{l} \displaystyle{ \lim_{\epsi\to 0}x^\epsi(t)=r(p,t)\,,}\\ \\ \displaystyle{ \lim_{\epsi\to 0}p^\epsi(t)=u(p,t)\,.} \end{array}$$ Here $r(p,t)$, $u(p,t)$ are solutions of $$\label{semiclassical_flow} \dot{r}=u,\quad \dot{u}=-\nabla W(r)$$ with initial conditions $r_0=0$, $u_0=p$. The initial condition $r_0=0$ reflects that $\vert\psi\vert^2$ looks like $\delta(r)$ on the macroscopic scale, provided that $\Vert\psi\Vert_2=1$. For general initial conditions, $r_0\ne 0$, we would have to shift the initial $\psi$ by $\epsi^{-1}r_0$. The strong operator convergence may look slightly abstract, but all the desired physical information can be deduced. E.g., the initial $\psi$ defines the momentum distribution $\vert\widehat{\psi}(k)\vert^2$ independent of $\epsi$ and the $\delta(r)$ spatial distribution in the limit $\epsi\to 0$. Then, according to \eqref{semiclassical_limit}, for small $\epsi$ the position distribution at time $t$ is given by \begin{eqnarray*} &\int_{\mathbb{R}^d}f(x)\vert\psi^\epsi(x,t)\vert^2\,dx=(\psi,f(x^\epsi(t))\psi)\\ &\simeq(\psi,f(r(p,t))\psi)=\int\vert\widehat{\psi}(k)\vert^2f(r(k,t))\,dk, \end{eqnarray*} which means that the phase space distribution $\delta(r)\vert\widehat{\psi}(k)\vert^2\,dr\,dk$ is transported according to the semiclassical flow \eqref{semiclassical_flow}. The spatial marginal of this distribution at time $t$ is the desired approximation to the true position distribution $\vert\psi^\epsi(x,t)\vert^2$. $\vert\psi^\epsi(x,t)\vert^2$ may oscillate rapidly on small scales and some averaging, as embodied by the test function $f$, is needed. In this paper we investigate the semiclassical limit \eqref{basic_dynamics}, \eqref{basic_generator} through the strong convergence of the position operator $x^\epsi(t)$. We will show that, in the limit $\epsi\to 0$, $x^\epsi(t)$ is diagonal with respect to the band index and in each band the structure is analogous to \eqref{semiclassical_limit} with $p$ replaced by the quasimomentum $k$ and \eqref{semiclassical_flow} replaced by \eqref{semiclassical_dynamics}. More generally we will consider the semiclassical limit of the Weyl quantized operators $a^W(\epsi x,p)$, whose classical symbol is periodic in $p$. To give a short outline: In the following section we collect some properties of periodic Schr\"{o}dinger operators. In Section \ref{sect_main_results} we state our main results, which are proved in Sections \ref{effective}, \ref{sect_asymptotic_position}, and \ref{PsiDO}, respectively. In Section \ref{explain_results} we discuss some implications for the position and quasimomentum distributions, and, more generally, for the band Wigner functions. The difficulties arising from band crossings are explained in Section \ref{outlook}. \section{Periodic Schr\"odinger operators}\label{sec_zwo} For the periodic potential $V$ we will need only some rather minimal assumptions, which we state as \begin{Con}[$\mathrm{C_{per}}$] Let $\Gamma\simeq\mathbb{Z}^d$ be the lattice generated by the basis $\{\gamma_1,\ldots,\gamma_d\}$, $\gamma_i\in\mathbb{R}^d$. Then $V(x+\gamma)=V(x)$ for all $x\in\mathbb{R}^d$, $\gamma\in\Gamma$. Furthermore, we assume $V$ to be infinitesimally operator bounded with respect to $H_0$. The last condition is satisfied, e.g., if $V\in L^p(M)$, where $M$ is the fundamental domain of\, $\Gamma$, and $p=2$ for $d\leq 3$ and $p>d/2$ for $d> 3$, respectively. \end{Con} \noindent ($\mathrm{C_{per}}$) will be assumed throughout. We recall the {\em Bloch-Floquet} theory for the spectral representation of $$\label{periodic_generator} H_\mathrm{per}=\frac{1}{2}p^2+V(x)\,.$$ The reciprocal lattice $\Gamma^*$ is defined as the lattice generated by the dual basis $\{\gamma_1^*,\ldots,\gamma_d^*\}$ determined by $\gamma_i\cdot\gamma_j^*=2\pi\delta_{ij}$, $i,j=1,\ldots,d$. The fundamental domain of $\Gamma$ is denoted by $M$, the one of $\Gamma^*$ by $M^*$. $M^*$ is usually referred to as {\em first Brillouin zone}. If we identify opposite edges of $M$, resp.\ $M^*$, then it becomes a flat $d$-torus denoted by $\mathbb{T}=\mathbb{R}^d/\Gamma$, resp.