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periodic potential, adiabatic decoupling
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\begin{document}



\title{\LARGE\bf Effective dynamics
 for Bloch electrons:
 Peierls substitution and beyond}

\author{ \large Gianluca Panati, Herbert Spohn, Stefan Teufel \medskip  \\
\normalsize Zentrum Mathematik and Physik Department,\\
\normalsize Technische Universit\"{a}t M\"{u}nchen, 80290 M\"{u}nchen, Germany   \medskip\\
\normalsize email: panati@ma.tum.de, spohn@ma.tum.de,
teufel@ma.tum.de }
\maketitle

\begin{abstract}  
We reconsider the longstanding  problem of an electron moving in a crystal  
under the  influence of weak external electromagnetic fields.
More precisely we analyze the dynamics generated by the Schr\"odin\-ger operator
$H = \frac{1}{2} ( -\I\nabla_x - A(\varepsilon x) )^2 + V(x) +
\phi(\varepsilon x)$, where $V$ is a lattice periodic potential and $A$ and
$\phi$ are external potentials which vary slowly on the scale set by the
lattice spacing. We study the limit $\varepsilon\to 0$ in several steps:
(i) Approximately invariant subspaces associated with isolated Bloch bands are
constructed. (ii) We derive an effective quantum Hamiltonian for states
inside such a decoupled subspace. The effective Hamiltonian has an asymptotic
expansion in $\varepsilon$, starting with the term given through the
Peierls
substitution. Our construction allows, in principle, to compute also all
higher order terms and we give the first order correction to the Peierls
substitution explicitly. (iii) The semiclassical limit of the effective
Hamiltonian yields the first order corrections to the ``semiclassical model''
of solid states physics, including a new term which has been missed in
earlier heuristic studies.
\end{abstract}




\newpage
\tableofcontents



\section{Introduction}



An outstanding problem of solid state physics is to understand the
motion of electrons in a 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{AsMe,Ko,Za}. One argues that for suitable wave packets,
which are spread over many lattice spacings, the main effect of a
periodic potential   $V_{\Gamma}$ on the
electron dynamics corresponds to changing the dispersion relation
from the free kinetic energy $E_{\rm free}(p) = \frac{1}{2}\, p^2$
to the modified kinetic energy $E_n(p)$ given by the $n^{\rm th}$
Bloch function. Otherwise the electron responds to slowly varying external
potentials $A$, $\phi$ as in the case of a vanishing periodic
potential. Thus the semiclassical equations of motion are
\be  \label{semiclassical_dynamics}
\dot q = \nabla  E_n(v) \,,\qquad \dot v = -\nabla \phi(q) +
\dot q \vp B(q)\,,
\ee
where $v = p - A(q)$ is the kinetic momentum and $B= \mathrm{curl
A}$ is the magnetic field. (We choose units in which 
the Planck constant $\hbar$,   the speed $c$ of light, and
the mass $m$   of the electron
 are equal to one, and absorb the charge $e$ into the potentials.)
The corresponding equations of motion
for the canonical variables $(q,p)$ are generated by the
Hamiltonian
\[
H_{\rm sc} (q,p) = E_n\big(p-A(q)\big) + \phi(q)\,,
\]
where $q$ is the position and $p$ the quasi-momentum of the
electron. Note that there is a semiclassical evolution for each
Bloch band separately.

In this article we use adiabatic perturbation theory in order to
understand on a mathematical level how these semiclassical
equations emerge from the underlying Schr\"odinger equation
\be \label{basic_dynamics}
\I\,  \partial_s\,\psi(x,s)= \left(
{\textstyle\frac{1}{2}}\big(-\I \nabla_x - A(\epsi x)\big)^2 +
V_{\Gamma}(x) + \phi(\epsi x)\right)
 \psi(x,s)
\ee
in the limit $\epsi\to 0$ at leading order. In addition, for the first time, the order
$\epsi$  correction to (\ref{semiclassical_dynamics}) is established, see Equation (\ref{Semi1}).
 

In (\ref{basic_dynamics}) the potential  $V_{\Gamma}:\R^d\to \R$ is
periodic with respect to some regular lattice $\Gamma$ generated
through the basis $\{\gamma_1,\ldots,\gamma_d\}$,
$\gamma_j\in\R^d$, i.e.\
\[
\Gamma =\Big\{ x\in\R^d: x= \textstyle{\sum_{j=1}^d}\alpha_j\,\gamma_j
\,\,\,\mbox{for some}\,\,\alpha \in \mathbb{Z}^d \Big\}
\]
and $V_{\Gamma}(\,\cdot + \gamma) = V_{\Gamma}(\cdot)$ for all
$\gamma\in\Gamma$. The lattice spacing defines the microscopic
spatial scale. The external potentials $A(\epsi x)$ and $\phi(\epsi
x)$, with $A:\R^d\to\R^d$ and $\phi:\R^d\to\R$, are slowly varying on
the scale of the lattice, as expressed through the  dimensionless
scale parameter $\epsi$, $\epsi\ll 1$. In particular, this means
that the external fields are weak compared to the fields generated
by the ionic cores, a condition which is
 satisfied for real metals  even for the strongest external electrostatic fields
available and for a wide range of  magnetic fields, cf.\
\cite{AsMe}, Chapter~12.


Note that the external forces due to $A$ and  $\phi$ are of order
$\epsi$ and therefore have to act over a time of order
$\epsi^{-1}$ to produce finite changes, which is taken as the definition of
the macroscopic time scale. Hence, we will be interested in solutions of (\ref{basic_dynamics})
for macroscopic times, but we will work mostly   in the microscopic
coordinates $(x,s)$. For sake of
comparison we recall that the macroscopic space-time scale
$(x',t)$ is defined through $x'=\epsi x$ and $t=\epsi s$. With
this change of scale  Equation  (\ref{basic_dynamics}) reads
\be  \label{scaled_dynamics}
\I\,\epsi\, \partial_{t }\,\psi^\epsi(x',t )= \left( {\textstyle
\frac{1}{2}}\big(-\I \epsi\nabla_{x'} - A(x')\big)^2 +
V_{\Gamma}(x'/\epsi) + \phi(x')\right)
 \psi^\epsi(x',t )
\ee
with initial conditions
$\psi^\epsi(x')=\epsi^{-d/2}\psi(x'/\epsi)$. If $V_{\Gamma}=0$,
Equation (\ref{scaled_dynamics}) is the usual semiclassical limit
with $\epsi$ set equal to $\hbar$.



The problem of deriving (\ref{semiclassical_dynamics})  from
the Schr\"odinger equation
(\ref{basic_dynamics}) in the limit $\epsi\to 0$ has been attacked
along several routes. In the physics literature
(\ref{semiclassical_dynamics}) is usually accounted for  by
constructing suitable semiclassical wave packets, cf.\
\cite{Ko,Za}. The few mathe\-matical approaches  to the
time-dependent problem   (\ref{scaled_dynamics}) extend 
techniques from semiclassical analysis, as the WKB ansatz
\cite{GRT,DGR}, or   Wigner measures \cite{GMMP}, the latter being carried out
only for vanishing external potentials.
  The large time asymptotics of the solutions to
(\ref{scaled_dynamics}) without external potentials is studied in \cite{AsKn}.

In the following we will try to convince the reader  that an improved
understanding of the approximation  must be based on the following
observations. The step from (\ref{basic_dynamics}) to
(\ref{semiclassical_dynamics}) involves actually two limits.
Semiclassical behavior can only emerge if a Bloch band is
separated by a gap from the other bands and thus the corresponding
subspace decouples adiabatically from its orthogonal complement.
Hence we must reformulate (\ref{basic_dynamics}) as a
space-adiabatic problem. This has been done in \cite{HST},
where the semiclassical model (\ref{semiclassical_dynamics}) is
derived for the case of zero
 magnetic field.
The present paper is, in spirit, a continuation of the program
started in \cite{HST} to the case of both, external magnetic and
electric fields. This becomes possible by reformulating the problem in
such a way that the general scheme of space-adiabatic perturbation theory
as developed in \cite{PST1} can be
applied, granted some crucial modifications. The  results we obtain in this way
constitute not only the derivation of the semiclassical
model in this generality, but they add, as we shall explain, new
insight to the structure of the problem. In particular, we are now able to
compute systematically higher order corrections in the small parameter $\epsi$
to the semiclassical equations (\ref{semiclassical_dynamics}).
The corrected equations including all terms of first order in
$\epsi$ read
\begin{eqnarray}
\dot q &=&\nabla_{ v } \Big( E_n( v ) - \epsi \,
B(q)\cdot M_n( v )\Big)
- \epsi\, \dot  v \vp \Omega_n( v )\,,\nonumber\\\label{Semi1}\\
\dot  v  &=&  -\nabla_q \Big(\phi(q) - \epsi\,
B(q)\cdot M_n( v )\Big) +\dot q \vp B(q) \,.\nonumber
\end{eqnarray}
The Berry connection of the eigenspace-bundle corresponding
to the Bloch band $E_n$ enters in a gauge-invariant way through
its curvature $\Omega_n = \D \A_n$ and through the effective magnetic moment $M_n$.
The precise definitions of the new terms are given below in Corollary \ref{MainCor}.



The first equation in (\ref{Semi1}) agrees with the expression
found by Sundaram and Niu \cite{SuNi}, while  the correction in
the second equation in (\ref{Semi1}) is new. We remark that  the
dynamical system (\ref{semiclassical_dynamics}) is of interest
  in its own, c.f.\ \cite{MaNo} and references therein, due
to the nontrivial topology of the underlying phase space, and we hope that
the corrections in (\ref{Semi1}) give rise to further investigations.
In our paper we discuss the equations (\ref{Semi1}) only shortly at the beginning of Section 4,
where we show in particular, that they are Hamiltonian with respect to a non-standard
symplectic form. One concrete physical application of the refined semiclassical equations (\ref{Semi1})
is a quantitative theory for the anomalous Hall effect  \cite{JNM}.

\begin{remark}In the presence of a strong external magnetic field with rational flux per unit cell
one formally obtains   semiclassical equations identical to (\ref{Semi1}), except that
the Bloch band $E_n$ must be replaced by one of the magnetic sub-bands.
As on striking  consequence   they  provide the semiclassical explanation
for the quantization of the Hall conductivity. More precisely, for spatial dimension $d=2$,
$\phi(q)= -\mathcal{E}\cdot q$,  $B(q)=0$, the equations of motion 
(\ref{Semi1}) become $\dot q = \nabla_vE_n(v) + \mathcal{E}^\perp\Omega_n(v)$, $\dot v = \mathcal{E}$,
where $\Omega_n$ is now scalar and $\mathcal{E}^\perp$ is $\mathcal{E}$ rotated by $\pi/2$.
We assume initially $v(0)=k$ and a completely filled band, which means to integrate with respect to $k$ over the first Brillouin zone.
Then the average current for band $n$ is given by 
\[
j_n = \int \D k \,\dot q(k) = \int \D k \,\big(\nabla_{ k }  E_n( k )-     \mathcal{E}^\perp \Omega_n( k
)\big) =  -  \mathcal{E}^\perp \int \D k\,\Omega_n( k)\,.
\]
   $\int \D k\,\Omega_n( k)$ is the Chern number of the
magnetic Bloch bundle and as such an integer.\er
\end{remark}

 Since the precise statements of our results
require considerable
 technical preparations, they are postponed  to Section~3.
At this point we only give an informal outline of the results,
concluding with the theorem connecting (\ref{basic_dynamics})
and~(\ref{Semi1}).

Under Assumption (A$_1$) the Hamiltonian
\be \label{Hamiltonian}
H^\epsi = {\textstyle\frac{1}{2}}\big(-\I \nabla_x - A(\epsi
x)\big)^2 + V_{\Gamma}(x) + \phi(\epsi x)\,.
\ee
is   self-adjoint on the domain $H^2(\R^d)$ and hence generates solutions to
(\ref{basic_dynamics}) in $\Hi:=L^2(\R^d)$ through the unitary group
 $\E^{-\I  H^\epsi s}$, $s\in \R$.
As a first step  we construct for each Bloch band $E_n(k)$, which does not cross or touch any
other Bloch band,  an orthogonal projector
$\Pi^\epsi_n$ such that the associated subspace $\Pi^\epsi_n\Hi$
of the full Hilbert space $\Hi$ is approximately invariant under
the time evolution. More precisely, $\Pi^\epsi_n$ satisfies
\[
\|\,[\,H^\epsi,\,\Pi^\epsi_n \,] \,\|_{\B(\Hi)} =
\Or(\epsi^\infty)
\]
and thus
\[
\|\,[\,\E^{-\I  H^\epsi s},\,\Pi^\epsi_n \,] \,\|_{\B(\Hi)} =
\Or(\epsi^\infty|s|) \,.
\]
  Hence transitions between
$\Pi^\epsi_n\Hi$ and its orthogonal complement
$(\Pi^\epsi_n\Hi)^\perp$ are asymptotically smaller than any power
of $\epsi$ {\em uniformly for all initial states}. In this sense
$\Pi^\epsi_n\Hi$ is an adiabatically decoupled subspace.

As the second step we construct an effective Hamiltonian which
approximately generates the dynamics inside the
band-subspace $\Pi^\epsi_n\Hi$. To this end we unitarily map $\Pi^\epsi_n\Hi$
to a suitable reference Hilbert space. In the reference
representation the effective Hamiltonian is given as a
 pseudodifferential operator which allows for an asymptotic expansion in powers of
$\epsi$. At leading order we reproduce the well known fact that
 the full Hamiltonian
$H^\epsi$  restricted to the decoupled subspace is
   given through the Peierls substitution, i.e.\
\be\label{Peierls}
H^\epsi \Pi^\epsi_n = \Big( E_n\big( -\I \nabla_x - A(\epsi
x)\big) + \phi(\epsi x)\Big)\,\Pi^\epsi_n + \Or(\epsi)\,,
\ee
where $\Or(\epsi)$ holds in the norm of  $\B(\Hi)$, the space of
bounded operators on $\Hi$. The operator  $E_n\big( -\I \nabla_x -
A(\epsi x)\big)$ has to be understood in the sense of Weyl
quantization. However, in order to approximate the unitary time-evolution
$\E^{-\I H^\epsi s}\Pi^\epsi_n$ on the decoupled subspace for finite macroscopic times $t=\epsi s$,
the error term in (\ref{Peierls}) is not good enough and
one needs in addition   at least   the terms of order $\epsi$.  For this reason
 we compute explicitly the
asymptotic expansion of the effective Hamiltonian up to first
order terms in $\epsi$.

To state our   theorem relating  the Schr\"odinger equation
(\ref{basic_dynamics}) and the corrected semiclassical model (\ref{Semi1})
we need a few extra notations. By $\Gamma^*$ we denote the
dual lattice of $\Gamma$ and
$\Phi^t_{n}:\R^{2d}\to\R^{2d}$ is the flow corresponding to
  (\ref{Semi1}). After  the change of coordinates $(q,v) \mapsto (q,p) =
(q,v+A(q))$ the same flow is described by
  \[
\overline \Phi^t_{n} (q,p) = \Big(
\Phi^t_{n\,q}\big(q,p-A(q)\big),\,\Phi^t_{n\,v}\big(q,p-A(q)\big)+A(q)\Big)\,.
\]
Our theorem says that   the semiclassical observables are given through
pseudodifferential operators with $\Gamma^*$-periodic symbols and
that the Heisenberg time-evolution of such an observable is
approximated by transporting the symbol along the flow  (\ref{Semi1}) up to an error of order $\epsi^2$.

\begin{theorem} \label{EgCor}

Let $E_n$ be an isolated, non-degenerate Bloch band,
see Definition~\ref{Isodef}, and let the potentials satisfy
Assumption (A$_1$).
Let $a\in C^\infty_{\rm b}(\R^{2d})$ be
$\Gamma^*$-periodic in the second argument, i.e.\ $a(q,p+\gamma^*) = a(q,p)$ for all
$\gamma^*\in\Gamma^*$, and $\widehat a = a( \epsi x,-\I\nabla_x)$ be its Weyl quantization.
Then for each finite time-interval $I\subset \R$ there is a
constant $C<\infty$ such that for $t\in I$
\[
 \left\|\, \Pi^\epsi_n\, \left(\, \E^{\I H^\epsi t/\epsi} \, \widehat a\,\,  \E^{-\I
  H^\epsi t/\epsi}\,-\, \widehat{ a\circ \overline \Phi^{t}_{n} }\, \right)\,\Pi^\epsi_n \,\right\|_{\B(L^2(\R^d))}
  \leq \epsi^2\,C\,.
\]
In particular, for $\psi_0 \in \Pi^\epsi_n \Hi$   we have that
\[
\big|\,\big\langle \psi_0, \,\E^{\I H^\epsi t/\epsi} \, \widehat a\,\,  \E^{-\I
  H^\epsi t/\epsi}\,\psi_0\big\rangle - \big\langle \psi_0,\,\widehat{ a\circ \overline \Phi^{t}_{n} }
  \,\psi_0\big\rangle \,\big| \leq \epsi^2\,C\,\|\psi_0\|^2\,.
\]
\end{theorem}

To our knowledge, Theorem
\ref{EgCor} is the first rigorous result
relating the full time-dependent Schr\"odinger equation
(\ref{basic_dynamics}) to the semiclassical model
(\ref{semiclassical_dynamics}) for general external magnetic and
electric fields and the first result to include the first order correction.

 
We remark that the time-{\em independent} problem  was solved in \cite{GMS}, with predecessors
\cite{BeRa, Bu, HeSj, Ne}. In this case the goal is to obtain an effective Hamiltonian
 with the same spectrum as $H^\epsi$ closed to  some prescribed energy.

We end the
introduction with a brief outline of the paper.  In Section~2 we
discuss the periodic Hamiltonian, i.e.\ (\ref{Hamiltonian})
without $A$ and $\phi$. In particular we
recall the unitary Bloch-Floquet transformation and explain our
assumptions in detail.
In Section~3 we apply the general scheme of space-adiabatic perturbation
theory as developed in \cite{PST1} to the present setting. This
contains the construction of the decoupled subspace and of the
effective Hamiltonian generating the intraband dynamics to all
orders in $\epsi$.
 The key observation for applying
space-adiabatic perturbation theory   is that the Hamiltonian
$H^\epsi$ can be written, after a suitable Bloch-Floquet
transformation, as the Weyl quantization of an operator-valued
symbol. However, the underlying Hilbert space is not of the form
$L^2(\R^d,\Hi_{\rm f})$, as for standard pseudodifferential operators, but
$L^2(B,\Hi_{\rm f})$, where $B$ is the first Brillouin zone, the
fundamental domain of the dual lattice $\Gamma^*$. Our symbols
are not functions on the phase space $\R^d\times \R^d$, but, roughly speaking,
 on $B\times \R^d$. Hence a suitable version of the parameter
dependent pseudodifferential calculus is developed in the
Appendix.
Finally, in Section~4 we  discuss the semiclassical limit of the
effective intraband Hamiltonian and prove, in particular,
Theorem~\ref{EgCor}.


\section{The periodic Bloch Hamiltonian}

In order to formulate our assumptions and our results we first
need to recall several well known facts about the periodic
Hamiltonian
\[
H_{\rm per} := -\frac{1}{2} \Lap + V_\Gamma\,,
\]
i.e.\ about (\ref{Hamiltonian}) without the
non-periodic perturbations $A$ and $\phi$.

The potential $V_\Gamma$ is periodic with respect to
the lattice $\Gamma$.
Its dual lattice $\Gamma^*$ is defined as the
lattice generated by the dual basis
$\{\gamma_1^*,\ldots,\gamma_d^*\}$ determined through the
conditions $\gamma_i\cdot\gamma_j^* = 2\pi \delta_{ij}$,
$i,j\indexd$. The centered \ix{fundamental domain} fundamental
domain of $\Gamma$ is denoted by
\[
 M = \Big\{ x\in\R^d: x=
\textstyle{\sum_{j=1}^d}\alpha_j\,\gamma_j \,\,\,\mbox{for}\,\,\alpha_j\in
[-\textstyle{\frac{1}{2},\frac{1}{2}}]
 \Big\}\,,
\]
 and analogously the centered fundamental domain of $\Gamma^*$ is
denoted by $M^*$. In solid state physics the set $M^*$ is called
\ix{Brillouin zone} the {\em first Brillouin zone}, and for this
reason we will denote it also as $B$. In the following $M^*$ is
always equipped with the {\em normalized} Lebesgue measure denoted by
$\D k$. We introduce the notation $x=[x] +\gamma$ for the a.e.\
unique decomposition of $x\in\R^d$ as a sum of $[x]\in M$ and
$\gamma\in\Gamma$. We use the same brackets for the analogous
splitting $k=[k] + \gamma^*$.

The Bloch-Floquet transform of a function $\psi\in\Sch(\R^d)$
is defined     as
\be \label{BFdef}
(\U\psi)(k,x):=\sum_{\gamma\in\Gamma} \E^{-\I (x+\gamma)\cdot
k}\psi(x+\gamma),\,\,\, (k,x)\in\R^{2d}
\ee
and one directly reads off from (\ref{BFdef}) the following
periodicity properties:
 \be
\label{BF1} \big(\U\psi\big) (k, y+\gamma) = \big( \U\psi\big)
(k,y)\quad \mbox{ for all} \quad \gamma\in\Gamma\,,
\ee
\be
\big(\U\psi\big) (k+\gamma^*, y) = \E^{-\I
y\cdot\gamma^*}\,\big( \U\psi\big) (k,y) \quad\mbox{ for all}
\quad \gamma^*\in\Gamma^*\,. \label{BF2}
\ee

From (\ref{BF1}) it follows that, for any fixed $k\in{\R^d}$,
$\big( \U\psi \big)(k,\cdot)$, is a $\Gamma$-periodic function and
can then be regarded as an element of $L^2(\T^d)$, $\T^d$ being
the flat torus $\R^d/\Gamma$.  

 Equation (\ref{BF2}) involves a unitary
representation of the group of lattice translation on $\Gamma^*$
(denoted again as $\Gamma^*$ with a little abuse of notation),
given by
\[
\tau:\Gamma^*\to\U(L^2(\T^d))\,,\quad\gamma^*\mapsto
\tau(\gamma^*)\,,
\]
where $ \tau(\gamma^*)$ is given by
multiplication with $\E^{\I\,y\cdot\gamma^*}$  in $L^2(\T^d,\D
y)$. It will prove convenient to introduce the Hilbert space
\be
\Hi_\tau :=\Big\{ \psi\in L^2_{\rm loc}(\R^d, L^2(\T^d)):\,\,
\psi(k -\gamma^*) = \tau(\gamma^*)\,\psi(k) \Big\}\,, \label {H
tau}
\ee
equipped with the inner product
\[
\langle \psi,\,\ph\rangle_{\Hi_\tau} = \int_{B}\D k\, \langle
\psi(k),\,\ph(k)\rangle_{L^2(\mathbb{T})}\,.
\]
Notice that if one considers the trivial representation, i.e.\
$\tau \equiv {\bf 1}$, then $\Hi_\tau$  is nothing but a space of
$\Gamma^{*}$-periodic vector-valued functions over $\R^d$.

Obviously, there is a natural isomorphism  between $\Hi_\tau$ and
$L^2(B,L^2(\T^d))$ given by restriction from $\R^d$ to $B$, and
with inverse given by $\tau$-equivariant  continuation, as
suggested by (\ref{BF2}). The reason for working with $\Hi_\tau$
instead of $L^2(B,L^2(\T^d))$ is twofold. First of all it allows
to apply the pseudodifferential calculus as developed in the
Appendix. On the other hand it makes statements about domains of
operators more transparent as we shall see.

The map defined by (\ref{BFdef}) extends to a unitary operator
\[
\U: L^2(\R^d)\to \Ht \cong L^2(B, L^2(\T^d)) \cong L^2(B)\otimes
L^2(\T^d)\,.
\]
   The facts that $\U$ is an isometry and that
$\U^{-1}$ given through
\be\label{BFinv}
\big(\U^{-1}\ph\big)(x) =
\int_{B}\D k\, \,\E^{\I x\cdot k}\,\ph(k,[x])
\ee
satisfies
$\U^{-1}\U\psi = \psi$ for $\psi\in\Sch(\R^d)$ can be checked by
direct calculation. It is also straightforward to check that
$\U^{-1}$ extends to an isometry from $\Ht$ to $L^2(\R^d)$. Hence
$\U^{-1}$ must be injective and as a consequence  $\U$ must be
surjective and thus unitary.

In order to determine the Bloch-Floquet transform for operators
like the full Hamiltonian (\ref{Hamiltonian}),   we need to
discuss how differential and multiplication operators behave under
Bloch-Floquet transformation. The following assertions follow in a
straightforward way from the definition (\ref{BFdef}). Let $P=-\I
\nabla_x$ with domain $H^1(\R^d)$ and $Q$ be multiplication with
$x$ on the maximal domain, then
\begin{eqnarray}\label{P trasf}
\U \,P\, \U^{-1} &=& {{\bf 1}}\otimes -\I \nabla_y^{\rm per} + k\otimes{{\bf 1}} \,,\\
\U \,Q\, \U^{-1} &= & \I\nabla^\tau_k   \,,\label{Q trasf}
\end{eqnarray}
where  $-\I  \nabla_y^{\rm per}$ is equipped with periodic
boundary conditions or, equivalently, operating on the domain
$H^1(\T^d)$. The domain of  $\I\nabla^\tau_k$ is $\Hi_\tau\cap
H^1_{\rm loc}(\R^d, L^2(\mathbb{T}))$, i.e.\ it consists of
distributions in  $H^1(B, L^2(\T^d))$ which satisfy the
$y$-dependent boundary condition associated with (\ref{BF2}).

The central feature of the Bloch-Floquet transformation is,
however, that multiplication with a $\Gamma$ periodic function
like $V_\Gamma$ is mapped into multiplication with the same
function, i.e.\
 $\U\,V_{\Gamma}(x)\,\U^{-1} = {\bf 1} \otimes V_\Gamma(y)$.

For later use we remark that the following relations can be
checked using the definitions (\ref{BFdef}) and (\ref{BFinv}):
\begin{eqnarray*}
\psi\in H^m(\R^d)\,,\,\,m \geq 0 &\quad\Longleftrightarrow\quad &
\U\psi \in L^2(B,H^m(\mathbb{T}))\,, \\
\langle x\rangle^m\psi(x)\in L^2(\R^d) \,,\,\,m\geq 0
&\quad\Longleftrightarrow\quad & \U\psi \in \Hi_\tau\cap H^m_{\rm
loc}(\R^d,L^2(\mathbb{T})) \,.
\end{eqnarray*}


\begin{remark}  Often the Bloch-Floquet transformation is defined
for $\psi\in\Sch(\R^d)$  as
\be \label{BFdef2}
(\widetilde\U\psi)(k,x):=\sum_{\gamma\in\Gamma} \E^{-\I  x \cdot
k}\psi(x+\gamma),\,\,\, (k,x)\in\R^{2d}\,.
\ee
In contrast to (\ref{BFdef}),  functions in the range of\
$\widetilde \U$ are periodic in $k$ and quasi-periodic in $y$:
 \be
\label{BF12} \big(\widetilde\U\psi\big) (k, y+\gamma) = \E^{\I
k\cdot\gamma}\,\big( \widetilde\U\psi\big) (k,y)\quad \mbox{ for
all} \quad \gamma\in\Gamma\,,
\ee
\be
\big(\widetilde\U\psi\big) (k+\gamma^*, y) = \big(
\widetilde\U\psi\big) (k,y) \quad\mbox{ for all} \quad
\gamma^*\in\Gamma^*\,. \label{BF22}
\ee
Our choice of $\U$ instead of $\widetilde \U$ comes from the fact,
that the transform of the gradient has a domain which is
independent of $k\in B$, cf.\ (\ref{P trasf}). This is, as we
shall see,  essential for an application of the pseudodifferential
calculus of the Appendix.\er
\end{remark}



 For the  Bloch-Floquet
transform of the free Hamiltonian one finds
 \[
 \U\,H_{\rm per}\,\U^{-1} = \int_{B}^\oplus\D k\,H_{\rm
per}(k)
\]
with
\be \label{H(k)}
H_{\rm per}(k) = \frac{1}{2}\big( -\I \nabla_y + k\big)^2 +
V_\Gamma(y)\,,\quad k\in B\,.
\ee
For fixed $k\in B$ the operator $H_{\rm per}(k)$ acts on
$L^2(\T^d)$ with domain $H^2(\T^d)$ independent of $k\in B$,
whenever the following assumption on the potential is satisfied.



 \begin{assumption}[A$_1$] We assume that $V_\Gamma$ is
infinitesimally bounded with respect to $-\Lap$ and that $\phi \in
C^\infty_{\rm b}(\R^d,\R)$ and $A_j\in C^\infty_{\rm b}(\R^d,\R)$
 for
any  $j\in\{ 1,\ldots,d \} $.
\end{assumption}

From this assumption it follows in particular that also the full Hamiltonian 
$H^\epsi$ is self-adjoint on $H^2(\R^d)$. The previous assumption excludes
the case of globally constant electric and magnetic field.
However, for the questions we shall address, locally constant
fields serve as well.


The band structure of the fibred spectrum of $H_{\rm per}$ is
crucial for the following and a more detailed discussion can be
found e.g.\ in \cite{Wi}. The resolvent $R_\lambda^0 = (H_0(k)
-\lambda)^{-1}$ of the
 operator $H_0(k)=\frac{1}{2}\big( -\I \nabla_y + k\big)^2$
is compact  for fixed $k\in B$. Since, by assumption, $R_\lambda
V_\Gamma$ is bounded, also $R_\lambda = (H_{\rm per}(k)
-\lambda)^{-1} = R_\lambda^0 + R_\lambda V_\Gamma R_\lambda^0$ is
compact. As a consequence $H_{\rm per}(k)$ has purely discrete
spectrum with eigenvalues
 of finite multiplicity which  accumulate at infinity.
For definiteness the eigenvalues are enumerated according to their
magnitude,  $E_1(k) \leq E_2(k) \leq E_3(k)\leq\ldots$ and
repeated according to their multiplicity. The corresponding
normalized eigenfunctions $\{\ph_n(k)\}_{n\in\N}\subset H^2(\T^d)$
are called Bloch functions \ix{Bloch functions} and form, for any
fixed $k$, an orthonormal basis of $L^2(\T^d)$. We will call
$E_n(k)$ the $n^{\rm th}$ band function. \ix{Bloch bands} Notice
that, with this choice of the labelling, $E_n(k)$ and
$\varphi_n(k)$ are generally \emph{not} smooth functions of $k$ due
to eigenvalue crossings. Since
\be
H_{\rm per}(k-\gamma^*) = \tau(\gamma^*)\,H_{\rm per}(k)
\,\tau(\gamma^*)^{-1}\,, \label{Hper equiv}
\ee
the band functions
$E_n(k)$ are periodic with respect to $\Gamma^*$.




\begin{definition}\label{Isodef}
We say that a band $E_n(k)$ or a group of bands $\{
E_n(k)\}_{n\in\mathcal I}$, $\mathcal{I}= [I_-,I_+]\cap \N $, is
an \textbf{isolated Bloch band} or \textbf{satisfies the gap condition}, if
\[
\inf_{k\in B} {\rm dist}
\Big(\, \bigcup_{n\in\mathcal{I}} \{E_n(k) \},\,
\bigcup_{m\notin\mathcal{I}} \{E_m(k)\}\,\Big)=: C_{\rm g}>0\,.
\]
\end{definition}





  For the following we fix an index set $\mathcal{I}\subset\N$
corresponding to an isolated group of bands.
Let $P_{\mathcal{I}}(k)$ be the spectral projector of $H_{\rm
per}(k)$ corresponding to the eigenvalues $\{
E_n(k)\}_{n\in\mathcal I}$,  then $P_{\mathcal{I}} :=
\int^\oplus_{B}\D k\, P_{\mathcal{I}}(k)$ is  the projector on the
given isolated Bloch band.