\ $\mathbb{T}^*=\mathbb{R}^d/\Gamma^*$. Let us introduce the Bloch-Floquet transformation, which should be viewed as a discrete Fourier transform, through $(\mathcal{U}\psi)(k,x):=\sum_{\gamma\in\Gamma} \mathrm{e}^{-i(x+\gamma)\cdot k}\psi(x+\gamma),\quad (k,x)\in\mathbb{R}^{2d}\,,$ for $\psi\in\mathcal{S}(\mathbb{R}^d)$. Clearly, $$\label{BC} \begin{array}{l} \displaystyle{ (\mathcal{U}\psi)(k,x^\prime+\gamma)=(\mathcal{U}\psi)(k,x^\prime)\,,} \\ \\ \displaystyle{ (\mathcal{U}\psi)(k^\prime+\gamma^*,x) =\mathrm{e}^{-ix\cdot \gamma^*}(\mathcal{U}\psi)(k^\prime,x)\,.} \end{array}$$ Therefore it suffices to specify $\mathcal{U}\psi$ on the set $M^*\times M$ and, if needed, extend it to all of $\mathbb{R}^{2d}$ by (\ref{BC}). The linear map $\mathcal{U}:L^2(\mathbb{R}^d)\supset\mathcal{S}(\mathbb{R}^d) \to\mathcal{H}:=\int^\oplus_{M^*}L^2(M)\,dk,$ with $dk$ the normalized Lebesgue measure on $M^*$, has norm one and can thus be extended to all of $L^2(\mathbb{R}^d)$ by continuity. $\mathcal{U}$ is surjective as can be seen from the inverse mapping $$(\mathcal{U}^{-1}\phi)(x):=\int_{M^*}\mathrm{e}^{ix\cdot k}\phi(k,x)\,dk,$$ which has norm one. Thus $\mathcal{U}:L^2(\mathbb{R}^d)\to\mathcal{H}$ is unitary. To transform $H_\mathrm{per}$ under $\mathcal{U}$, we first note that $\tilde p = \mathcal{U}p\mathcal{U}^{-1} = D_x + k$, with $D_x = -i\nabla_x$. Therefore $\tilde{H}_{\mathrm{per}}:=\mathcal{U}H_{\mathrm{per}}\mathcal{U}^{-1}= \int_{M^*}^\oplus H_{\mathrm{per}}(k)\,dk\,,$ and $H_\mathrm{per}(k)=\frac{1}{2}(D_x+k)^2+V(x),\quad k\in\mathbb{R}^d\,.$ $H_\mathrm{per}(k)$ acts on $L^2(M)$ with $k$-independent domain $D:=H^2(\mathbb{T})$. $\psi\in D$ is periodic in $x$. $H_\mathrm{per}(k)$ is a semi-bounded self-adjoint operator, since by condition ($\mathrm{C_{per}}$) $V$ is infinitesimally operator bounded with respect to $-\Delta$ \cite{cycon-froese-kirsch-simon}. In particular, $H_{\mathrm{per}}(k)$ is an entire analytic family of type (B) in the sense of Kato for $k\in\mathbb{C}^d$. Since the resolvent of $H_0(k)=\frac{1}{2}(D_x+k)^2$ is compact, the resolvent $R_\lambda(H_{\mathrm{per}}(k)):=(H_{\mathrm{per}}(k)-\lambda)^{-1}$, $\lambda\ne\sigma(H_{\mathrm{per}}(k))$, is also compact, and $H_{\mathrm{per}}(k)$ has a complete set of (normalized) eigenfunctions $\varphi_n(k)\in H^2(\mathbb{T})$, $n\in\mathbb{N}$, called {\em Bloch functions}. The corresponding eigenvalues $E_n(k)$, $n\in\mathbb{N}$, accumulate at infinity and we enumerate them according to their magnitude and multiplicity, $E_1(k)\leq E_2(k)\leq\ldots$\ . $E_n(k)$ is called the $n$-th {\em band function}. We note that $H_\mathrm{per}(k)=\mathrm{e}^{-ix\cdot\gamma^*} H_\mathrm{per}(k+\gamma^*)\mathrm{e}^{ix\cdot\gamma^*}$. Therefore $E_n(k)$ is periodic with respect to $\Gamma^*$. If $E_{n-1}(k)0$ be arbitrary, $B_R=\{p\mid\vert p\vert\leq R\}$. We start with \begin{eqnarray*} \lefteqn{\int\limits_{M^*} \left\| \,\int\limits_{B_R}\widehat{W}^\epsi(p) \frac{\vert p\vert}{\epsi}\Bigg(\frac{P_n(k-p)-P_n(k)}{\vert p\vert}\right.} \\ & & \left. \hspace{4cm}+\, e_p\cdot\nabla_k P_n(k)\Bigg)\psi(k-p)\,dp \right\|_{L^2(M)}\,dk \\ &\leq& \sup_{k\in M^*}\sup_{p\in B_R}\left\|| p|^{-1}\left(P_n(k-p)-P_n(k)\right) +\,\,e_p\cdot\nabla_k P_n(k)\right\|\\ & &\hspace{2cm}\times\,\int_{B_R}\left|\widehat{W}^\epsi(p)\right|\frac{| p|}{\epsi} \int_{M^*}\left\|\psi(k-p)\right\|_{L^2(M)}\,dk\,dp\\ &\leq & \left\| \psi\right\|_\mathcal{H} \big\|\widehat{F}^\epsi\big\|_{L^1} \\ &&\hspace{1cm}\times \, \sup_{k\in M^*,\,p\in B_R}\left\|| p|^{-1}(P_n(k-p)-P_n(k))+e_p\cdot\nabla_k P_n(k)\right\|\,. \end{eqnarray*} Since $\Vert\widehat{F}^\epsi\Vert_{L^1}$ does not depend on $\epsi$ and since the difference quotient approaches the derivative uniformly on the compact domain $M^*$, the $\mathcal{H}$-norm of the first part tends to zero uniformly. For the remaining part we have \begin{eqnarray*} \lefteqn{ \int\limits_{M^*} \left\|\,\int\limits_{|p|>R}\widehat{W}^\epsi(p)\frac{| p|}{\epsi} \Bigg(\frac{P_n(k-p)-P_n(k)}{| p|} \right.}\\ && \left. \hspace{4cm}+ \,\, e_p\cdot\nabla_k P_n(k)\Bigg)\psi(k-p)\,dp\right\|_{L^2(M)} \, dk \\ &\leq & \left\|\psi\right\|_{\mathcal{H}} \big\| \widehat{F}^\epsi\big\|_{L^1(B_R^c)} \\ & &\hspace{1cm} \times\, \sup_{k\in M^*,\,p\in\mathbb{R}^d}\left\| | p|^{-1}(P_n(k-p)-P_n(k))+e_p\cdot\nabla_k P_n(k) \right\| \,, \end{eqnarray*} which tends to zero uniformly as $\epsi\to 0$, since $\Vert\widehat{F}^\epsi\Vert_{L^1(B_R^c)}\to 0$ for any fixed $R>0$. \end{proof} As a consequence of Lemma \ref{first_order} the difference of the two unitary groups in Eq.