In terms of Bloch functions, one has that
 $P_{\mathcal{I}}(k)= \sum_{n \in \mathcal{I}}
|\ph_n(k)\rangle\langle\ph_n(k)|$. However, in general, $\ph_n(k)$
are not smooth functions of $k$ at eigenvalue crossings, while $P_{\mathcal{I}}(k)$ is
 a smooth
function of $k$ because of the gap condition. Moreover, from (\ref{Hper equiv}) it follows that
\[
 P_{\mathcal{I}}(k-\gamma^*) = \tau(\gamma^*)\,
P_{\mathcal{I}}(k) \,\tau(\gamma^*)^{-1}\,.
\]

For the mapping to the reference space we will need the following
assumption.

 \begin{assumption}[A$_2$] \hspace{-1.3pt}If $d>1$ and if
 the isolated group of Bloch bands
$\{E_n\hspace{-1pt}(k)\}_{n\in\mathcal I}$ is degenerate in the sense that $\ell:=|\mathcal{I}| >1$, then  we assume that there exists an
orthonormal basis  $\left\{ \psi _{j}(k) \right\}_{j=1}^{\ell}$ of
${\rm Ran}P_{\mathcal{I}}(k)$ whose elements are smooth and
 $\tau$-equivariant with respect to $k$, i.e.\
 $\psi_j(k-\gamma^*)= \tau(\gamma^*) \psi_j(k)$ for all
 $j\in\{1,\ldots,\ell\}$ and $\gamma^*\in\Gamma^*$.
 \end{assumption}

  In the special but important case in
which the relevant band consist of an isolated $\ell$-fold degenerate
eigenvalue (i.e. $E_n(k)= E_{*}(k)$ for every $n \in \mathcal{I},
\, |\mathcal{I}|= \ell $), Assumption (A$_2$) is equivalent to the
existence of an orthonormal basis consisting of smooth and
$\tau$-equivariant Bloch functions.
In the general case, in
which \textit{eigenvalue crossings} inside the relevant band are
present, Assumption (A$_2$) is weaker, since it is not  required that
$\psi_{j}(k)$ is an eigenfunction of the free
Hamiltonian $H_{\rm per}(k)$, but only of the corresponding
eigenprojection $P_{\mathcal{I}}(k)$.

We expect that Assumption (A$_2$) is generically satisfied, i.e.\  
the eigenvector bundles corresponding to isolated  Bloch bands are trivial.
For $d=1$ this follows from the fact that all $U(n)$-bundles with base space  $S^1$ are trivial.
For $\ell=1$ the bundle must be trivial, since the first Chern class vanishes as a consequence 
of time-reversal invariance. The latter statements are a geometric reformulation of a, infact slightly stronger,
result   by  Nenciu \cite{Ne}. A more detailed study of the geometry of Bloch bundles 
exceeds the scope of this paper and is postponed to a forthcoming paper.

   In the presence of a strong
external magnetic field the Bloch bands split into magnetic
sub-bands. Generically, their first Chern number does not vanish and therefore 
Assumption (A$_2$) fails. As well understood, the nonvanishing of the first Chern number is directly linked to 
the integer quantum Hall effect \cite{TKNN, Si}, hence our interest in extending Theorem 3 to magnetic Bloch bands.
The required modifications of our theory  will be discussed in  \cite{PST3}.


Let $P_n(k)=  |\ph_{n}(k)\rangle\langle\ph_{n}(k)|$, then the
projector on the $n^{\rm th}$ band subspace is given through $P_n
= \int^\oplus_{B}\D k\, P_n(k)$. By construction the band subspaces
are invariant under the dynamics generated by $H_{\rm per}$,
\[
\Big[ \,\E^{-\I  \U H_{\rm per}\U^{-1}\,s},\,P_n\,\Big] = \Big[
\,\E^{-\I  E_n(k) s},\,P_n\,\Big] = 0\quad\mbox{for all}
\,\,n\in\N\,,\,\, s\in\R\,.
\]
Notice, however, that $P_n$ is not a spectral subspace of
$H_{\rm per}$, in general.

According to Proposition \ref{PerObsProp},  in the original
representation $H_{\rm per}$ acts on the $n^{\rm th}$ band
subspace as
\[
H_{\rm per} \psi = \U^{-1} (E_n(k)\otimes{\bf 1})\U \,\psi =
E_n(-\I \nabla_x)\,\psi\,,
\]
where $\psi\in \U^{-1}P_n\,\U \,L^2(\R^d)$. In other words, under the time evolution
generated by the periodic Hamiltonian wave
functions in the $n^{\rm th}$ band subspace propagate freely but
with a modified dispersion relation given through the  $n^{\rm
th}$ band function $E_n(p)$.

\section{Space-adiabatic perturbation for Bloch bands}

In the presence of non-periodic external fields the subspaces $P_n\Hi$
are no longer invariant, since the external fields induce
transitions between different band subspaces. If the potentials
are varying slowly, these transitions are small and one expects
that there  still exist approximately invariant subspaces associated
with isolated Bloch bands. The dynamics of states inside the
decoupled subspaces should be generated by an effective Hamiltonian given
through the Peierls substitution as in (\ref{Peierls}).

In this section we apply the general scheme of adiabatic
perturbation theory as developed in \cite{PST1} to the problem of
perturbed Bloch bands in order to rigorously justify the heuristic
picture.

We first present a theorem which summarizes the main results
of this section. The remaining parts   give the
results and the proofs of the three main steps in space-adiabatic
perturbation theory: In Section 3.1 we construct the almost
invariant subspaces associated with isolated Bloch bands. In
Section 3.2 we explain how to unitarily map the decoupled
subspace  to a suitable reference Hilbert space. In this reference
representation the action of the full Hamiltonian is given through
a semiclassical pseudodifferential operator, whose expansion can
be computed to any order in $\epsi$. This effective Hamiltonian is
constructed in Section 3.3 and we compute its asymptotic expansion
explicitly including the subprincipal symbol. For a detailed
presentation of the general idea we refer to \cite{PST1} and \cite{Te}.
The main
technical innovation necessary in order to apply the scheme to the
present case is the development of a pseudo\-differential calculus
for operators acting on sections of a bundle over the flat torus $B$, or, equivalently,
acting on the space $\Ht$. This is done
in  the Appendix.

Before going into the details of the construction we present a
theorem which encompasses the main results of this section.
Generalizing from (\ref{H tau}) it is convenient to introduce the
following notation. For any separable Hilbert space $\Hi_{\rm f}$
and any unitary representation $\tau:\Gamma^*\to\U(\Hi_{\rm f})$,
one defines the Hilbert space
\[
L^2_\tau(\R^d, \Hi_{\rm f})
:=\Big\{ \psi\in L^2_{\rm loc}(\R^d, \Hi_{\rm f}):\,\, \psi(k
-\gamma^*) = \tau(\gamma^*)\,\psi(k) \Big\}\,,
\]
equipped with
the inner product
\[
\langle \psi,\,\ph\rangle_{L^2_\tau} = \int_{B}\D k\, \langle
\psi(k),\,\ph(k)\rangle_{\Hi_{\rm f}}\,.
\]
Using the results of the previous section and imposing Assumption
(A$_1$), the Bloch-Floquet transform of the full Hamiltonian
(\ref{Hamiltonian}) is given through
\be H^\epsi_{\rm BF} :=
\U\,H^\epsi\,\U^{-1} = \frac{1}{2}\Big( -\I \nabla_y + k -
A\big(\I\epsi\nabla_k^\tau\big)\Big)^2 + V_\Gamma(y) +
\phi\big(\I\epsi\nabla_k^\tau\big) \label{Ham BF}
\ee
with domain $L^2_{\tau}(\R^d, H^2(\T^d))$.

The application of space-adiabatic perturbation theory   to an isolated group
of bands $\{E_n(k)\}_{n\in\mathcal I}$ yields the following result, where
the reference Hilbert space for the effective dynamics is
 $\K := L^2(\T^d) \otimes \C^\ell $ with $\ell:= {\rm dim} P_{\mathcal{I}}(k)$.


\begin{theorem}[Peierls substitution and higher order corrections]
\label{Th main}
 Let $\{E_n\}_{n\in\mathcal{I}}$ be an isolated group of
bands, see\ Definition \ref{Isodef}, and let Assumptions (A$_1$)
and (A$_2$) be satisfied.
Then there exist
\begin{enumerate}
\item
 an orthogonal
projection $\Pi \in \B(\Ht)$,
\item
a unitary map $U \in \B(\Pi\Ht,\K)$, and
\item
a self-adjoint operator $ \widehat{h}  \in \B(\K)$
\end{enumerate}
 such that
\[
\big\|\,\big[ \, H^\epsi_{\rm
BF}, \,\Pi \,\big] \,\big\|= \Or(\epsi^\infty)
\,, \qquad \|\,\Pi - P_\mathcal{I} \,\|=\Or(\epsi)
 \]
and
\[
\big\|\,\big( e^{-i H^\epsi_{\rm BF} t} -U^{*}\ e^{-i \widehat{h}  t}\ U
\big) \Pi\, \big\|=\Or (\epsi ^{\infty }(1+|t|)).
\]
The effective Hamiltonian  $\widehat{h} $ is the Weyl quantization   of a
semiclassical symbol $h  \in S^{1}_{\tau \equiv {\bf 1}}(\epsi,
\B(\C^\ell))$ with    an asymptotic expansion
  which can be computed to any order.
The $\B(\C^\ell)$-valued principal symbol $h_0(k,r)$ has matrix-elements
\be
h_0(k,r)_{\alpha \beta}= \big\langle \psi_\alpha(k-A(r)), H_0(k,r)\, \psi_\beta(k-A(r))\big\rangle\,,
\ee
where $\alpha, \beta \in \{1, \ldots, \ell \}$ and $H_0(k,r)$ is defined in (\ref{H symbol}).
\end{theorem}
The general formula for the subprincipal symbol of the effective Hamiltonian can be found in \cite{PST1}.
The structure and the interpretation of the
effective Hamiltonian are most transparent   for the case of a single isolated band.

\begin{corollary} \label{MainCor}
For an isolated $\ell$-fold degenerate eigenvalue $E
(k)$  the $\B(\C^\ell)$-valued symbol $h (k,r) = h_0(k,r) + \epsi
h_1(k,r) + \Or_0(\epsi^2)$ constructed in Theorem~\ref{Th main} has matrix-elements
\be
 h_0(k,r)_{\alpha \beta}= \big( E(k - A(r)) + \phi(r)\big)
\delta_{\alpha \beta} \label{h0 specialbis}
\ee
 and
\begin{eqnarray}
\label{h1 special}
 \lefteqn{ \hspace{-.5cm}h_1(k,r)_{\alpha
\beta} =  \big( \nabla\phi(r) - \nabla E(\widetilde k) \times B(r)\big) \cdot \A (\widetilde k)_{\alpha
\beta}- B(r)\cdot M(\widetilde k)_{\alpha
\beta}}\\ \nonumber\\
&:=&
\Big( \partial_j \phi(r) -   \partial_l E(\widetilde{k})\,\big( \partial_j A_l(r)
-  \partial_l A_j(r)\big)\Big) \, \A_j(\widetilde{k})_{\alpha\beta}
 \nonumber  \\
&&-   \big(\partial_j A_l  - \partial_l
A_j \big)(r)  \,\, {\rm Re} \left[ {\textstyle \frac{\I}{2}} \,  \big\langle \partial_l\psi_{\alpha}(\widetilde k),
  (H_{\rm per} - E)(\widetilde k) \
\partial_j \psi_{\beta} (\widetilde k) \big\rangle_{\Hi_{\rm f}}\right]\,,   \nonumber 
\end{eqnarray}
 where  summation over indices appearing twice is implicit, $\widetilde k = k - A(r)$, and $\alpha, \beta \in \{1, \ldots, \ell \}$.
  The  coefficients of the Berry connection are
\be\label{5AgeoDef}
 \A (k)_{\alpha \beta} = \I \,\big\langle \psi_{\alpha}(k),
\nabla \psi_{\beta}(k) \big\rangle_{\Hi_{\rm f}} \,.
\ee
 \end{corollary}



In dimension $d=3$ the subprincipal symbol (\ref{h1 special}) has a straight forward
physical interpretation. The 2-forms $B$ and $M$ are naturally
identified with the vectors $B= {\rm curl} A$ and
\[
M(k)_{\alpha
\beta} =  \textstyle{\frac{\I}{2}}\,\big\langle
\nabla\psi_{\alpha}(k), \ \vp (H_{\rm
per}(k) -
E(k))\nabla\psi_{\beta}(k)
\big\rangle_{\Hi_{\rm f}}\,.
\]
Then the symbol of the effective Hamiltonian has a form reminiscent of the
one obtained from a multipole expansion for a classical charge distribution
in weak external fields. In this sense one can interpret $\A (k)$
as an effective electric dipole moment and $M(k)$ as  an effective magnetic
dipole moment. 
We do not know if this analogy carries on to the higher order terms.

\begin{remark}\label{NotRem}
Our results hold for arbitrary dimension $d$. However, to simplify presentation,   
we use a notation motivated
by the vector product and the duality between 1-forms and 2-forms for $d=3$.
If $d\not= 3$, then $B$, $\Omega_n$ and $M_n$ are 2-forms.
The inner product of 2-forms is
\[
B \cdot M :=  *^{-1}(B \wedge *M)    =\sum_{j=1}^d \sum_{i=1}^d  B_{ij} M_{ij}\,,
\]
where $*$ denotes the Hodge duality induced by the euclidian metric,
and for a vector field $w$ and a 2-form $F$ the ``vector product'' is
\[
(w \vp F)_j := (*^{-1}(w \wedge *F))_j = \sum_{i=1}^d  w_i
F_{ij}\,,
\]
where the duality between 1-forms and vector fields was used implicitly.  \er
\end{remark}

  We remark  that E.\ I.\ Blount \cite{Bl1,Bl2}
derived the same effective Hamiltonian for isolated Bloch bands
through a formal diagonalization of the full Hamiltonian in the
Bloch-Floquet representation. It seems that his work remained largely 
unnoticed due to its complexity.





Theorem \ref{Th main}  is a direct consequence of the results
proved in Propositions \ref{Prop Invariant subspace}, \ref{Prop
unitaries} and \ref{Prop Heff}. The proof of Corollary \ref{MainCor}
is given at the end of this section.

 As explained before,
the main idea of the proof is to adapt the general scheme of
space-adiabatic perturbation theory of \cite{PST1} to the case of
the Bloch electron.
While formally this seems straightforward,   one must overcome
two mathematical problems. First of all,   in the present case the
symbols are \emph{unbounded}-operator-valued functions. One can deal with
unbounded-operator-valued symbols by considering them as bounded
operators from their domain equipped with the graph norm into the
Hilbert space, cf.\ e.g.\ \cite{DiSj}.

The second, more serious problem consists in setting up a Weyl
calculus for operators acting on  spaces like
$L^2_\tau(\R^d,\Hi_{\rm f})$.
This is done in  the Appendix and we will use in this section terminology and
notations introduced there.

The results of the Appendix allow us to   write the Hamiltonian $H^\epsi_{\rm BF}$ as the
Weyl quantization $H_0(k,\I\epsi\nabla_k)$ of the $\tau$-equivariant symbol
\be
H_0(k,r)=\frac{1}{2}\left( -\I\nabla _{x}+k- A(r)\right)
^{2}+V_{\Gamma}(x)+ \phi(r) \label{H symbol}
\ee
acting on the Hilbert space $\Hi_{\rm
f}:=L^{2}(\mathbb{T}_{x}^{d},dx)$ with constant domain $\Do :=
H^2(\T^d)$. For sake of   clarity, we spend two more words on
this point. For any fixed $(k,r) \in \R^{2d}$, $H_0(k,r)$ is regarded
as a bounded operator from $\Do$ to $\Hi_{\rm f}$ which is
$\tau$-equivariant with respect to the \emph{bounded}
representation $ \tau_{1} := \tau|_{\Do}$ acting on $\Do$ and the
{\em unitary} representation $ \tau_{2} := \tau$ acting on $\Hi_{\rm
f}$, cf.\  Definition \ref{ABDefequivsymbol}. Then the general
theory developed in  the Appendix can be applied: The usual Weyl
quantization of $H_0$ is an operator from $\Sch'(\R^d, \Do)$ to
$\Sch'(\R^d, \Hi_{\rm f})$ given by
\be \widehat{H}_0 = \frac{1}{2}\Big( -\I \nabla_y + k -
A\big(\I\epsi\nabla_k \big)\Big)^2 + V_\Gamma(y) + \phi
\big(\I\epsi\nabla_k \big) \label{Weyl H}\,.
\ee
 Then $\widehat{H}_0$ can be restricted to $L^2_{\rm loc}(\R^d,
\Do)$, since $A$ and $\phi$ are smooth and bounded.  Since $H_0$ is
a $\tau$-equivariant symbol,   $\widehat{H}_0$ preserves
$\tau$-equivariance and can then be restricted to an operator from
$L^2_{\tau}(\R^d, \Do)$ to $L^2_{\tau}(\R^d, \Hi_{\rm f})$. To
conclude that (\ref{Weyl H}), restricted to $L^2_{\tau}(\R^d,
\Do)$, agrees with (\ref{Ham BF}), it is enough to recall that $\I
\nabla_k^{\tau}$ is defined as $\I \nabla_k$ restricted to $H^1
\cap \Ht$ and to use the spectral calculus.