\ \eqref{difference_1} can be written as \begin{eqnarray} \label{difference_2} \lefteqn{ \tilde{U}^\epsi(t/\epsi)-\tilde{U}^{\epsi,n}_\mathrm{diag} (t/\epsi)} \nonumber\\ & = \,i\epsi\int_0^{t/\epsi}\tilde{U}^\epsi(\epsi^{-1}t-s) \left( \tilde{Q}_n\nabla_k\tilde{P}_n + \tilde{P}_n\nabla_k\tilde{Q}_n \right) \cdot\tilde{F}^\epsi\, \tilde{U}^{\epsi,n}_\mathrm{diag}(s)\,ds+o(1)\,. \end{eqnarray} We have to estimate the integral without losing one order of $\epsi$ from the integration over time. As in the proof in \cite{avron-elgart} of the adiabatic theorem the idea is to rewrite the integrand as a time derivative, i.e.\ as a commutator of $\tilde{H}^n_\mathrm{diag}$ with an appropriately chosen operator $A$, at least up to an unavoidable error $o(1)$. Let us define for $n\in\mathcal{I}$ $$B_n(k)=R^2_{E_n(k)}(H_\mathrm{per}(k))Q_n(k)(D_x+k)P_n(k)\,.$$ \begin{Lem} For $n\in\mathcal{I}$ we have $\tilde{Q}_n\nabla_k\tilde{P}_n + \tilde{P}_n\nabla_k\tilde{Q}_n =[\tilde{B}_n+\tilde B^*_n, \tilde{H}_\mathrm{per}]\,.$ \end{Lem} \begin{proof} Using the spectral decomposition and recalling $Q_n(k)\nabla_k P_n(k)=-R_{E_n(k)}(H_\mathrm{per}(k))Q_n(k)(D_x+k)P_n(k)$ from Lemma \ref{smooth_projection}, one directly computes \begin{eqnarray*} \lefteqn{ B_n(k)H_\mathrm{per}(k)-H_\mathrm{per}(k)B_n(k)}\\ &=&-(H_\mathrm{per}(k)-E_n(k))R^2_{E_n(k)}(H_\mathrm{per}(k))Q_n(k)(D_x+k)P_n(k)\\ &=&-R_{E_n(k)}(H_\mathrm{per}(k))Q_n(k)(D_x+k)P_n(k)\\ &=&Q_n(k)\nabla_k P_n(k)\,. \end{eqnarray*} The lemma then follows from $\tilde{P}_n\nabla_k\tilde{Q}_n = - (\tilde{Q}_n\nabla_k\tilde{P}_n)^*$. \end{proof} \begin{Lem} \label{commu} $\left[B_n + B_n^* ,\tilde{W}^{\epsi,n}_\mathrm{diag}\right]\to 0$ in $B(\mathcal{H},\mathcal{H})$ as $\epsi$ tends to zero. \end{Lem} \begin{proof} To have a concise notation in the following, expressions like $\tilde{W}^{\epsi,n}_\mathrm{diag}P_n(k)$ are understood in the sense that $\tilde{W}^{\epsi,n}_\mathrm{diag}$ acts on all $k$-depend\-ing objects on its right hand side. We recall that $\tilde{W}^{\epsi,n}_\mathrm{diag}= \tilde{P}_n\tilde{W}^\epsi\tilde{P}_n+\tilde{Q}_n\tilde{W}^\epsi\tilde{Q}_n$. Hence \begin{eqnarray*} \lefteqn{\Big[ B_n(k), \tilde{W}^{\epsi,n}_\mathrm{diag}\Big] }\\ &=& Q_n(k)\Big[R^2_{E_n(k)}(H_\mathrm{per}(k))Q_n(k)(D_x+k)P_n(k), \tilde{W}^{\epsi}\Big]P_n(k)\,. \end{eqnarray*} We now examine the commutators $[P_n(k),\tilde{W}^\epsi]$, $[D_x+k,\tilde{W}^\epsi]$ and $[R^2_{E_n(k)}Q_n(k), \tilde{W}^\epsi]$ one by one. It follows from the proof of Lemma \ref{first_order} that $[P_n(k),\tilde{W}^\epsi]$ vanishes as $\epsi\to 0$ and the analogous statement for $[R^2_{E_n(k)}Q_n(k), \tilde{W}^\epsi]$ can be shown to hold by a similar argument. Thus it remains to discuss the commutator $[D_x+k,\tilde{W}^\epsi]$. For $\psi\in H^1(\mathbb{R}^d)$ we compute \begin{eqnarray*} \lefteqn{ (2\pi)^{d/2}([D_x+k,\tilde{W}^\epsi]\mathcal{U}\psi)(k)} \\ &=&\int_{\mathbb{R}^d}\widehat{W}^\epsi(p)(((D_x+k)-(D_x+k-p))\mathcal{U}\psi)(k-p)\,dp\\ &=&\epsi\int_{\mathbb{R}^d}\widehat{W}^\epsi(p)\epsi^{-1}p(\mathcal{U}\psi) (k-p)\,dp\\ &=&\epsi(\tilde{F}^\epsi\mathcal{U}\psi)(k)\,, \end{eqnarray*} which clearly vanishes uniformly for $\psi \in L^2$ as $\epsi\to 0$, since $F\in\mathcal{S}(\mathbb{R}^d,\mathbb{R}^d)$. \end{proof} In summary we have shown that $\left( \tilde{Q}_n\nabla_k\tilde{P}_n + \tilde{P}_n\nabla_k\tilde{Q}_n \right) \cdot\tilde{F}^\epsi =\left(\left[\tilde{B}_n + \tilde B_n^*, \tilde{H}^n_\mathrm{diag}\right]+o(1)\right)\cdot \tilde{F}^\epsi\,,$ and it remains to check \begin{Lem} $\left[\tilde{H}^n_\mathrm{diag},\tilde{F}^\epsi\right]\to 0$ in $B(\mathcal{U}H^1, \mathcal{H})$ as $\epsi$ tends to zero. \end{Lem} \begin{proof} The commutator $[H_\mathrm{per},F^\epsi] = -\frac{1}{2}\epsi^2 (\Delta F^\epsi) -\frac{1}{2}\epsi (\nabla F^\epsi)\cdot\nabla-\frac{1}{2}\epsi (\nabla\cdot F^\epsi)\nabla$ vanishes in $B(H^1,L^2)$ as $\epsi\to 0$. The commutator $[\tilde{W}^{\epsi,n}_\mathrm{diag},\tilde{F}^\epsi]$ vanishes in $B(\mathcal{H},\mathcal{H})$, since the commutator of $\tilde{P}_n$ and $\tilde{Q}_n$ with $\tilde{F}^\epsi$ are both of uniform order $o(1)$ (in $B(\mathcal{H},\mathcal{H})$) and $[\tilde{W}^\epsi,\tilde{F}^\epsi]$ vanishes identically. \end{proof} Defining $\tilde A_n = \left(\tilde B_n +\tilde B_n^*\right) \cdot \tilde F^\epsi\,,$ it follows that the integrand in (\ref{difference_2}) can be written as $\left(\tilde{Q}_n\nabla_k\tilde{P}_n + \tilde{P}_n\nabla_k\tilde{Q}_n \right) \cdot\tilde{F}^\epsi =\left[\tilde{A}_n, \tilde{H}^n_\mathrm{diag}\right]+o(1)\,,$ where $o(1)$ is in the norm of $B(\mathcal{U}H^1,\mathcal{H})$. (Note that for $A^\epsi\in B(L^2,L^2)$, $\lim_{\epsi\to 0}$ $A^{\epsi}=0$ in $B(L^2,L^2)$ implies, in particular, that also $\lim_{\epsi\to 0}A^{\epsi}=0$ in $B(H^1,L^2)$). We are now ready for the \begin{proof}[Proof of Theorem \ref{MT1}] Since ${U}^{\epsi,n}_\mathrm{diag}(t):H^1\to H^1$ is bounded uniformly in $t$ and $\epsi$ (cf.\ Section \ref{sect_asymptotic_position}), we obtain for the difference \eqref{difference_2} of the unitary groups, \begin{eqnarray} \label{last_difference} \displaystyle{ \lefteqn{\Big(\tilde{U}^\epsi(t/\epsi)-\tilde{U}^{\epsi,n}_\mathrm{diag} (t/\epsi)\Big)}} \nonumber\\ & \displaystyle{ =\,-\,i\epsi\int_0^{t/\epsi}\tilde{U}^\epsi(\epsi^{-1}t-s) \left[\tilde{A}_n,\tilde{H}^n_\mathrm{diag}\right] \tilde{U}^{\epsi,n}_\mathrm{diag}(s) \,ds+o(1)\,.} \end{eqnarray} Abbreviating $X^n(s)=\tilde{U}^\epsi(-s)\tilde{U}^{\epsi,n}_\mathrm{diag}(s)$ and $\tilde{A}_n(s) = \tilde{U}^{\epsi,n}_\mathrm{diag}(-s) \tilde{A}_n \tilde{U}^{\epsi,n}_\mathrm{diag}(s)$, we get, using partial integration in \eqref{last_difference}, \begin{eqnarray*}\lefteqn{\hspace{-1cm} -i\epsi\int_0^{t/\epsi}\tilde{U}^\epsi(t/\epsi)\,X^n(s)\, \tilde{U}^{\epsi,n}_\mathrm{diag}(-s)\left[\tilde{A}_n,\tilde{H}^n_\mathrm{diag}\right] \tilde{U}^{\epsi,n}_\mathrm{diag}(s)\,ds}\\ &=&\epsi\,\tilde{U}^\epsi(t/\epsi)\int_0^{t/\epsi}\,X^n(s) \left(\frac{d}{ds}\tilde{A}_n(s)\right)\,ds\\ &=&\epsi\left(\tilde A_n\,\tilde{U}^{\epsi,n}_\mathrm{diag}(t/\epsi)- \tilde{U}^\epsi(t/\epsi)\,\tilde{A}_n\right)\\ &&-\,\epsi\,\tilde{U}^\epsi(t/\epsi)\int_0^{t/\epsi} \left(\frac{d}{ds}X^n(s)\right)\tilde{A}_n(s)\,ds\\ &=&\epsi\left(\tilde{A}_n\,\tilde{U}^{\epsi,n}_\mathrm{diag}(t/\epsi)- \tilde{U}^\epsi(t/\epsi)\,\tilde{A}_n\right)\\ &&-\,i\epsi\,\tilde{U}^\epsi(t/\epsi)\int_0^{t/\epsi}\tilde{U}^\epsi(-s)\, \tilde{W}^{\epsi,n}_{\rm od}\,\tilde{A}_n\, \tilde{U}^{\epsi,n}_\mathrm{diag}(s)\,ds\,. \end{eqnarray*} For $\epsi\to 0$ the first term vanishes since $\tilde{A}_n$ is bounded and the second term vanishes, since $W^{\epsi,n}_{\rm od}$ tends to zero uniformly according to Lemma \ref{first_order}. \end{proof} \section{Convergence of the position operator}\label{sect_asymptotic_position} In this section we will study the asymptotics of the position operator $x^\epsi(t)$. As in the case of the unitaries we have to establish that the off-diagonal contributions to $x^\epsi(t)$ vanish in the limit $\epsi\to 0$. \begin{proof}[Proof of Theorem \ref{MT2}] Let $\psi\in D(|x|)\cap H^2$ and $n\in \mathcal{I}$. Then \begin{eqnarray} \lefteqn{ \left\| \left(x^\epsi (t) - x^{\epsi,n}_{\rm diag} (t)\right) \psi\right\|}\nonumber\\ \label{X1} & \leq & \left\| \left(x^\epsi (t) - U_{\rm diag}^{\epsi,n} (-t/\epsi)\, x^\epsi\, U_{\rm diag}^{\epsi,n} (t/\epsi) \right) \psi\right\| \\ \label{X2} &&+ \left\| \left( U_{\rm diag}^{\epsi,n} (-t/\epsi) \,x^\epsi\, U_{\rm diag}^{\epsi,n} (t/\epsi) - x^{\epsi,n}_{\rm diag} (t)\right) \psi\right\|\,. \end{eqnarray} In order to estimate (\ref{X1}), note that we have $$\label{mean_momentum} x^\epsi(t)\psi=\epsi x\psi+\epsi\int_0^{t/\epsi}U^\epsi(-s) D_xU^\epsi(s)\psi\,ds$$ and \begin{eqnarray*}\lefteqn{ U_{\rm diag}^{\epsi,n} (-t/\epsi)\, x^\epsi\, U_{\rm diag}^{\epsi,n} (t/\epsi) }\\ &=& \epsi x \psi + \epsi \int_0^{t/\epsi} U_{\rm diag}^{\epsi,n} (-s) \left( D_x + i \left[ W^{\epsi,n}_{\rm diag} , x \right] \right)U_{\rm diag}^{\epsi,n} (s) \psi\,ds\\ &=&\epsi x \psi + \epsi \int_0^{t/\epsi} U_{\rm diag}^{\epsi,n} (-s) D_x U_{\rm diag}^{\epsi,n} (s) \psi\,ds + o(1)\,. \end{eqnarray*} The last equality holds, since $[ W^{\epsi,n}_{\rm diag} , x] = o(1)$ in $B(L^2)$, as follows immediately from the fact that $[ W^\epsi, P_n] = o(1)$ and $[ W^\epsi, Q_n] = o(1)$, cf.