Moreover, if one introduces the order function $w(k,r) :=
(1+k^2)$, then $H_0 \in S^w_\tau(\B(\Do,\Hi))$. More generally, we will
give the proofs for any symbol $H \in
S^w_\tau(\epsi,\B(\Do,\Hi))$, whose principal symbol is then
denoted by $H_0$.

We now proceed along the lines of the general scheme of \cite{PST1}.
The basic strategy of the proof remains unchanged, but
several important technical details are different.



\subsection{The almost invariant subspace}

In this section we construct the adiabatically decoupled subspace
associated with an isolated group of bands. Similar constructions
have a considerable history and we refer to \cite{MaSo,NeSo,PST1} for
references.

Given an isolated group of bands $\{ E_n(k)\}_{n\in\mathcal I}$,
we change notation and
define   $\pi_{0}(k,r)  = P_{\mathcal{I}}(k - A(r))$.  It
follows from the $\tau$-equivariance of $H_0$ and from the gap
condition  that $\pi_{0} \in S_{\tau}^{1}( \B(\Hi_{\rm f}))$. We
also define the shorthand $A(\epsi)=\Or_0( \epsi^n)$, where the subscript $0$  expresses that
a family   $A(\epsi)\subset\B(\Hi)$ is $\Or( \epsi^n)$ in the norm of
bounded operators.


\begin{proposition}

Let $\{E_n\}_{n\in\mathcal{I}}$ be an isolated group of bands  and let Assumption (A$_1$) be
satisfied. Then there exists an orthogonal projection $\Pi \in
\B(\Ht)$ such that
\be
\big[ \, H^\epsi_{\rm BF}, \,\Pi \,\big] =
\Or_0(\epsi^\infty) \label{HP commutator}
\ee
and $\Pi
=\widehat{\pi }+\Or(\epsi^{\infty })$, where $\widehat{\pi }$ is
the Weyl quantization of a $\tau$-equivariant semiclassical symbol
\[
\pi \asymp \sum_{j\geq 0}\varepsilon ^{j}\pi _{j}\quad
\mathrm{in} \quad S^{1}_{\tau }(\epsi, \B(\Hi_{\rm f}))\,,
\]
whose principal part $\pi _{0}(k,r)$ is the spectral projector of
$H_{0}(k,r)$ corresponding to the given isolated group of bands.
\label{Prop Invariant subspace}
\end{proposition}



\begin{proof}
We first construct $\pi$ on a formal symbol
 level.



\begin{lemma}\label{Lemma Moyal proj}
Let $w(k,r)=(1 + k^2)$. There exists a {\em unique} formal symbol
\[
\pi = \sum_{j=0}^\infty \epsi^j\pi_j \quad \in \,M^{1}_\tau(\epsi,
\B(\Hi_{\rm f}))\cap M^{w}_\tau(\epsi,\B(\Hi_{\rm f},\Do))
\]
such that $\pi_0(k,r) = P_{\mathcal{I}}\big(k-A(r)\big)$ and
\begin{enumerate}
\item $\pi\,\sharp  \,\pi = \pi$,
\item $\pi^* = \pi$,
\item $H\,\sharp  \,\pi - \pi\,\sharp  \, H = 0$.
\end{enumerate}
\end{lemma}


\begin{proof}

We construct the formal symbol $\pi$ locally in
phase space and obtain by uniqueness, which can be proved as in
\cite{PST1}, a globally defined formal symbol.

Fix a point $z_0=(k_0,r_0) \in \R^{2d}$. From the continuity of
the map $z\mapsto H(z)$ and the gap condition it follows that
there exists a neighborhood $\U_{z_0}$ of $z_0$ such that for
every $z\in\U_{z_0}$ the set $\{E_n(z)\}_{n\in\mathcal{I}}$ can be
enclosed by a positively-oriented circle $\Lambda(z_0)\subset \C$
independent of $z$ in such a way that $\Lambda(z_0)$ is symmetric
with respect to the real axis,
\be\label{5Cg}
{\rm
dist}\big(\Lambda(z_0), \sigma(H(z))\big) \geq \frac{1}{4}C_{\rm
g} \quad\mbox{for all} \quad z\in\U_{z_0}
\ee
and
\be\label{5Cr}
{\rm Radius}(\Lambda(z_0))\leq C_{\rm r}\,.
\ee
The constant
$C_{\rm g}$ appearing in (\ref{5Cg}) is the same as in Definition
\ref{Isodef} and the existence of a constant $ C_{\rm r}$
independent of $z_0$ such that (\ref{5Cr}) is satisfied follows
from the periodicity of $ \{E_n(z)\}_{n\in\mathcal{I}}$ and the
fact that $A$ and $\phi$ are bounded. Indeed, $\Lambda$ can be chosen
$\Gamma^*$-periodic, i.e.\ such that $\Lambda(k_0+\gamma^*,r_0)=
\Lambda(k_0,r_0)$ for all $\gamma^*\in \Gamma^*$.

%%%%%%

Let us choose any $\zeta\in \Lambda(z_0)$ and restrict all the
following expressions to $z\in\U_{z_0}$. We will construct a
formal symbol $R(\zeta)$ with values in $\B(\Hi_{\rm f},\Do)$ ---
the local Moyal resolvent of $H$ --- such that
\be\label{5res}
(H-\zeta)\,\sharp  \,R(\zeta)= {\bf 1}_{\Hi_{\rm f}}
\quad\mbox{and}\quad R(\zeta)\,\sharp  \,(H-\zeta) = {\bf 1}_\Do
\quad\mbox{on}\,\,\U_{z_0}\,.
\ee
To this end let
\[
R_0(\zeta) = (H-\zeta)^{-1}\,,
\]
where  according to (\ref{5Cg})  $R_0(\zeta)(z)\in \B(\Hi_{\rm
f},\Do)$ for all $z\in\U_{z_0}$, and, using differentiability of
$H(z)$, $\partial^\alpha_z R_0(\zeta)(z)\in \B(\Hi_{\rm f},\Do)$
 for all $z\in\U_{z_0}$. By construction one has

\[
(H-\zeta)\,\sharp  \,R_0(\zeta) = {\bf 1}_{\Hi_{\rm f}} +
\Or_0(\epsi)\,,
\]
where the remainder is $\Or(\epsi)$ in the $\B(\Hi_{\rm f})$-norm.
We proceed by induction. Suppose that
\[
R^{(n)}(\zeta) = \sum_{j=0}^n \epsi^j R_j(\zeta)
\]
with $R_j(\zeta)(z) \in \B(\Hi_{\rm f},\Do)$ for all $z\in\U_{z_0}
$ satisfies the first equality in (\ref{5res}) up to
$\Or(\epsi^{n+1})$, i.e.
\be
\label{R H} (H-\zeta) \,\sharp  \,R^{(n)} (\zeta) = {\bf 1}_{\Hi_{\rm
f}} + \epsi^{n+1} E_{n+1}(\zeta) + \Or_0(\epsi^{n+2})\,,
\ee
 where
$E_{n+1}(\zeta)(z) \in \B(\Hi_{\rm f})$. By choosing
\be  \label{R n+1}
R_{n+1}(\zeta) =  - R_0(\zeta)\, E_{n+1}
\ee
we obtain that $R^{(n+1)}(\zeta) = R^{(n)}(\zeta) +
\epsi^{n+1}R_{n+1}(\zeta)$ takes values in $\B(\Hi_{\rm f},\Do)$
and satisfies  the first equality in (\ref{5res}) up to
$\Or(\epsi^{n+2})$. Hence the formal symbol $R(\zeta)=
\sum_{j=0}^\infty \epsi^j R_j(\zeta)$ constructed that way
satisfies  the first equality in (\ref{5res}) exactly. By the same
argument one shows that there exists a formal symbol
$\widetilde{R}(\zeta)$ with values in $\B(\Hi_{\rm f}, \Do)$ which
exactly satisfies the second equality in (\ref{5res}). By the
associativity of the Moyal product, they must agree:
\[
\widetilde{R}(\zeta) = \widetilde{R}(\zeta)\, \sharp  
\,(H-\zeta)\,\sharp  \,R(\zeta) = R(\zeta) \qquad \mbox{on  } \U_{z_0}.
\]



  Equations  (\ref{5res}) imply that $R(\zeta)$ satisfies the
resolvent equation
\be\label{5reseq}
R(\zeta) - R(\zeta') = (\zeta-\zeta')\,R(\zeta)
\,\sharp  \,R(\zeta')\quad\mbox{on}\,\,\U_{z_0}
\ee
for any
$\zeta,\zeta'\in \Lambda(z_0)$.  From the resolvent equation it
follows as in \cite{PST1} that the
 $\B(\Hi_{\rm f},\Do)$-valued formal symbol $\pi = \sum_{j=0}^\infty
\epsi^j\pi_j$ defined through
\be\label{5pidef}
\pi_j(z) :=
\frac{\I}{2\pi}\oint_{\Lambda(z_0)} \D\zeta\,
R_j(\zeta,z)\quad\mbox{on}\,\,\U_{z_0}
\ee
satisfies (i) and (ii)
of Lemma \ref{Lemma Moyal proj}. As for (iii) a little bit of care
is required. Let $J:\Do \to\Hi_{\rm f}$ be the continuous
injection of $\Do$ into $\Hi_{\rm f}$. Using (\ref{5pidef}) and
(\ref{5reseq}) it follows that $\pi\, J\,\sharp  \,R(\zeta) =
R(\zeta)\,J\,\sharp  \,\pi$ for all $\zeta\in \Lambda(z_0)$.
Moyal-multiplying from left and from the right with $H-\zeta$ one
finds $H\,\sharp  \,\pi\, J= J\, \pi \,\sharp  \,H$ as operators in
$\B(\Do,\Hi_{\rm f})$. However, by construction $H\,\sharp  \,\pi$ takes
values in $\B(\Hi_{\rm f})$ and, by density of $\Do$, the same
must be true for $\pi\,\sharp  \,H$.

We are left to show that $\pi \in \,M^1_\tau(\epsi, \B(\Hi_{\rm
f}))\cap M_\tau^w(\epsi, \B(\Hi_{\rm f},\Do))$. To this end notice
that by construction $\pi$ inherits the $\tau$-equivariance of
$H$, i.e.\
\[
\pi_j (k-\gamma^*,q) = \tau(\gamma^*)\,\pi_j(k,q)\,
\tau(\gamma^*)^{-1}\,.
\]
From (\ref{5pidef}) and (\ref{5Cr}) we conclude that for each
$\alpha\in\N^{2d}$ and $j \in \N$ one has
 \be \label{5bound1}
\|(\partial^\alpha_z\pi_j)(z)\|\leq 2\pi C_{\rm
r}\,\sup_{\zeta\in\Lambda(z_0)}\| (\partial^\alpha_z
R_j)(\zeta,z)\|\,,
\ee
where $\|\cdot\|$ stands either for the
norm of $\B(\Hi_{\rm f})$ or for the norm of $\B(\Hi_{\rm
f},\Do)$. In order to show that  $\pi \in \,M^{1}_\tau(\epsi,
\B(\Hi_{\rm f}))$ it suffices to consider  $z =(k,r) \in
B\times\R^d$ since $\tau(\gamma^*)$ is unitary and thus the
$\B(\Hi_{\rm f})$-norm of $\pi$ is periodic. According to
(\ref{5bound1}) we must show that
\be \label{Rj goal} \|(\partial_z^\alpha R_j)(\zeta,z)
\|_{\B(\Hi_{\rm f})} \leq C_{\alpha j}\quad \forall \, z\in
\U_{z_0}, \, \zeta\in\Lambda(z_0)
\ee
with $C_{\alpha j}$
independent of $z_0 \in B \times \R^d $.

We prove (\ref{Rj goal}) by induction. Assume, by
induction hypothesis, that for any $j \leq n$ one has that
 \be
\label{Inductive hp} R_j(\zeta) \in S_{\tau}^1(  \B(\Hi_{\rm
f}))\cap S_{\tau}^w( \B(\Hi_{\rm f}, \Do))
\ee
uniformly in $\zeta$,
in the sense that the Fr\'echet semi-norms are bounded by
$\zeta$-independent constants. Then, according to Proposition
\ref{ABProSymbComp}, $E_{n+1}(\zeta)$, as defined by (\ref{R H}),
belongs to $S_{\tau}^{w^2}(  \B(\Hi_{\rm f}))$ uniformly in
$\zeta$. By $\tau$-equivariance, the norm of $E_{n+1}(\zeta)$ is
periodic and one concludes that $E_{n+1}(\zeta) \in S_{\tau}^1(
\B(\Hi_{\rm f}))$ uniformly in $\zeta$. It follows from (\ref{R
n+1}) that (\ref{Inductive hp}) is satisfied for $j = n+1$.

  We are left to show that (\ref{Inductive hp}) is fulfilled
for $j = 0$. We notice that according to (\ref{5Cg}) one has for
all $z\in\R^{2d}$
\[
\|R_0(\zeta)\|_{\B(\Hi_{\rm f})}= \| (H(z) -
\zeta)^{-1}\|_{\B(\Hi_{\rm f})}= \frac{1}{{\rm dist} (\zeta,
\sigma(H(z)))} \leq \frac{4}{C_{\rm g}}\,.
\]
By the chain rule,
\be  \|(\partial_z R_0)(\zeta,z) \|_{\B(\Hi_{\rm
f})} = \| \big( R_0(\zeta) (\partial_z H_0)  R_0(\zeta) \big)(z)
\|_{\B(\Hi_{\rm f})}\,.
\ee
Since $\partial_z H_0 \, R_0(\zeta) $ is
a $\tau$-equivariant $\B(\Hi_{\rm f})$-valued symbol, its norm is
periodic. Therefore it suffices to estimate its norm for $z \in B \times \R^d$, which yields
the required bound. For a general $\alpha \in \N^{2d}$,
the norm of $\partial_z^{\alpha} R_0(\zeta)$ can be bounded in a
similar way. This proves that $R_0(\zeta)$ belongs to $S_{\tau}^1(
\B(\Hi_{\rm f}))$ uniformly in $\zeta$.




  On the other hand
\begin{eqnarray*}
\|R_0(k,r)\|_{\B(\Hi_{\rm f},\Do)}&=& \| (1+\Lap_x)\,
R_0([k]+\gamma^*,r) \|_{\B(\Hi_{\rm f})} \\ &=&  \| (1+\Lap_x)\,
\tau(\gamma^*) R_0([k],r)
\tau^{-1}(\gamma^*) \|_{\B(\Hi_{\rm f})}\\
&\leq&  C \, \| (1 + {\gamma^*}^2) (1+\Lap_x)\, R_0([k],r)
\|_{\B(\Hi_{\rm f})} \\&\leq&  C' (1 + {\gamma^*}^2)  \leq  2C' (1 + k^2)\,,
\end{eqnarray*}
where we used the fact that $ \| (1+\Lap_x) R_0(z) \|_{\B(\Hi_{\rm
f})} $ is bounded for $ z \in B \times \R^d$. The previous
estimate and the fact that $\partial_z H_0 \,\, R_0(\zeta)\in
S_{\tau}^1(  \B(\Hi_{\rm f}))$ yield
\begin{eqnarray*}
\|(\partial_z R_0)(\zeta,z) \|_{\B(\Hi_{\rm f}, \Do)}& = &\| \big(
R_0(\zeta) (\partial_z H_0) R_0(\zeta) \big)(z) \|_{\B(\Hi_{\rm
f}, \Do)}  \\&\leq& C (1 + k^2) \,.
\end{eqnarray*}
Higher order derivatives, are bounded by the same argument,
yielding that $R_0(\zeta)$ belongs to $S_{\tau}^w( \B(\Hi_{\rm f},
\Do))$ uniformly in $\zeta$. This concludes the induction
argument.

From the previous argument it follows moreover that
\be \label{Rj goal2}
\|(\partial_z^\alpha R_j)(\zeta,z) \|_{\B(\Hi_{\rm f},
\Do)} \leq C_{\alpha j} \ w(z)    \quad \forall \ z \in \U_{z_0},
\ \zeta\in\Lambda(z_0)
\ee
with $C_{\alpha j}$ independent of $z_0
\in \R^{2d}$. By (\ref{5bound1}), this implies   $\pi \in
\,M^{w}_\tau(\epsi, \B(\Hi_{\rm f}, \Do))$ and concludes the
proof.
\end{proof}


\textit{Proof of Proposition \ref{Prop Invariant subspace}.}
   From the projector constructed in Lemma \ref{Lemma Moyal
proj} one obtains, by resummation, a semiclassical symbol $\pi \in
S^{1}_\tau(\epsi,\Hi_{\rm f})$ whose asymptotic expansion is given
by $\sum_{j\geq 0}\varepsilon ^{j}\pi _{j}$. Then  according to
Proposition \ref{ABThCaldVailltorus} Weyl quantization  yields a
bounded operator $\widehat{\pi }\in \B(\Ht)$, which is
approximately  a
 projector  in the sense that
\[
\widehat{\pi }^{2}=\widehat{\pi }+\Or_0(\epsi^{\infty })\, \, \,
\mathrm{and\, \, \, }\widehat{\pi }^{*}=\widehat{\pi }\,.
\]

We notice that   Proposition \ref{ABProSymbComp} implies that $ H
\, \widetilde{\sharp  } \, \pi \in S_{\tau}^{w^2}( \epsi,\B(\Hi_{\rm
f}))$. But $\tau$-equivariance implies that the norm is periodic
and then $H \, \widetilde{\sharp  } \, \pi$ belongs to
$S_{\tau}^1(\epsi, \B(\Hi_{\rm f}))$. Then $ \pi \, \widetilde{\sharp  }
\, H = \big( H \, \widetilde{\sharp  } \, \pi \big)^{*}$ belongs to the
same class, so that $[H, \pi]_{\widetilde{\sharp  }} \in
S_{\tau}^1(\epsi,\B(\Hi_{\rm f}))$. This \textit{a priori}
information on the symbol class, together with Lemma \ref{Lemma
Moyal proj}.(iii), assures that
\be
\lbrack \widehat{H},\widehat{\pi } \rbrack =\Or_0(\epsi^{\infty })
\label{pi commutator}
\ee
with the remainder bounded in the $\B(\Ht)$-norm.



In order to get a true projector, we proceed as in \cite{NeSo}.
For $\epsi$ small enough, let
\be \label{Riesz2}
\Pi :=\frac{\I}{2\pi }\int_{|\zeta -1|=\frac{1}{2}}\D\zeta
\,(\widehat{\pi }-\zeta )^{-1}\,.
\ee
Then it follows   that $ \Pi^2=\Pi $, $\Pi
=\widehat{\pi }+\Or_0(\epsi^{\infty })$ and
\[
\| \,\lbrack \widehat{H},\Pi ]\,\| _{\B(\Ht)}\leq C  \|\,
[\widehat{H},\widehat{\pi }]\,\| _{ \B(\Ht)}=\Or (\epsi^{\infty
})\,.
\]
\end{proof}


\subsection{The intertwining unitaries}

After we determined the decoupled subspace associated with an
isolated group of bands, we aim at an effective description of the
dynamics inside this subspace. In order to get a nice and workable
formulation of the effective dynamics, it is convenient to map the
decoupled subspace to a simpler reference space. The natural reference
Hilbert space for the effective dynamics is
$\K := L^2(\T^d) \otimes \C^\ell $ with $\ell:= {\rm dim} P_{\mathcal{I}}(k)$.
Notation will be simpler in the following, if we think of the fibre $\C^\ell$ as
a subspace of $\Hi_{\rm f}$.
In order to construct such a unitary mapping, we reformulate
 Assumption (A$_2$).

 \begin{assumption}[A$_2'$]
 Let
 $\{E_n(k)\}_{n\in\mathcal I}$ be an isolated group of bands and
  let $\pi_\mathrm{r} \in \B(\Hi_{\rm f})$ be an orthogonal projector with  ${\rm dim}\pi_{\rm
  r}=\ell$.
  There is a
unitary-operator-valued map $ u_{0}: \R^{2d} \rightarrow
\U(\Hi_{\rm f}) $ so that
\be
 u_{0}(k,r) \, \pi_{0}(k,r) \, u_{0}^{*}(k,r)=
\pi_{\mathrm{r}} \label{intertwining}
\ee
for any $(k,r) \in
\R^{2d}$,
\be
u_{0}(k+\gamma^*, r)=u_{0}(k,r) \tau (\gamma^* )^{-1} \,,
\label{T-covariance right}
\ee
and $u_{0}$ belongs to $S^1(  \B(\Hi_{\rm f}))$. \er
\end{assumption}

  Clearly,
\be
u_{0}^{*}(k + \gamma^* ,r)=\tau (\gamma^* )u_{0}^{*}(k,r).
\label{T-covariance left}
\ee
  An operator-valued symbol satisfying (\ref{T-covariance
left}) (resp.\ (\ref{T-covariance right})) is called left $\tau
$-covariant (resp.\ right $\tau $-covariant).


The equivalence of (A$_2$) and (A$'_2$) can be seen as follows.
According to Assumption (A$_2$), there exists an orthonormal basis
$\left\{ \psi _{j}(k) \right\}_{j=1}^{\ell}$ of ${\rm
Ran}P_{\mathcal{I}}(k)$ which is smooth and $\tau$-equivariant
with respect to $k$. Let $\pi_{\rm r}:= \pi_0(k_0, r_0)$ for any
fixed point $(k_0, r_0)$. By the gap condition, ${\rm dim}\pi_{\rm
r} = {\rm dim}P_{\mathcal{I}}(k) $. Then for
any orthonormal basis $\left\{ \chi _{j}\right\}^{\ell}_{j=1}$ for
${\rm Ran}\pi_{\rm r}$, the formula
\be
\label{u0 explicit} \widetilde{u}_{0}(k,r):=\sum_{j = 1}^{\ell}
\left| \chi_{j}\right\rangle \left\langle \psi_{j}(k -A(r))\right|
\ee
 defines a partial isometry which can be extended to a unitary
operator $u_0(k,r) \in \U(\Hi_{\rm f})$. The fact that $\left\{
\psi _{j}(k) \right\}_{j=1}^{\ell}$ spans ${\rm
Ran}P_{\mathcal{I}}(k)$ implies (\ref{intertwining}), and the
$\tau$-equivariance of $\psi_j(k)$ reflects in (\ref{T-covariance
right}).

Viceversa, given $u_{0}$ fulfilling Assumption (A$'_2$), one can
check that the formula
\[
\psi_j(k - A(r)) := u_0^*(k,r) \chi_j,
\]
with $\left\{ \chi _{j}\right\} _{j =1}^{\ell}$ spanning ${\rm Ran}
\pi_{\rm r}$, defines an orthonormal basis for ${\rm Ran}
P_{\mathcal{I}}(k)$ which satisfies Assumption (A$_2$).



After these remarks recall that the goal of this section is to
construct a unitary operator which allow us to map the intraband
dynamics from ${\rm Ran}\Pi$ to an $\epsi$-independent reference
space $\K \subset \Href$. Since all the twisting of $\Ht$ has been
absorbed in the $\tau$-equivariant basis 
$\left\{ \psi_{j}\right\}_{j =1}^{\ell}$, or equivalently in $u_0$, the space
$\Href$ can be chosen to be a space of \emph{periodic}
vector-valued functions, i.e.
\[
\Href := L^2_{\tau \equiv {\bf 1}}(\R^d, \Hi_{\rm f}) \cong
L^2(\T^d, \Hi_{\rm f}).
\]
As in \cite{PST1} we introduce the orthogonal projector $\Pi_{\rm r}
:= \hat{\pi}_{\rm r} \in \B(\Href)$ since the effective intraband
dynamics can be described in
\[
\K := {\rm Ran}\Pi_{\rm r} \cong L^2_{\tau \equiv {\bf 1}}(\R^d,
\C^\ell) \cong L^2(\T^d, \C^\ell)
\]
as it will become apparent later on. Recall that
$\ell = {\rm dim}P_{\mathcal{I}}(k) = {\rm dim} \pi_{\rm r}$.


\begin{proposition}
\label{Prop unitaries} Let $\{E_n\}_{n\in\mathcal{I}}$ be an
isolated group of bands, see\ Definition \ref{Isodef}, and let
Assumptions \emph{(A$_1$)} and \emph{(A$'_2$)} be satisfied. Then
there exists a unitary operator $U:$ $  \Ht \rightarrow \Href$ \
such that
\be
U\, \Pi \, U^{*}=\Pi _{\mathrm{r}}  \label{Intertwines}
\ee
and $U=\hat{u}+\Or_0(\epsi^{\infty })$, where $u\asymp \sum_{j\geq
0}\varepsilon ^{j}u_{j}$ belong to $S^{1}(\epsi ,\B(\Hi_{\rm
f}))$, is right $\tau$-covariant at any order and has principal
symbol $u_{0}$.
\end{proposition}

\begin{proof}
   Since
$u_{0}$ is right $\tau $-covariant, one proves by induction that
the same holds true for any $u_{j}$. Indeed, by referring to the
notation in \cite{PST1}, one has that
\[
u_{n+1}=(a_{n+1}+b_{n+1})u_{0}
\]
with $a_{n+1}=-\frac{1}{2}A_{n+1}$ and
 $b_{n+1}=[\pi_{r},B_{n+1}]$. From the defining equation
\[
u^{(n) }\ \sharp  \ u^{(n)*}-1=\varepsilon
^{n+1}A_{n+1}+\mathcal{O}(\varepsilon ^{n+2})
\]
and the induction hypothesis, it follows that $A_{n+1}$ is a
periodic symbol. Then $w^{(n)}:=u^{(n)}+\varepsilon
^{n+1}a_{n+1}u_{0}$ is right $\tau $-covariant. Then the defining
equation
\[
w^{(n) }\ \sharp  \ \pi \ \sharp  \ w^{(n)*}-\pi _{\mathrm{r }}=\varepsilon
^{n+1}B_{n+1}+\mathcal{O}(\varepsilon ^{n+2})
\]
shows that $B_{n+1}$ is a  periodic symbol, and so is $b_{n+1}$.
Hence $u_{j}$ is right $\tau $-covariant, and there exists a
semiclassical symbol $u\asymp \sum_{j}\varepsilon ^{j}u_{j}$ so
that $u\in S^{1}(\epsi,\B(\Hi_{\rm f}))$.

   One notices that right
$\tau$-covariance is nothing but a special case of $(\tau_1,
\tau_2)$-equivariance, for $\tau_2 \equiv {\bf 1}$ and $\tau_1 =
\tau $. Thus it follows from Proposition \ref{ABThCaldVailltorus}
that the Weyl quantization of $u$ is a bounded operator
$\widehat{u}\in \B( \Ht,\Href) $\ such that:

\begin{enumerate}
\item  $\widehat{u}\,\widehat{u}^{*}={\bf 1}_{\Href}+\Or_0(\epsi^{\infty
})$\quad and \quad $\widehat{u}^{*}\widehat{u}= {\bf
1}_{\Ht}+\Or_0(\epsi ^{\infty })$,
\item  $\widehat{u}\,\Pi \, \widehat{u}^{*}=\Pi_{\rm r}+\Or_0(\epsi ^{\infty })$.
\end{enumerate}

Finally we modify $\widehat{u}$ as in \cite{PST1} by an
$\Or_0(\epsi^{\infty})$-term in order to get the   unitary
operator $U\in \U(\Ht, \Href)$.    \end{proof}


\subsection{The effective Hamiltonian}


The  final step in space-adiabatic perturbation theory is to
define and compute the effective Hamiltonian for the intra-band
dynamics. This is done, in principle, by projecting the full
Hamiltonian $H^\epsi_{\rm BF}$ to the decoupled subspace and
afterwards rotating to the reference space.


\begin{proposition}
\label{Prop Heff} Let $\{E_n\}_{n\in\mathcal{I}}$ be an isolated
group of bands and let Assumptions $(A_1)$ and $(A_2)$ be
satisfied. Let $h$ be a resummation in $S^{1}_{\tau \equiv {\bf
1}}(\epsi,\B(\Hi_{\rm f}))$ of the formal symbol
\be
h=u  \, \sharp   \, \pi  \, \sharp   \, H \, \sharp   \, \pi \, \sharp   \, u^{*} \,\in\,
M^{1}_{\tau \equiv {\bf 1}}(\epsi,\B(\Hi_{\rm f})) \,.
\label{heffective2}
\ee
 Then $\widehat h  \in \B(\Href)$,
$ [  \widehat h,\Pi_{\rm r} ] =0$ and
\be
\big( \E^{-\I  H^\epsi_{\rm BF} t} -U^{*}\, \E^{-\I \widehat h
t}\, U \big) \Pi  =\Or_0(\epsi^{\infty }(1+|t|))\,. \label{Heff
unitaries}
\ee
\end{proposition}

\begin{remark}
The definition of the effective Hamiltonian is not entirely unique in the sense
that any $H_{\rm eff}$ satisfying (\ref{Heff unitaries}) would 
serve as well as an effective Hamiltonian. However, the asymptotic expansion of $H_{\rm eff}$
is unique and therefore it is most convenient to define the effective Hamiltonian
through (\ref{heffective2}).\er
\end{remark}

\begin{proof}
In the proof we denote $\Ham$ as $\widehat{H}$ to emphasize the
fact that it is the Weyl quantization of $H \in S_{\tau}^w(\epsi,
\B(\Do, \Hi_{\rm f}))$.

First note that (\ref{heffective2}) follows from the following
facts: according to Lemma  \ref{Lemma Moyal proj} and Proposition
\ref{ABProSymbComp} we have that
\[
\pi\,\sharp  \, H \, \sharp  \, \pi \in M_{\tau}^{w^2} (\epsi, \B(  \Hi_{\rm
f})) = M_{\tau}^{1} (\epsi, \B(  \Hi_{\rm f}))\,,
\]
where we used that $\tau$ is a unitary representation.
With Proposition \ref{Prop unitaries} it follows that $h\in
M_{\tau\equiv 1}^{1} (\epsi, \B(  \Hi_{\rm f}))$. Therefore
$\widehat h \in \B(\Href)$ follows from Proposition
\ref{ABThCaldVailltorus}, while $[ \widehat h,\Pi_{\rm r} ] =0$ is
satisfied by construction.

It remains to check (\ref{Heff unitaries}):
\begin{eqnarray*}
\big(   \E^{-\I  \widehat{H} t}- U^{*}\, \E^{-\I \widehat h t}\,U
 \big) \Pi &=& \big(   \E^{-\I  \widehat{H} t}-  \E^{-\I U^{*} \,\widehat h \,Ut}
 \big) \widehat \pi +\Or_0(\epsi^\infty) \\
 &=&
 \big(   \E^{-\I  \widehat \pi \widehat{H}\widehat \pi t}-  \E^{-\I U^{*} \,\widehat h \,Ut}
 \big) \widehat \pi +\Or_0(\epsi^\infty)\\&=& \Or(\epsi^\infty(1+|t|))\,,
\end{eqnarray*}
where the last equality follows from the usual Duhammel argument and the
fact that the difference of the generators is
$\Or_0(\epsi^\infty)$ in the norm of bounded operators by
construction.
\end{proof}

Since $[ \widehat{h} ,\Pi_{\rm r }] =0$, the effective Hamiltonian
will be regarded, without differences in notation, either as an
element of $\B(\Href)$ or as an element of $\B(\K)$.