\ proof of Lemma \ref{first_order}. Hence, using (\ref{mean_momentum}), the remaining term from (\ref{X1}) is \begin{eqnarray}\lefteqn{ \epsi\int_0^{t/\epsi}\left(U^\epsi(-s)D_xU^\epsi(s)- U^{\epsi,n}_\mathrm{diag}(-s)D_xU^{\epsi,n}_\mathrm{diag}(s)\right)\psi\, ds}\nonumber\\ &=&\int_0^{t}\left(U^\epsi(-s/\epsi)- U^{\epsi,n}_\mathrm{diag}(-s/\epsi) \right)D_x U^{\epsi,n}_\mathrm{diag}(s/\epsi)\psi\,ds \label{zweiter_term}\\ &&+\,\int_0^{t}U^\epsi(-s/\epsi)D_x \left(U^\epsi(s/\epsi)- U^{\epsi,n}_\mathrm{diag}(s/\epsi)\right) \psi\,ds\,.\label{dritter_term} \end{eqnarray} Using the fact that $V$ and $W$ are infinitesimally operator bounded with respect to $-\frac{1}{2}\Delta$ and that $\psi\in H^2$, we get for $\psi(s):= U^{\epsi,n}_\mathrm{diag}(s/\epsi)\psi$ \begin{eqnarray*} \left\|D_x^2 \psi(s) \right\| & \leq & \left\|H^{\epsi,n}_{\rm diag} \psi(s)\right\| + \left\|(V+W^{\epsi,n}_{\rm diag})\psi(s)\right\| \\ & \leq & \left\|H^{\epsi,n}_{\rm diag} \psi \right\| + c_1 \left\|D_x^2\psi(s)\right\| + c_2\left\|\psi\right\|\,, \end{eqnarray*} with $c_1<\frac{1}{2}$ and $c_2<\infty$. Hence $\|D_x U^{\epsi,n}_\mathrm{diag}(s/\epsi)\psi\|_{H^1}\leq c\|\psi\|_{H^2}$ with $c$ independent of $s$ and $\epsi$ and we can apply Theorem \ref{MT1} to conclude that the operator acting on $\psi$ in (\ref{zweiter_term}) vanishes in $B(H^2,L^2)$ as $\epsi\to 0$. We come to (\ref{dritter_term}). Let $\psi(s)=({U}^\epsi(s/\epsi)- {U}^{\epsi,n}_\mathrm{diag}(s/\epsi))\psi$, then, by Cauchy-Schwarz, $\left\| {D}_x\psi(s)\right\|^2 = \left(\psi(s),{D}^2_x\psi(s)\right) \leq \left\|\psi(s)\right\| \, \left\|{D}^2_x \psi(s)\right\|\,.$ The first factor tends to zero by Theorem \ref{MT1} whereas the second is uniformly bounded by the same argument as in the treatment of (\ref{zweiter_term}) a few lines above. Next we rewrite (\ref{X2}) as $\epsi U_{\rm diag}^{\epsi,n} (-t/\epsi) \, x_{\rm od}^n\, U_{\rm diag}^{\epsi,n} (t/\epsi)$ with $x_{\rm od}^n := Q_n x P_n + P_n x Q_n$. This certainly vanishes as $\epsi\to 0$ if $x_{\rm od}^n$ can be shown to be a bounded operator. To see this, note that in Bloch representation $x$ acts as $i\nabla_k$. Hence $(\mathcal{U} Q_n x P_n \psi)(k) = i Q_n(k) \nabla_k P_n(k)(\mathcal{U}\psi)(k) = i Q_n(k) (\nabla_k P_n(k)) (\mathcal{U}\psi)(k)$ and thus $\|Q_n x P_n\| = \|\tilde Q_n \nabla_k\tilde P_n \|$. Finally also $P_n x Q_n$ is bounded, since it is the adjoint of $Q_n x P_n$. \end{proof} \section{Semiclassical equations of motion for the position operator}\label{PsiDO} As we have shown, on the macroscopic scale the position and quasimomentum operators commute with the projection on isolated bands. Thus it remains to investigate the semiclassical limit for each isolated band separately. For this purpose we note that any $\psi \in \tilde P_n\mathcal{H}$ is of the form $\psi_n(k)\varphi_n(x,k)$ with $\psi_n\in L^2(M^*)$. Since $\varphi_n$ already satisfies \eqref{BC}, we have to extend the Bloch coefficients periodically. We determine now how $H^{\epsi,n}_{\rm diag}$ acts on $L^2(M^*)$. We have $[H_{\rm per},\tilde P_n]=0$ and therefore $H_{\rm per}$ acts as multiplication by $E_n(k)$. For $W^{\epsi,n}_{\rm diag}$ we have \begin{eqnarray}\lefteqn{ \hspace{-.5cm} \left( \tilde P_n \tilde W^\epsi \tilde P_n \mathcal{U} \psi \right) (k,x) }\nonumber \\ & = &(2\pi)^{-d/2}\int_{\mathbb{R}^d} \widehat W^\epsi (p) \big( \varphi_n(k),\varphi_n(k-p)\big)_{L^2(M)} \psi_n(k-p) \,dp\,\varphi_n(k,x)\nonumber\\ & =: & (\tilde W^{\epsi,n}\psi_n)(k)\varphi_n(x,k)\,. \end{eqnarray} Thus $H^{\epsi,n}_{\rm diag}$ restricted to $\tilde P_n\mathcal{H}$ is unitarily equivalent to $H^{\epsi,n} := E_n(k) + \tilde W^{\epsi,n}$. To be able to use techniques from semiclassics we next approximate $\tilde W^{\epsi,n}$ by the operator $\tilde W^{\epsi,n}_{\rm sc} = W(-i\epsi\nabla_k)$, where $\nabla_k$ is understood with periodic boundary conditions on $\R^d/\Gamma^*$. \begin{Lem}\label{difference_eps} For any $n\in\mathcal{I}$ $$\label{Wdiag} \tilde W^{\epsi,n} = \tilde W^{\epsi,n}_{\rm sc} + o(\epsi)$$ in $B(L^2(M^*))$. \end{Lem} \begin{proof} By definition we have $\left( \tilde W^{\epsi,n}_{\rm sc} \psi \right)(k) = (2\pi)^{-d/2} \int_{\mathbb{R}^d} \widehat W^\epsi (p)\psi_n(k-p) \,dp\,,$ and therefore \begin{eqnarray}\label{wdiff} \lefteqn{\left( \left( \tilde{W}^{\epsi,n} - \tilde{W}^{\epsi,n}_{\rm sc} \right) \psi\right) (k) =} \nonumber\\ & = (2\pi)^{-d/2} {\displaystyle \int_{\mathbb{R}^d}} \widehat W^{\epsi,n}(p) \Big( \big(\varphi_n(k),\varphi_n(k-p)\big)_{L^2(M)} -1\Big)\psi(k-p)\, dp\,. \end{eqnarray} As to be shown, there exists a constant $c$ such that $$\label{skalar} \left|\big(\varphi_n(k),\varphi_n(k-p)\big)_{L^2(M)} -1\right| \leq c|p|^2$$ for Lebesgue almost all $k$. Therefore we conclude \begin{eqnarray*} \left\| \left(\tilde{W}^{\epsi,n} - \tilde{W}^{\epsi,n}_{\rm sc}\right) \psi \right\|_{L^2(M^*)} & \leq & c\epsi^2\left\| \int \left| \widehat W^\epsi (p) \frac{|p|^2}{\epsi^2}\right| |\psi(k-p)|\, dp\,\right\|_{L^2(M^*)} \\ &\leq& c' \epsi^2 \|\psi\|_{L^2(M^*)}\,. \end{eqnarray*} To show \eqref{skalar} note that one can chose $\varphi_n(k)$ such that the map $k\mapsto \varphi_n(k)\in L^2(M)$ is smooth Lebesgue almost everywhere. This is because according to Lemma \ref{smooth_projection} the projections $P_n(k)$ depend smoothly on $k$ and hence one can locally define $\varphi_n(k) = P_n(k) \varphi_n(k_0)/\| P_n(k) \varphi_n(k_0) \|$. Now we can cover $M^*$ by finitely many open disjoint sets $U_i$ such that $M^*\setminus \cup_i U_i$ is a set of Lebesgue measure zero and $\varphi_n(k)$ can be defined on the closure of each $U_i$ in the way described above. One obtains a family $\varphi_n(k)$ of eigenfunctions which is smooth except at the boundaries between the sets, where we pick $\varphi_n(k)$ with an arbitrary phase. Wherever $\varphi_n(k)$ is smooth, Taylor expansion yields $\varphi_n(k-p) = \varphi_n(k) -p \cdot \nabla_k \varphi_n(k) + \frac{1}{2} p\cdot\mathbb{H}(\varphi_n)(k')p$, where $\mathbb{H}(\varphi_n)$ denotes the Hessian and $\frac{1}{2} p\cdot \mathbb{H}(\varphi_n)(k')p$ is the Lagrangian remainder. In view of $(\varphi_n(k),\nabla_k\varphi_n(k))_{L^2(M)}=0$, which follows from comparing (\ref{gradient_projection}) with $(\nabla_k P_n \psi)(k) = ( \varphi_n(k),\psi (\cdot,k)) \nabla_k \varphi_n(k) + ( \nabla_k \varphi_n(k),\psi(\cdot,k))\varphi_n(k)\,,$ we obtain \begin{equation*} \left|\big(\varphi_n(k),\varphi_n(k-p)\big)_{L^2(M)} -1\right| \leq c(k)|p|^2\,. \end{equation*} Here $c(k) = \frac{1}{2} \sum_{i,j} |(\varphi_n(k'),\partial_{k_i}\partial_{k_j} \varphi_n(k'))|$. However, $c(k)$ is bounded uniformly in $k$, since $\varphi_n(k)$ is smooth on each compact $\bar {U_i}$. \end{proof} We define now the semiclassical Hamiltonian $H^{\epsi,n}_{\rm sc}$ $$H^{\epsi,n}_{\rm sc} = E_n(k) + W(-i\epsi\nabla_k)$$ acting on $L^2(M^*)$. Then Lemma \ref{difference_eps} shows that the difference $H^{\epsi,n} - H^{\epsi,n}_{\rm sc}$ is of order $o(\epsi)$ uniformly in $B(L^2(M^*))$ and hence (cf.\ Section \ref{effective}) the difference of the corresponding unitary groups approaches zero as $\epsi\to 0$. \begin{Cor}\label{ucor} Let $U^{\epsi,n}_\mathrm{sc}(t)= \mathrm{e}^{-it{H}^{\epsi,n}_\mathrm{sc}}$ and ${U}^{\epsi,n} (t)= \mathrm{e}^{-it{H}^{\epsi,n}}$, then $\lim_{\epsi\to 0}\big({U}^{\epsi,n}(t/\epsi)- {U}^{\epsi,n}_\mathrm{sc}(t/\epsi)\big)=0$ in $B(L^2(M^*))$. \end{Cor} The semiclassical limit for $U^{\epsi,n}_{\rm sc}(t/\epsi)$ on $L^2(\mathbb{T}^*)$ is well studied. We refer to \cite{folland,hoevermann,robert}. As a consequence the strong limits \begin{eqnarray}\label{62} \lim_{\epsi\to 0} U^{\epsi,n}_{\rm sc}(- t/\epsi)\,(-i\epsi \nabla_k)\, U^{\epsi,n}_{\rm sc}(t/\epsi) & = & r_n(t;k)\,, \\ \label{61} \lim_{\epsi\to 0} U^{\epsi,n}_{\rm sc}(- t/\epsi)\,k\, U^{\epsi,n}_{\rm sc}(t/\epsi) & = & k_n(t;k) \end{eqnarray} exist on $H^1(\mathbb{T}^*)$. $r_n$ and $k_n$ act as multiplication operators and are defined as in (\ref{dynamical_system}) with initial conditions $(r_n(0),k_n(0))=(0,k)$. Since the restriction of $\epsi x^{n}_{\rm diag}$ to the $n$-th band subspace is unitarily equivalent to $- i\epsi \nabla_k$ on $L^2(\mathbb{T}^*)$, we can, in view of Theorem \ref{MT2}, conclude the proof of Theorem \ref{MT3} by showing \begin{Lem} In $B(L^2(M^*))$ we have $$\label{xx1} \lim_{\epsi\to 0} \big( U^{\epsi,n}(- t/\epsi)(-i\epsi \nabla_k) U^{\epsi,n}(t/\epsi) - U^{\epsi,n}_{\rm sc}(- t/\epsi)(-i\epsi \nabla_k) U^{\epsi,n}_{\rm sc}(t/\epsi) \big) =0\,.