We now compute the principal and the subprincipal symbol of $\widehat
h$ for
   the special but most relevant case of an
isolated $\ell$-fold degenerate eigenvalue, i.e. $E_n(k)\equiv E(k)$
for every $n \in \mathcal{I}, \, |\mathcal{I}|= \ell $. Recall that
in this special case Assumption ($A_2$) is equivalent to the
existence of an orthonormal system of smooth and
$\tau$-equivariant Bloch functions corresponding  to the eigenvalue
$E(k)$. If $\ell=1$ then Assumption (A$_2$) is always
satisfied.
The part of $u_0$ intertwining $\pi_0$ and $\pi_{\rm
r}$ is given by equation (\ref{u0 explicit}) where $\psi_j(k)$ are
now Bloch functions, i.e. eigenvectors of $H_{\rm per}(k)$
  to the eigenvalue $E(k)$.


\begin{proof}[Proof of Corollary \ref{MainCor}]
In the following  $h$ is identified with $ \pi_{\rm
r}h  \pi_{\rm r}$ and regarded as a $\B(\C^\ell)$-valued symbol. We
consider the matrix elements
\[
h(k,r)_{\alpha
\beta} := \langle \chi_{\alpha}, h (k,r)
\chi_{\beta} \rangle
\]
for $\alpha, \beta \in \{1, \ldots, \ell
\}$, where we recall that $\chi_\alpha = u_0(k,r) \psi_\alpha(k-A(r))$.
 Equation (\ref{h0 specialbis}) follows immediately from the fact
that $h_0 = u_0 \, H_0 \, u_0^{*}$ and that $\psi_{\alpha}$ are
Bloch functions.  As for $h_1$, we use the general formula
  of \cite{PST1}, which reads applied to the present
setting as
\begin{eqnarray}\label{h1 specialgen}
h_{1\,\alpha \beta }(k,r) &=& -\I\,\big\langle \psi_\alpha(\widetilde k), \,\{E(\widetilde k) +\phi(r) , \psi_\beta(\widetilde k)
\}\big\rangle \nonumber\\&&-\,\textstyle{\frac{\I}{2}}\big\langle \psi_\alpha(\widetilde k) ,\{(H_{\rm per}(\widetilde k)-E(\widetilde k)
),\psi_\beta(\widetilde k) \}\big\rangle \,.
\end{eqnarray}
Here $\{A,\ph\}= \nabla_r A\cdot \nabla_k \ph - \nabla_k A\cdot \nabla_r \ph$ are the Poisson brackets
for an operator-valued function $A(k,r)$ acting on a vector-valued function $\ph(k,r)$.
We need to evaluate (\ref{h1 specialgen}).  Inserting (\ref{u0 explicit}) and
performing a straightforward computation  the first term in
(\ref{h1 specialgen}) gives the first term in (\ref{h1 special})
while the second term contributes to the $\alpha
\beta$ matrix element with
\[
\frac{\I}{2} \sum_{j,l=1}^{d} \big(\partial_j A_l  - \partial_l
A_j \big)(r) \,\big\langle \psi_{\alpha}(\widetilde k),
\partial_l (H_{\rm per} - E)(\widetilde k) \
\partial_j \psi_{\beta} (\widetilde k) \big\rangle_{\Hi_{\rm f}}\,.
\]
The
derivative on $(H_{\rm per} - E)$ can be moved to the first
argument of the inner product by noticing that
\[
0 = \nabla
\big\langle \psi_{\alpha}, (H_{\rm per} - E) \phi \big\rangle =
\big\langle \nabla \psi_{\alpha}, (H_{\rm per} - E) \phi
\big\rangle + \big\langle \psi_{\alpha}, \nabla (H_{\rm per} - E)
\phi \big\rangle
\]
since $\psi_{\alpha}$ is in the kernel of
$(H_{\rm per} - E)$.  Finally the imaginary part of 
\[
\frac{\I}{2} \sum_{j,l=1}^{d} \big(\partial_j A_l  - \partial_l
A_j \big)(r) \,\big\langle\partial_l  \psi_{\alpha}(\widetilde k),
\,(H_{\rm per} - E)(\widetilde k) \
\partial_j \psi_{\beta} (\widetilde k) \big\rangle_{\Hi_{\rm f}}
\]
vanishes, as can be seen  by direct computation, concluding the proof.
\end{proof}



\section{Semiclassical dynamics  for   Bloch electrons}

Up to now we approximated the full quantum mechanical time-evolution
generated by (\ref{Hamiltonian}) by a quantum mechanical time-evolution
on a smaller Hilbert space and with the effective Hamiltonian $\widehat h$ as a simpler
generator.  The effective Hamiltonian is, in general, a pseudodifferential operator
with a matrix-valued symbol. Whenever the isolated group of bands under
consideration contains   just a single eigenvalue of multiplicity
$\ell$, then its principal symbol  is
a scalar multiple of the identity. It is well known, as discussed e.g.\ in
\cite{PST1}, how to perform the semiclassical limit for such
Hamiltonians. 

In this section we take a slightly different attitude and  show
how to incorporate the first order correction into the
$\epsi$-dependent classical flow generated by the dynamical
equations (\ref{Semi1}). This
program is performed in two steps. We first prove an Egorov
theorem for observables in the reference representation and then,
in order to obtain the result for the physical observables   in Theorem~\ref{EgCor}, 
we translate the results from the reference
representation on $\Hi_{\rm ref}$ back to the original
representation on $L^2(\R^d)$. The latter task turns
out to be computationally quite involved. However, since the beautiful equations (\ref{Semi1})
are new, with predecessors in \cite{SuNi}, and since it is not obvious 
how to base their derivation on WKB 
techniques, see
\cite{DGR}, we provide the details of the computation.

For the same reason it is worthwhile, before performing this
program, to show that the semiclassical equations (\ref{Semi1})
are Hamiltonian equations with respect to a suitable
symplectic structure. The Hamiltonian formulation has the
advantage that the existence of global solutions of (4) follows
immediately, and that it becomes straightforward to deal with
questions related to symmetries and conserved quantities.


The dynamical equations (\ref{Semi1}), which define the
$\epsi$-corrected semiclassical model, are given by
\begin{equation} \label{SC model}
\begin{array}{ccc}
    \dot q & = & \nabla_v H_{\rm sc}(q,v)  -   \epsi \, \dot v \times \Omega_n(v)\,, \\
    &&\\
    \dot v & = & \hspace{-3mm}-  \nabla_q H_{\rm sc}(q,v) +   \dot q \times B(q)
\end{array}
\end{equation}
with Hamiltonian
\[ \label{SC energy}
H^{\epsi}_{\rm sc}(q,v)= E_n(v) + \phi(q) - \epsi \, M_n(v) \cdot
B(q).\] 
  Recall that we are using the notation
introduced in Remark \ref{NotRem} and that $B$ and $\Omega_n$ are
 the 2-forms corresponding  to the magnetic field and
to the curvature of the Berry connection, i.e.\ in components 
\[
B(q)_{ij} = \big( \partial_{i} A_j - \partial_j A_i
\big)(q)
\]
for $i,j \indexd$, and  
\[
\Omega_n(v)_{ij} = \big( \partial_{i} \mathcal{A}_j - \partial_j
\mathcal{A}_i \big)(v)\,.
\]
Let
us fix a system of coordinates $z=(q,v)$ in $\R^{2d}$. The
standard symplectic form $\Theta_0 = \Theta_{0}(z)_{lm} \ dz_m
\wedge dz_l $, where $l,m \in\{ 1,\ldots ,2d\}$, has coefficients given by the constant matrix
\[ \label{Symplectic standard} \Theta_{0}(z)= \left(
\begin{array}{cc}
  0 & -\mathbb{I} \\
  \mathbb{I} & 0
\end{array}\right)\,,
\] 
where $\mathbb{I}$ is the identity matrix in
$\mathrm{Mat}(d,\R)$. 
The symplectic form, which turns (\ref{SC model})
into Hamilton's equation of motion for $H_{\rm sc}$,
is given by the  2-form $\Theta_{B,\,
\epsi}= \Theta_{B,\,\epsi}(z)_{lm} \ dz_m \wedge dz_l$ with
coefficients  
\be
\label{Symplectic matrix} \Theta_{B,\, \epsi}(q,v)= \left(
\begin{array}{cc}
  B(q) & -\mathbb{I} \\
  \mathbb{I} &  \epsi \ \Omega_n(v)
\end{array}\right)\,.
\ee 
For
$\epsi=0$ the 2-form $\Theta_{B,\, \epsi}$ coincides with the
magnetic symplectic form $\Theta_{B}$ usually employed to
describe in a gauge-invariant way  the motion of a particle in a
magnetic field (\cite{MaRa}, Section~6.6). For $\epsi$
small enough, the matrix (\ref{Symplectic matrix})   defines a
symplectic form, i.e.\ a closed non-degenerate 2-form. Indeed,
since $\det \Theta_{B} = 1$ it follows that, for $\epsi$ small
enough, $\Theta_{B,\, \epsi}$ is   not degenerate. In
particular it is sufficient to choose
\[
\epsi < \sup_{q,v \in \R^d} \big( \|  B(q) \,
\Omega_n(v) \| + \|  \Omega_n(v) \| \big).
\] The closedness of $\Theta_{B,\, \epsi}$ follows from the fact that
$B$ and $\Omega_n$ correspond to closed 2-forms over $\R^d$.
 
 
 With these definitions the corresponding Hamiltonian equations are
\[ \Theta_{B,\, \epsi}(z) \ \dot{z} = \D H_{\rm sc}(z)\,,
\]
or equivalently
\[
\left(
\begin{array}{cc}
  B(q) & -\mathbb{I} \\
  \mathbb{I} & \epsi \ \Omega_n(v)
\end{array}\right)
\left(
\begin{array}{c}
  \dot q \\
  \dot v
\end{array}\right) =
\left(
\begin{array}{c}
  \nabla_q H(q,v) \\
  \nabla_v H(q,v)
\end{array}\right)\,,
\] 
which agrees with (\ref{SC model}). We notice that
our discussion remains valid if  $\Omega_n$ admits a potential only locally,
as it happens generically for magnetic Bloch bands.


We now turn to the derivation of the semiclassical model (\ref{Semi1}).
Hence we assume that the isolated group of bands
consists of a single non-degenerate Bloch band $E(k)$.  
We start with Egorov's theorem for observables in the reference
space. As the only difference to the standard presentation of
Egorov's theorem, as e.g.\ in \cite{Ro}, we treat the first order
corrections by considering an $\epsi$-dependent Hamiltonian flow
instead of having a separate dynamics for the subprincipal symbol
of an observable.

\begin{proposition} \label{Egorov}
Let $E$ be a simple isolated Bloch band and let $\widehat h$ be
the effective Hamiltonian constructed in Theorem \ref{Th main},
which acts on the reference space $\K = L^2_{\tau \equiv
1}(\R^d)$ of\, $\Gamma^*$-periodic $L^2_{\rm loc}$-functions.
Let $\widetilde \Phi^t: \R^{2d}\to \R^{2d}$ be the
Hamiltonian flow generated by the   Hamiltonian function
\[
h_{\rm cl}(k,r) = h_0(k,r) + \epsi h_1(k,r)\,.
\]
Then for any semiclassical observable $\widehat a =
a_0(k,\I\epsi\nabla_k) + \epsi a_1(k,\I\epsi\nabla_k)$ with $a\in S^{1}(\epsi,\C)$ we
have that
\be
\big\|\, \E^{\I\widehat h t/\epsi}\,\widehat a\,\E^{-\I\widehat h
t/\epsi}\,-\, \widehat{ a \circ \widetilde \Phi^t  }\,\big\|
= \Or(\epsi^2)
\ee
uniformly for any finite interval in time.
\end{proposition}
\begin{proof}
Since the Hamiltonian function is bounded with bounded derivatives,
it follows immediately that $a \circ \widetilde \Phi^t  \in S^{1}(\epsi)$
and that $\frac{\D}{\D t} \,( a \circ \widetilde \Phi^t ) \in S^{1}(\epsi)$.
Therefore the proof is just the standard computation
\begin{eqnarray*}\lefteqn{
\E^{\I\widehat h t/\epsi}\,\widehat a\,\E^{-\I\widehat h
t/\epsi}\,-\, \widehat{ a \circ \widetilde \Phi^t  }    =\int_{0}^{t}\,\D t'\,\frac{\D}{\D
t'}\left( \E^{\I \widehat{h}t'/\epsi }\,  \big( \widehat{ a \circ \widetilde \Phi^{t-t'}  }  \big)   \,\E^{-\I\widehat{h}t'/\epsi
}\right) } \\
&=&\int_{0}^{t}\,\D t'\,\E^{\I\widehat{h}t'/\epsi }\,
\left( \frac{\I}{\epsi }\left[ \,\widehat{h},
  \big( \widehat{ a \circ \widetilde \Phi^{t-t'}  }  \big) \,\right]
 -
 \Big(  \textstyle{\frac{\D}{\D t'}} \,( a \circ \widetilde \Phi^{t-t'})\Big)^{\widehat{\,\,}}\,  \right)
\E^{-\I\widehat{h}t'/\epsi }\,,
\end{eqnarray*}
together with the fact that the integrand is $\Or(\epsi^2)$ in the norm of bounded
operators, since
by construction
\[
 \frac{\D}{\D t'} \,( a \circ \widetilde \Phi^{t-t'} ) =
 \big\{ \, h_{\rm cl},\,a \circ \widetilde \Phi^{t-t'} \big\}
\]
and, computing the expansion of the Moyal product,
\[
\frac{\I}{\epsi }
\left[ \,h ,
     a \circ \widetilde \Phi^{t-t'}  \,\right]_{\widetilde \sharp  }=\big\{ \, h_{\rm cl},\,a \circ \widetilde \Phi^{t-t'}
     \big\}
     +\Or(\epsi^2)\,.
\]
\end{proof}

In order to obtain the Egorov theorem for the physical
observables, we need to undo the transformation to the reference
space and the Bloch-Floquet transformation. We start with the
simpler observation on how the Bloch-Floquet transformation
maps semiclassical observables.


\begin{proposition}\label{PerObsProp}
 Let $a\in S^1(\epsi,\C)$ be
  $\Gamma^*$-periodic, i.e.\ $a(q,p+\gamma^*) = a(q,p)$ for all
  $\gamma^*\in\Gamma^*$.
Let $b(k,r) = a(r,k)$ then $b\in S^1_\tau(\epsi,\C )$
and
\[
 \widehat  a = \U^*\,\widehat b\,\U\,,
\]
where  the Weyl quantization   is in the sense of\, $\widehat
a= a(\epsi x, -\I\nabla_x)$ acting on $L^2(\R^d)$ and $\widehat b = b(k,\epsi\I\nabla_k)$ acting
on $\Ht$.
\end{proposition}



\begin{remark}   
 An analogous statement \emph{cannot} be true for   general
operator-valued $\tau$-equivariant symbols. For example, the
symbol $b(k,r):= H_{\rm per}(k - A(r))$ is $\tau$-equivariant and in
particular a semiclassical observable. However, the corresponding
operator in the original representation is
\[
\U^{*}\,\widehat b \,\U = - \frac{1}{2} \big(-\I \nabla_x -
A(\epsi x)\big)^2 + V_{\Gamma}(x)
\]
which cannot be written as a $\epsi$-pseudodifferential operator
with scalar symbol. \er
\end{remark}

\begin{proof} We give the proof for $a(\cdot,p)\in\Sch(\R^d)$. The
general result follows from standard density arguments, cf.\
\cite{DiSj}.
For $\psi\in \Sch(\R^d)$ we have according to (\ref{ABKer2}) the explicit formula
\be \label{a quantized}
\big(a(\epsi x, -\I  \nabla_x) \psi \big)(x)=
\frac{1}{(2\pi)^{d/2}}\, \sum_{\gamma\in\Gamma}\int_{\R^{d}}
\D\eta   \big( \mathcal{F}a \big) (\eta, \gamma) \ \E^{\I \epsi
(\eta \cdot \gamma)/2} \E^{\I \epsi \eta \cdot x} \psi(x +
\gamma)\,.
\ee
On the other hand for $(\U\psi)(k,r)=: \ph(k,r)$ by definition it
holds that
\be \label{B quantized}
\big(b(k, \I \epsi \nabla_k) \ph \big)(k,r)= \sum_{\gamma
\in \Gamma} \int_{\R^{d}} \D\eta \big( \F b\big) (\gamma, \eta) \
\E^{- \I \epsi (\eta \cdot \gamma)/2} \E^{\I  \gamma \cdot k}
\ph(k - \epsi \eta, r)\,.
\ee
The assumptions on $a$ and $\psi$ guarantee that all the integrals
and sums in the following expressions are absolutely convergent
and thus that interchanges in the order of integration are
justified by Fubini's theorem.

We compute the inverse Bloch-Floquet transform of (\ref{B
quantized}) using (\ref{BFinv}),
\begin{eqnarray} \lefteqn{
 \big( \U^{-1} \widehat b\, \ph \big)(x)=} \label{Last observ} \\
&=& \sum_{\gamma \in \Gamma} \int_{B} \D k  \int_{\R^{d}} \D\eta\,
\big( \mathcal{F}b \big) (\gamma, \eta) \ \E^{\I k \cdot x} \E^{-
\I \epsi (\eta \cdot \gamma)/2} \E^{\I  \gamma \cdot k} \ph(k -
\epsi
\eta, [x]) \nonumber \\
&=& \sum_{\gamma \in \Gamma} \int_{\R^{d}} \D\eta \,\big(
\mathcal{F}b \big) (\gamma, \eta) \E^{\I \epsi (\eta \cdot
\gamma)/2} \E^{\I \epsi \eta \cdot x} \int_{B} \D k \, \E^{\I (k -
\epsi \eta) \cdot (x + \gamma)}  \ph(k - \epsi \eta,
[x])\,.\nonumber
\end{eqnarray}
The $\tau$-equivariance of $\ph$ implies that the function
$f(k,y):= \E^{\I k \cdot y} \ph(k, [y]) $ is exactly periodic in
the first variable. Then the integral in $\D k$ can be shifted by
an arbitrary amount, so that
\[
\int_{B} \D k \, \E^{\I (k - \epsi \eta) \cdot (x + \gamma)} \ph(k
- \epsi \eta, [x]) =  \int_{B} \D k \, \E^{\I k \cdot (x +
\gamma)} \ph(k, [x + \gamma]) = \psi(x + \gamma)\,.
\]
Inserting this expression in the last line of (\ref{Last observ})
and comparing with (\ref{a quantized})   concludes the proof.
\end{proof}


Before we arrive at the proof of Theorem \ref{EgCor}, we must also
understand how the unitary map constructed in Section 3.2 maps
observables in the Bloch-Floquet representation to observables in
the reference representation.

\begin{proposition} \label{traPro}
Let $\widehat b = b_0(k,\epsi\I\nabla_k) +\epsi \,b_1(k,\epsi\I\nabla_k) $ with
symbol $b\in S^1(\epsi,\C)$ which is $\Gamma^*$-periodic in the first argument.
Let $U: \Pi\Ht \to \K$ be the unitary map constructed in Section 3.2.
Then
\[
U\,\Pi\,\widehat b\,\Pi\,U^* = \widehat c + \Or(\epsi^2)\,,
\]
where $c(\epsi,k,r) =\big( b \circ T\big)(k,r)$ with
\[
T:  \R^{2d}\to\R^{2d}\,, \quad (k,r) \mapsto
\Big(k +\epsi\,  \A_m \big(k-A(r)\big)  \nabla A_m(r) ,\,r+\epsi \A
\big(k-A(r)\big)\Big)\,.
\]
Here and in the following summation over indices appearing
twice is implicit.
\end{proposition}

\begin{proof}
In order to compute $c = u\,\sharp  \,\pi\,\sharp  \,b\,\sharp \, \pi\,\sharp  \,u^*$, observe that, since $b$ is
scalar-valued,
the principal symbol remains unchanged, i.e.\ $c_0 = u_0\,\pi_0\,b_0\,\pi_0\,u_0^* =
b_0$. For the subprincipal symbol
we use the general transformation formula (\ref{h1 specialgen}) obtained for the Hamiltonian,
which applies to all operators whose principal symbol commutes
with $\pi_0$. In this case the eigenvalue $E$ in (\ref{h1
specialgen}) must be replaced by the corresponding principal
symbol and a term for the subprincipal symbol $b_1$ must be added, cf.\ \cite{PST1}. Hence we find that
\begin{eqnarray*}
c_1(k,r) &= &- \I \,\big\langle \psi  (k-A(r)) ,\{ b_0(k,r) ,\psi
(k-A(r))
\}\big\rangle \\&& +\, \langle \psi  (k-A(r)) , b_1(k,r)\psi  (k-A(r))\rangle\\
&=& \partial_{k_n} b_0(k,r) \,  \I \,\big\langle \psi  (k-A(r)) , \partial_m \psi
(k-A(r))\big\rangle\,
\partial_n A_m(r)\\
&& +\,\partial_{r_n} b_0(k,r) \,  \I \,\big\langle \psi  (k-A(r)) , \partial_n \psi
(k-A(r))\big\rangle + b_1(k,r)\\&=&
 \partial_{k_n} b_0(k,r) \,\A_m   (k-A(r)) \,
\partial_n A_m(r)  \\&&+\,  \partial_{r_n} b_0(k,r)  \,\A_n  (k-A(r))
 + b_1(k,r)\,,
\end{eqnarray*}
where summation over indices appearing twice is implicit. Now a
comparison with the
Taylor expansion of  $\big( b \circ T \big)(k,r)$ in powers of $\epsi$  proves the
claim.
\end{proof}

We have now all the ingredients needed for the

\begin{proof}[Proof of Theorem \ref{EgCor}]
Let $a\in C^\infty_{\rm b}(\R^{2d})$ be
$\Gamma^*$-periodic in the second argument, then according to
Proposition \ref{PerObsProp} we have
\be\label{trafo3}
\Pi^\epsi_n\,  \E^{\I H^\epsi t/\epsi} \, \widehat a\,\,  \E^{-\I
  H^\epsi t/\epsi}\,\Pi^\epsi_n = \U^*\,\Pi \,  \E^{\I H^\epsi_{\rm BF} t/\epsi} \, \widehat b\,\,  \E^{-\I
  H^\epsi_{\rm BF} t/\epsi}\,\Pi\,\U
\ee
with $b(k,r) = a(r,k)$. With Theorem \ref{Th main} and Proposition \ref{traPro} we find that
\be\label{trafo1}
\Pi \,  \E^{\I H^\epsi_{\rm BF} t/\epsi} \, \widehat b\,\,  \E^{-\I
  H^\epsi_{\rm BF} t/\epsi}\,\Pi = U^*\, \E^{\I \widehat h t/\epsi} \, \widehat c\,\,  \E^{-\I
  \widehat h t/\epsi}\,U + \Or(\epsi^2)\,,
\ee
where $c(\epsi,k,r) =\big( b \circ T\big)(k,r)$. Now we can
apply Proposition \ref{Egorov} to conclude that
\[
 \E^{\I \widehat h t/\epsi} \, \widehat c\,\,  \E^{-\I
  \widehat h t/\epsi} = \widehat{ \big( c\circ \widetilde
  \Phi^t\big) } + \Or(\epsi^2).
\]
Since, for $\epsi$ sufficiently small,  $T$
is a diffeomorphism, we can write
\[
c\circ \widetilde \Phi^t  = c\circ T^{-1}\circ T\circ \widetilde \Phi^t \circ T^{-1}\circ T
=: c\circ T^{-1}\circ \overline \Phi^t \circ T  = b\circ \overline \Phi^t \circ T \,,
\]
where the flow $\overline \Phi^t_\epsi$ in the new coordinates
will be computed explicitly below.
Inserting the results into (\ref{trafo1}), we obtain
\begin{eqnarray*}
\Pi \,  \E^{\I H^\epsi_{\rm BF} t/\epsi} \, \widehat b\,\,  \E^{-\I
  H^\epsi_{\rm BF} t/\epsi}\,\Pi &= &U^*\, \widehat{\big( b\circ \overline \Phi^t \circ T\big)}
   \,U + \Or(\epsi^2)\\
   &=& \Pi \, \widehat{\big( b\circ \overline \Phi^t \big)}\,\Pi  + \Or(\epsi^2)\,,
\end{eqnarray*}
where we used Proposition \ref{traPro} for the second equality.
Inserting into (\ref{trafo3}) we finally find that
\be
\Pi^\epsi_n\,  \E^{\I H^\epsi t/\epsi} \, \widehat a\,\,  \E^{-\I
  H^\epsi t/\epsi}\,\Pi^\epsi_n = \Pi^\epsi_n \, \widehat{\big( a\circ \overline \Phi^t  \big)}\,\Pi^\epsi_n
  + \Or(\epsi^2)\,,
\ee
where we did not make the exchange of the order of the arguments
in $a$ explicit.

Since we can compute the flow only approximately and only through
its vector field, we make use of the following lemma.

\begin{lemma} Let $\Phi_i:\R^{2d} \times \R\to \R^{2d}$ be
the flow associated with the vector field $v_i\in
C^\infty_{\rm b}(\R^{2d}, \R^{2d})$, $i=1,2$.
\begin{enumerate}
\item If for all $\alpha\in\N^{2d}$ there is a $c_\alpha<\infty$ such that
\[
\sup_{x\in\R^{2d}} |\,\partial^\alpha\,(v_1-v_2)(x) | \leq
c_\alpha \,\epsi^2\,,
\]
then  for each bounded interval $I\subset \R$ there are constants $C_{I,\alpha}<\infty$
such that
\be\label{lem1}
\sup_{t\in I, x\in\R^{2d}} |\,\partial^\alpha\,(\Phi^t_1-\Phi^t_2)(x) | \leq
C_{I,\alpha} \,\epsi^2\,.
\ee
\item Let $a\in S^1(\epsi,\C)$. If (\ref{lem1}) holds for the
flows $\Phi_1,\Phi_2$, then there is a constant $C<\infty$, such
that for all $t\in I$
\[
\big\|\, \widehat{a\circ\Phi^t_1} -
\widehat{a\circ\Phi^t_2}\,\big\|_{\B(L^2(\R^d))}\leq C\,\epsi^2\,.
\]
\end{enumerate}
\end{lemma}
\begin{proof}
Assertion (i) is just a simple application of Gronwall's lemma. Assertion
(ii) follows from the fact that the norm of the quantization of a
symbol in $S^1$ is bounded by a constant times the sup-norm of
finitely many derivatives of the symbol, which are $\Or(\epsi^2)$
according to (\ref{lem1}).
\end{proof}


According to assertion (ii) of the lemma  it suffices to show that
\[
\overline \Phi^t  (q,p) = \Big(
\Phi^t_{ n\,q}(q,p-A(q)),\,\Phi^t_{n\,v}(q,p-A(q))+A(q)\Big)+\Or(\epsi^2)
\]
in the above sense, where $\Phi^t_n$ is the flow of (\ref{Semi1}).
And from assertion (i) we infer that it suffices to prove the
analogous properties on the level of the vector fields.

Through a subsequent change of coordinates we aim at computing the
vector field of $\Phi^t_n$ up to an error of order $\Or(\epsi^2)$.
We start with the vector field of $\widetilde \Phi^t$.
The effective Hamiltonian on the reference space including first order
terms reads
\begin{eqnarray} \label{53Ham}
h( r,k)& =&   E(k - A(r)) + \phi(r)\\&& -\, \epsi\,\Big( F_{\rm
Lor}(r, \nabla E (k - A(r)))\cdot \A (k - A(r))  + B(r)\cdot M(k -
A(r))\Big)\,, \nonumber
\end{eqnarray}
with the Lorentz force
\[
F_{\rm Lor}(r, \nabla E(k - A(r))) = -\nabla\phi(r) + \nabla E(k - A(r)) \vp
B(r)\,.
\]
To simplify the computation we switch
to the kinetic momentum $\widetilde k = k- A(r)$. A straightforward 
computation, which is explained below, yields
\begin{eqnarray}
\dot r &=& \nabla E(\widetilde k) - \epsi \nabla_{\widetilde k}
\Big( \A (\widetilde k) \cdot F_{\rm Lor}(r, \widetilde k)
+  B(r)\cdot M(\widetilde k) \Big)\,,\nonumber\\ \label{53E1}\\
\dot{\widetilde k}&=& -\nabla \phi(r) + \dot r\vp B(r) +
\epsi\nabla_r \Big( \A (\widetilde k) \cdot F_{\rm
Lor}(r,\widetilde k)
+ B(r)\cdot M(\widetilde k) \Big)\,. \nonumber
\end{eqnarray}
As the next step we perform the change of coordinates induced by
$T$,
\be\label{53Qtrans}
q = r+ \epsi \A  (\widetilde k )\,,\qquad
 p= \widetilde k -A(r) + \epsi  \nabla_r \big( \A  (\widetilde k )\cdot   A(r)
 \big)\,,
\ee
and then switch to the kinetic momentum
\begin{eqnarray}\label{53Pitrans}
v & =& p - A(q) \nonumber \\&=& \widetilde k  + \epsi   \A_l
(\widetilde k)\nabla A_l  (r)   - \epsi   \A_l
(\widetilde k)\partial_l  A  (r)+\Or(\epsi^2)\nonumber\\
&=& \widetilde k + \epsi \,\A  (\widetilde k)\vp
B(r)+\Or(\epsi^2)\,,
\end{eqnarray}
where we used Taylor expansion.
The  inverse transformations are
\begin{eqnarray*}
r &= &q - \epsi \,\A  (v) +\Or(\epsi^2)\,,
\\
 \widetilde k& = &  v  - \epsi
\,\A  (v)\vp B(q) + \Or(\epsi^2)\,,
\end{eqnarray*}
which   inserted into (\ref{53E1}) yield (\ref{Semi1}).


This concludes our proof. However,
since the corrected semiclassical equations (\ref{Semi1})
constitute a novel result, we supply  the
  details of the computations skipped before.
The canonical equations of motion of the Hamiltonian (\ref{53Ham})
are, componentwise,
\begin{eqnarray*}
\dot r_j & = & \partial_{k_j}  h(r,k)   =   \partial_{k_j} E(
k-A(r)) \\
&& - \epsi \,\partial_{k_j}\Big( F_{\rm Lor}(r,  k -
A(r) )\cdot \A (k -
A(r))  +   B(r)\cdot M(k - A(r))\Big)\,,
\end{eqnarray*}
\begin{eqnarray*}\lefteqn{
\dot k_j  =  -\partial_{r_j} h(r,k) = -\partial_{j}\phi(r) +
\partial_{l } E(k-A(r))\partial_{j} A_l  (r) } \\
&& -\,\epsi\, \partial_{k_l }\Big( \A (k-A(r))\cdot F_{\rm
Lor}(r,  k - A(r) ) + B(r)\cdot M(k-A(r))
\Big)\,\partial_{ j}A_l (r)\\
&& -\,\epsi \,\A_l (k-A(r))
\Big( \partial_j\partial_l \phi(r) -
\big(\nabla E(k-A(r))\vp \partial_j B(r)\big)_l  \Big)
\\&&+\, \epsi\,\partial_j
 B (r)\cdot M (k-A(r)) \,,
\end{eqnarray*}
with the convention to sum over repeated indices.
Substituting $\widetilde k = k- A(r)$ one obtains
\[
\dot r_j    =   \partial_{j} E( \widetilde k) - \epsi\,
\partial_{\widetilde k_j}\Big( F_{\rm Lor}(r, \widetilde k)\cdot \A (\widetilde k)
  + B(r)\cdot M(\widetilde k)\Big)
\]
and
\begin{eqnarray*}
\dot{\widetilde k}_j &=& \dot k_j -\partial_l  A_j(r)\,\dot
r_l \\&=& -\,\partial_{j}\phi(r) +
\partial_{l } E(\widetilde k)\,\partial_{j} A_l  (r) \\&&-\,
\epsi\, \partial_{k_l }\Big( \A (\widetilde k)\cdot
F_{\rm Lor}(r, \widetilde k) + M(\widetilde k)\cdot
B(r)\Big)\,\partial_{ j}A_l (r) \\
&& +\,\epsi \,\A_l (\widetilde
k) \,\partial_{r_j} F_{{\rm Lor\,} l }(r, \widetilde k)
+\, \epsi\,\partial_j
  B (r)\cdot M (k-A(r)) -\partial_l  A_j(r)\,\dot r_l
\\&=&
-\,\partial_{j}\phi(r) + \dot r_l  \Big( \partial_j A_l (r) -
\partial_l  A_j(r)\Big) \\&&+\,\epsi \,\A_l (\widetilde
k)\, \partial_{r_j} F_{{\rm Lor\,} l }(r, \widetilde k) + \epsi\, \partial_j B (r)\cdot M (\widetilde
k)
\\
&=& -\,\partial_{j}\phi(r)  + \big( \dot r\vp B(r)\big)_j +\epsi \,\A_l (\widetilde
k)\, \partial_{r_j} F_{{\rm Lor\,} l }(r, \widetilde k) +
\epsi\, \partial_j B (r)\cdot M (\widetilde k)\,,
\end{eqnarray*}
as claimed in (\ref{53E1}).