$$ \end{Lem} \begin{proof} The proof of (\ref{xx1}) is analogous to the proof of Theorem \ref{MT2} in Section 5, however, simpler. As in (\ref{mean_momentum}) we have $U_{\rm sc}^{\epsi,n}(- t/\epsi)(-i\epsi \nabla_k) U_{\rm sc}^{\epsi,n}(t/\epsi) = -i\epsi\nabla_k + \epsi \int_0^{t/\epsi} U_{\rm sc}^{\epsi,n}(-s) \big[ -i\nabla_k, H_{\rm sc}^{\epsi,n} \big] U_{\rm sc}^{\epsi,n}(s)\,ds$ and \begin{eqnarray*}\lefteqn{ U^{\epsi,n}(- t/\epsi)(-i\epsi \nabla_k) U^{\epsi,n}(t/\epsi) =}\\ & = & -i\epsi\nabla_k + \epsi \int_0^{t/\epsi} U^{\epsi,n}(-s) \big[ -i\nabla_k, H^{\epsi,n} \big] U^{\epsi,n}(s)\,ds\\ & = & -i\epsi\nabla_k + \epsi \int_0^{t/\epsi} U^{\epsi,n}(-s) \Big( \big[ -i\nabla_k, H^{\epsi,n}_{\rm sc} \big]+ \big[ -i\nabla_k, \Delta \tilde W^{\epsi,n}\big] \Big) U^{\epsi,n}(s)\,ds\,, \end{eqnarray*} where $\Delta \tilde W^{\epsi,n} :=\tilde{W}^{\epsi,n} - \tilde{W}^{\epsi,n}_{\rm sc}$. Now $[ -i\nabla_k, H^{\epsi,n}_{\rm sc}] = -i\nabla_k E_n(k)$ is bounded, and (\ref{xx1}) follows from Corollary \ref{ucor} if we can show that $[ -i\nabla_k, \Delta \tilde W^{\epsi,n}] = o(1)$ in $B(L^2(M^*))$. Noting that $(\Delta \tilde W^{\epsi,n}\psi)(k)$ is given by (\ref{wdiff}), this can be shown by an argument similar to the one in Lemma \ref{difference_eps}. %Finally, (\ref{kk1}) immediately follows from Corollary \ref{ucor}, since %multiplication by $k$ is bounded on $L^2(M^*)$. \end{proof} \section{Semiclassical equations of motion for general observables} \label{MT4S} We proceed to more general semiclassical observables. First note that Theorem \ref{MT5} follows immediately from the results obtained so far (Theorem \ref{MT1}, Corollary \ref{ucor} and \eqref{61}), since multiplication with $k$ in Bloch representation is bounded. Hence we now have that $$\lim_{\epsi \to 0} \|x^\epsi(t)\psi - \mathcal{U}^{-1}R(t)\mathcal{U}\psi\|=0$$ for all $\psi \in \mbox{Ran} P_\mathcal{I}\cap D(|x|)\cap H^2$ and that $$\lim_{\epsi \to 0} \|k^\epsi(t)\psi - \mathcal{U}^{-1} K(t)\mathcal{U}\psi\| = 0$$ for all $\psi \in \mbox{Ran} P_\mathcal{I}$. We next consider bounded continuous functions of $x^\epsi(t)$ and $k^\epsi(t)$: \begin{Lem} \label{fl} Let $f\in C_\infty(\R^d)$ and $g\in C(M^*)$. Then for all $\psi\in\mbox{Ran}P_\mathcal{I}$ we have $$\label{fxcon} \lim_{\epsi\to 0}\|\left( f(x^\epsi(t)) - \mathcal{U}^{-1}f(R(t))\mathcal{U}\right) \psi\|=0$$ and $$\label{fkcon} \lim_{\epsi\to 0}\|\left( g(k^\epsi(t)) - \mathcal{U}^{-1}g(K(t))\mathcal{U}\right) \psi\|=0\,.$$ \end{Lem} \begin{proof} We will sketch the proof for $x^\epsi(t)$. First note that $\bar R(t):=\mathcal{U}^{-1}R(t)\mathcal{U}$ is a bounded self-adjoint operator and commutes with $P_\mathcal{I}$. Hence the sets $D_\pm := (\bar R(t)\pm i)(\mbox{Ran} P_\mathcal{I}\cap D(|x|)\cap H^2)$ are dense in $P_\mathcal{I}$ (Since $R$ and $x^\epsi$ are vectors of operators in $\R^d$, note that this and the following statements hold component wise). For $\psi \in D_\pm$ we have $$\label{Rescon} \left[ (x^\epsi(t) \pm i)^{-1} - (\bar R(t) \pm i)^{-1}\right]\psi = (x^\epsi(t) \pm i)^{-1} (\bar R(t) - x^\epsi(t))\varphi$$ for $\varphi = (\bar R(t) \pm i)^{-1} \psi \in \mbox{Ran} P_\mathcal{I}\cap D(|x|)\cap H^2$. Thus, by Theorem \ref{MT1}, \eqref{Rescon} strongly approaches zero as $\epsi\to 0$ and, since $D_\pm$ are dense in $P_\mathcal{I}$, $(x^\epsi(t) \pm i)^{-1}$ strongly approach $(\bar R(t) \pm i)^{-1}$ on $P_\mathcal{I}$. Using the fact that polynomials in $(x_j\pm i)^{-1}$, $j=1,\ldots,d$, are dense in $C_\infty(\R^d)$ one concludes that the convergence $x^\epsi(t)\to\bar R(t)$ on Ran$P_\mathcal {I}$ in the strong resolvent sense'' implies $\lim_{\epsi\to 0} \|\left(f(x^\epsi(t))-f(\bar R(t))\right)\psi\| =0$ for all $f\in C_\infty(\R^d)$ and $\psi \in \mbox{Ran}P_\mathcal{I}$ (cf.\ Theorem VIII.20 in \cite{reed-simon-i}). However, by the functional calculus for self-adjoint operators we have $f(\mathcal{U}^{-1}R(t)\mathcal{U}) = \mathcal{U}^{-1}f(R(t))\mathcal{U}$ and \eqref{fxcon} follows. Clearly \eqref{fkcon} follows analogously. \end{proof} \begin{proof}[Proof of Theorem \ref{MT4}] Let $a\in \mathcal{O}(0)$. Referring again to the general Stone-Weier\-stra\ss\ theorem we can uniformly approximate $a(x,\xi)$ by a sum of products, i.e.\ $a(x,\xi)= \sum_{i=0}^\infty a_i f_i(x)g_i(\xi)$ with $f_i\in C_\infty(\R^d)$, $g_i\in C(M^*)$, $\sum |a_i|<\infty$ and $\sup_{i\in \mathbb{N}, x\in \R^d, \xi \in M^*}$ $|f_i(x)g_i(\xi)|<\infty$. Hence in order to prove Theorem \ref{MT4} we are left to show that for arbitrary $f\in C_\infty(\R^d)$ and $g\in C(M^*)$ we have $$\label{fe} \left(f(x)g(\xi)\right)^{W,\epsi}(t) \to \mathcal{U}^{-1}f(R(t)) g(K(t))\mathcal{U}$$ strongly on Ran$P_\mathcal{I}$. To see this recall the so called product rule for quantum observables (cf.\ \cite{robert}). It states, in particular, that for two symbols $A,B\in \mathcal{O}(0)$ $\lim_{\epsi\to 0} \left\|\left( (AB)^{W,\epsi} - A^{W,\epsi}B^{W,\epsi}\right)\psi\right\| = 0\,.$ Applied to our case this yields $\left(f(x)g(\xi)\right)^{W,\epsi}(t) \to \left(f(x)^{W,\epsi}g(\xi)^{W,\epsi}\right)(t) = f(x^\epsi(t))g(k^\epsi(t))\,.$ Finally, since $f$ and $g$ are bounded, Lemma \ref{fl} implies \eqref{fe} and thus Theorem \ref{MT4}. \end{proof} \section{Band crossings}\label{outlook} We proved the semiclassical limit for isolated bands only. In principle, there are two distinct mechanisms of how this assumption could be violated. First of all a band could be isolated but have a constant multiplicity larger than one. This occurs, e.g., for the Dirac equation where because of spin the electron and positron bands are both two-fold degenerate. A systematic study is only recent \cite{gerard-markowich-mauser-poupaud,spohn2} and leads to a matrix valued symplectic structure for the semiclassical dynamics. For periodic potentials degeneracies are the exception. They form a real analytic subvariety of the Bloch variety $B=\{(k,\lambda)\in\mathbb{R}^d\times\mathbb{R}\mid\exists f\in L^2(M): H_{\mathrm{per}}(k)f=\lambda f\}$ and have a dimension at least one less than the dimension of $B$ \cite{kuchment,wilcox}. Thus points of band crossings have a $k$-Lebesgue measure zero. From the study of band structures in solids one knows that band crossings indeed occur. Thus it is of interest to understand the extra complications coming from band crossings. There are two types of band crossings. The first one is removable through a proper analytic continuation of the bands. In a way, removable band crossings correspond to a wrong choice of the fundamental domain. E.g.\ for $V=0$ we may artificially introduce a lattice $\Gamma$. The bands touch then at the boundary of $M^*$. Upon analytic continuation we recover the single band $E_1(k)=k^2/2$ with $M^*=\mathbb{R}^d$. In one dimension all band crossings can be removed \cite{reed-simon-iv}. Thus, with the adjustment discussed, our result fully covers the case $d=1$. For $d\geq 2$ generically band crossings cannot be removed. It is then of great physical interest to understand how a wave packet tunnels into a neighboring band through points of degeneracy (or almost degeneracy). For a careful asymptotic analysis in particular model systems we refer to the monumental work of G.\ Hagedorn \cite{hagedorn}. Gerard \cite{gerard} considers a model system with two bands in two dimensions, i.e., the role of $-\frac{1}{2}\Delta+V$ is taken by $\left(\begin{array}{cc} k_1 & k_2 \\ k_2 & -k_1 \end{array}\right)$. He investigates the semiclassical limit and proves that the particle may tunnel to the other band with a probability which depends on how well the initial wave packet is concentrated near a semiclassical orbit hitting the singularity. \section*{Acknowledgments} FH gratefully acknowledges the financial support by the Deutsche Forschungsgemeinschaft via the Graduiertenkolleg {\em Mathematik im Bereich ihrer Wechselwirkung mit der Physik} at the LMU M\"{u}nchen. \begin{thebibliography}{99} \bibitem{asch-knauf} J.\ Asch and A.\ Knauf. {\em Motion in Periodic Potentials}, Nonlinearity {\bf 11}, 175-200 (1998). \bibitem{ashcroft-mermin} N.\ W.\ Ashcroft and N.\ D.\ Mermin. {\em Solid State Physics}, Saunders (1976). \bibitem{avron-elgart} J.\ E.\ Avron and A. 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