Next we   substitute (\ref{53Qtrans}) and
(\ref{53Pitrans}). In the following computations we frequently use
Taylor expansion to first order and   drop terms of order
$\epsi^2$  without notice. In particular in the terms of order $\epsi$ one can replace
$r$ by $q$ and $\widetilde k$ by $v$. We find that
\begin{eqnarray*}
\dot q_j &=& \dot r_j + \epsi \,\dot \A_j(v)\\
&=&  \partial_{j} E( v) -\epsi\,\Big(\A (v)\vp
B(q)\Big)_l  \partial_l \partial_j E(v) \\&& - \,\epsi\,
\partial_{v_j}\Big( \big(-\nabla\phi(q) + \nabla E(v)\vp
B(q)\big)_l  \A_l (v)
  + B(q)\cdot M(v)\Big)\\&& +\,\epsi\partial_l  \A_j
  \dot v_l \\&=&
 \partial_{j} E( v) - \epsi \dot v_l \Big( \partial_j \A_l -\partial_l  \A_j\Big) -\epsi\, B
 (q)\cdot
 \partial_j M (v) \\
 &=&
 \partial_{j} E( v) - \epsi \big(\dot v \vp \Omega(v)\big)_j   -\epsi\, B
 (q)\cdot
 \partial_j M  (v)\,,
\end{eqnarray*}
where we used already that $\dot v = F_{\rm Lor} + \Or(\epsi)$. Thus we
obtained the first equation of (\ref{Semi1}). For the second
equation we find
\begin{eqnarray*}
\dot v_j &= &\dot{\widetilde k}_j + \epsi \frac{\D}{\D
t}\,\Big(\A (v)\vp B(q)\Big) \\
&=&
-\,\partial_{j}\phi(q) + \epsi \A_l (v)\partial_l \partial_j \phi(q)
\\&&+\, \big( \dot q\vp B(q)\big)_j-
\epsi\big( \dot \A (v)\vp B(q)\big)_j - \epsi \Big(\dot q\vp \big(
\A_l (v)\partial_l  B(q)\big)\Big)_j
\\&&
+\,\epsi \,\A_l (v) \partial_{q_j} F_{{\rm Lor\,} l }(q, v) +
\epsi\,\partial_j B  (q)\cdot M (v)\\&&+\,
\epsi\big( \dot \A (v)\vp B(q)\big)_j+ \epsi \Big(\A (v)\vp \big( \dot
q_l \partial_l  B(q) \big)\Big)_j\\ &=&
-\,\partial_{j}\phi(q)+ \big( \dot q\vp B(q)\big)_j +
\epsi\, \partial_j B (q)\cdot M (v)\,,
\end{eqnarray*}
where the fact that
\[
\epsi \,\A(v) \Big( \partial_{q_j} F_{{\rm Lor\,} l }(q,
v)+\partial_l \partial_j \phi(q)\Big) = \epsi\, \A_l
(v) \Big( \dot q\vp \partial_j B(q)\Big)_l  + \Or(\epsi^2)
\]
cancels the remaining two terms is not so obvious, but can   be checked
by direct computation.
\end{proof}



\appendix
\section{Operator-valued Weyl calculus for $\tau$-equi\-var\-iant symbols}

The quantization of symbols which are functions not on the phase
space $\R^{2d}$, but on the cotangent bundle of a more general
configuration manifold is a widely studied topic. However, since geometric quantization
approaches do not yield a Weyl-Moyal calculus, they are of little use
in the adiabatic perturbation theory as developed in \cite{PST1}. On the
other hand,   it is rather straightforward to translate the well
developed theory for the configuration space $\R^{d}$ to a flat
torus by restricting to periodic functions and symbols. This
approach is used by G\'erard and Nier \cite{GeNi} in the context
of scattering theory in periodic media.

In this appendix we present a similar approach to Weyl quantization of
operator-valued symbols which are not exactly periodic, but
$\tau$-equivariant with respect to some nontrivial representation
$\tau$ of the group of lattice translations. We obtain a
pseudodifferential and semiclassical calculus which can be applied
to $\tau$-equivariant symbols like the Schr\"odinger Hamiltonian
with periodic potential in the Bloch-Floquet representation.
In particular, the full computational power of the
usual Weyl calculus is retained. The strategy is  to use the
strong results available for phase space $\R^{2d}$ by restricting
to functions which are $\tau$-equivariant in the configurational
variable.


Let $\Gamma \subset \R^d$ be a regular lattice generated through
the basis $\{\gamma_1,\ldots,\gamma_d\}$, $\gamma_j\in\R^d$, i.e.\
\[
\Gamma =\Big\{ x\in\R^d: x= \textstyle{\sum_{j=1}^d}\alpha_j\,\gamma_j
\,\,\,\mbox{for some}\,\,\alpha \in \mathbb{Z}^d \Big\}\,.
\]
Clearly the translations on $\R^d$ by elements of $\Gamma$ form an
abelian group isomorphic to $\mathbb{Z}^d$. The centered
fundamental cell of $\Gamma$ is denoted as
\[
B=\Big\{ x\in\R^d: x= \textstyle{\sum_{j=1}^d}\alpha_j\,\gamma_j
\,\,\,\mbox{for}\,\,\alpha_j\in
[-\textstyle{\frac{1}{2},\frac{1}{2}}]
 \Big\}\,.
\]


Let $\Hi$ be a separable Hilbert space and let $\tau$ be a
representation of $\Gamma$ in $\B^{*}(\Hi)$, the group of
invertible elements  of $\B(\Hi)$ , i.e.\ a group homomorphism
\[
\tau: \Gamma \to \B^*(\Hi),\qquad  \gamma\mapsto \tau(\gamma)\,.
\]
 If more than one Hilbert space appears, then $\tau$ denotes a
collection of such representations, i.e.\ one on each Hilbert
space. 

\noindent {\bf Warning:} In the application of the results of this appendix to Bloch electrons the lattice $\Gamma$ corresponds 
to the dual lattice $\Gamma^*$ in momentum space $\R^d$,  the extended zone scheme. 
 

Let $L_\gamma$ be the operator of translation by $\gamma\in\Gamma$
on $\Sch(\R^d,\Hi)$, i.e.\ $(L_{\gamma}\ph)(x)= \ph(x-\gamma)$,
and extend it by duality to  distributions, i.e.\ for
$T\in\Sch'(\R^d,\Hi)$ let $(L_\gamma T)(\ph)= T(L_{-\gamma}\ph)$.

\begin{definition}\label{ABequivdef}
A tempered distribution $T \in \Sch'(\R^d,\Hi)$ is said to be {\em
$\tau$-equivari\-ant} if
\[
L_\gamma T= \tau(\gamma) T \quad
\mbox{for all}\,\,\gamma\in\Gamma\,,
\]
where
$\big(\tau(\gamma)T\big)(\ph) = T\big(\tau(\gamma)^{-1} \ph\big)$
for $\ph\in\Sch(\R^d,\Hi)$. The subspace of $\tau$-equivari\-ant
distributions is denoted as $\Sch'_\tau$. Analogously we define
\[
\Hi_\tau = \Big\{ \psi\in L^2_{\rm loc}(\R^d,\Hi): \,
\psi(x-\gamma) = \tau(\gamma)\,\psi(x)\quad \mbox{for
all}\,\,\gamma\in\Gamma\Big\}\,,
\]
which, equipped with the
inner product
\[
\langle \ph,\psi\rangle_{\Hi_\tau} = \int_B \D x\, \langle
\ph(x),\psi(x)\rangle_{\Hi}\,,
\]
is a Hilbert space. Clearly
\[
C^\infty_\tau = \Big\{ \psi\in C^\infty(\R^d,\Hi): \,
\psi(x-\gamma) = \tau(\gamma)\,\psi(x)\quad \mbox{for
all}\,\,\gamma\in\Gamma\Big\}\,,
\]
is a dense subspace of $\Hi_\tau$.\er
\end{definition}

Notice that if $\tau$ is a unitary representation, then for any
$\ph, \psi \in \Hi_\tau$ the map $x\mapsto \langle
\ph(x),\psi(x)\rangle_{\Hi}$ is periodic, since
\[
 \langle  \ph(x-\gamma),\psi(x-\gamma)\rangle_{\Hi}
=  \langle \tau(\gamma) \ph(x), \tau(\gamma)\psi(x)\rangle_{\Hi} =
 \langle  \ph(x),\psi(x)\rangle_{\Hi}\,.
\]
Now that we have $\tau$-equivariant functions, we define
$\tau$-equivariant symbols. To this end we first recall the
definition of the standard symbol classes.

\begin{definition}
A function $w: \R^{2d} \to [0,+ \infty)$ is said to be an
\textbf{order function}, if there exist constants $C_0 > 0$ and
$N_0 > 0$ such that
\[
w(x) \leq C_0 \ \langle x-y \rangle^{N_0} \
w(y)
\]
 for every $x,y \in \R^{2d}$. \label{AADef order}\er
\end{definition}

It is obvious and will be used implicitly that the product of two
order functions is again an order function.

\begin{definition}
A function $A\in C^{\infty}(\R^{2d},\B(\Hi_1, \Hi_2))$ belongs to
the symbol class $S^w(\B(\Hi_1, \Hi_2))$ with order function $w$, if for every $\alpha
,\beta \in \N^{d}$ there exists a positive constant $C_{\alpha
,\beta }$ such that
\be\label{AASeminorms2}
\left\| (\partial_{q}^{\alpha
}\partial_{p}^{\beta}A)(q,p)\right\|_{\B(\Hi_1, \Hi_2)}\leq
C_{\alpha ,\beta }\ w(q,p)
\ee
for every $q,p \in \R^{d}$. \label{AADef Symb}\er
\end{definition}

\begin{definition} \label{AASemiSymbDef}
A map $A:[0,\epsi_{0})\rightarrow S_w(\B(\Hi_1,\Hi_2)),\epsi
\mapsto A_\epsi$ is a semiclassical symbol of
  order $w$, if there exists a sequence
$\{ A_j \}_{j\in \N}\subset A_j\in S^w(\B(\Hi_1,\Hi_2))$ such
that
\[
A \asymp \sum_{j=0}^\infty \epsi^j\,A_j\quad
\mbox{in}\quad S^w(\B(\Hi_1,\Hi_2))\,,
\]
which means that
for every $n\in \N$  and
for all $\alpha,\beta\in \N^d$ there exists a constant
$C_{\alpha ,\beta,n }$ such that for any $\epsi \in [0,\epsi_0)$
one has
\be \label{AASemisymb}
\Big\|\partial_{q}^{\alpha
}\partial_{p}^{\beta} \Big( A_\epsi(q,p) -\sum_{j=0}^{n-1}\,\epsi^j A_j(q,p) \Big) \Big\|_{\B(\Hi_1, \Hi_2)}
 \leq  \epsi^n \, C_{\alpha ,\beta,n }\ w(q,p)\,.
\ee
The space of semiclassical symbols of   order $w$
is denoted as $S^w(\epsi,\B(\Hi_1,\Hi_2) )$ or, if
clear from the context or if no specification is required, as  $S^w(\epsi)$.
The space of formal power series with coefficients in $S^w(\B(\Hi_1,\Hi_2))$ is
denoted as $M^w(\epsi,\B(\Hi_1,\Hi_2) )$.
\end{definition}\er


\begin{definition}
\label{ABDefequivsymbol}  A symbol $A_\epsi \in
S^w(\epsi,\B(\Hi_1,\Hi_2))$ is {\em $\tau$-equivariant} (more
precisely $(\tau_1, \tau_2)$-equivariant), if
\[
A_\epsi(q-\gamma,p) =
\tau_2(\gamma)\,A_\epsi(q,p)\,\tau_1(\gamma)^{-1} \quad \mbox{for
all}\,\,\gamma\in\Gamma\,.
\]
The space of $\tau$-equivariant symbols is denoted as $ S_{\tau}^w
(\epsi, \B(\Hi_1,\Hi_2))$.\er
\end{definition}

 Notice that  the coefficients in the asymptotic expansion of a
$\tau$-equivariant semiclassical symbol must be as well
$\tau$-equivariant, i.e.\ if $A_{\epsi}\asymp
\sum_{j=0}^\infty\epsi^j A_j$, $A_{\epsi}\in
S^w_{\tau}(\epsi,\B(\Hi_1,\Hi_2))$, then $A_j\in S_{\tau}^w(
\B(\Hi_1,\Hi_2))$.




Given any $\tau$-equivariant  symbol $A\in S_{\tau}^w (
\B(\Hi_1,\Hi_2))$, one can consider the usual Weyl quantization
$\widehat A$, regarded as an operator acting on
$\Sch'(\R^d,\Hi_1)$ with distributional integral kernel
\be
K_{A}(x,y)=\frac{1}{(2\pi \epsi)^{d}} \int_{\R^d}\D \xi\,
A\big({\textstyle \frac{1}{2}} (x+y),\xi \big)\ \E^{\I \xi \cdot
(x-y)/\epsi}\, . \label{ABKer2}
\ee
Notice that integral kernel associated to a
$\tau$-equivariant symbol $A$ is $\tau$-equivari\-ant in the
following sense:
\be\label{ABKperiodic}
K_A(x-\gamma,y-\gamma) =
\tau_2(\gamma)\,K_A(x,y)\,\tau_1(\gamma)^{-1} \quad \mbox{for
all}\,\,\gamma\in\Gamma\,.
\ee


The simple but important observation is that the space of
$\tau$-equivariant distributions is invariant under the action of
pseudodifferential operators with $\tau$-equivariant symbols.

\begin{proposition}\label{ABInvProp}
Let $A\in  S_{\tau}^w ( \B(\Hi_1,\Hi_2))$, then
\[
\widehat A\,
 \Sch'_{\tau_1}(\R^d,\Hi_1)
 \subset
\Sch'_{\tau_2}(\R^d,\Hi_2)\,.
\]
\end{proposition}
\begin{proof}
Since $\widehat A$ maps $\Sch'(\R^d,\Hi_1)$ continuously into
$\Sch'(\R^d,\Hi_2)$, we only need to show that $(L_\gamma \widehat
A T)(\ph) = (\tau_2(\gamma) \widehat A T)(\ph)$ for all $T\in
\Sch'_{\tau_1}(\R^d,\Hi_1)$ and $\ph\in\Sch(\R^d,\Hi_2)$.

To this end notice that as acting on $\Sch(\R^d,\Hi_2)$ one finds
by direct computation using (\ref{ABKer2}) that
$\widehat{A^*}\,L_\gamma=L_\gamma
\,(\tau_1(\gamma)^{-1})^*\,\widehat{
  A^*}\,\tau_2(\gamma)^*$. Indeed, let $\psi\in \Sch(\R^d,\Hi_2)$,
then
\begin{eqnarray*}
\big(\widehat{ A^*}\,L_\gamma\,\psi\big)(x) &=& \int_{\R^d} \D y\,
K_{A^*} (x,y) \,\psi(y-\gamma) = \int_{\R^d} \D y\, K_{A^*}
(x,y+\gamma)
\,\psi(y) \\
&=&  \int_{\R^d} \D y\, (\tau_1(\gamma)^{-1})^*\, K_{A^*}
(x-\gamma,y)\, \tau_2(\gamma)^*
\,\psi(y)\\
&=& \big( L_\gamma\, (\tau_1(\gamma)^{-1})^*\,\widehat{ A^*}\,
\tau_2(\gamma)^*\,\psi\Big)(x)
\end{eqnarray*}
Hence, using the fact that $\tau$ is a representation and that
$L_\gamma T=\tau_1(\gamma)T$,
\begin{eqnarray*}
(L_\gamma \widehat A T)(\ph) &=& T(\widehat A^*\,L_{-\gamma}\,\ph)
= T( L_{-\gamma}\,\tau_1(\gamma)^*\,\widehat
A^*\,(\tau_2(\gamma)^{-1})^*\,\ph)
\\& =& (\tau_2(\gamma)\,\widehat A\,\tau_1(\gamma)^{-1}\,L_\gamma
\,T)(\ph)=
 (\tau_2(\gamma)\,\widehat A\,T)(\ph)\,.
\end{eqnarray*}
  \end{proof}

For the convenience of the reader we also recall the definition
and the basic result about the Weyl product of semiclassical
symbols. For a proof see e.g.\ \cite{DiSj}.

\begin{proposition}\label{AAPropMoyalproduct}
Let $A\in S^{w_1} (\epsi, \B(\Hi_2,\Hi_3))$ and $B\in S^{w_2}
(\epsi,\B(\Hi_1,\Hi_2))$, then $\widehat A\widehat B = \widehat C$, with
$C\in S^{w_1 w_2}(\epsi ,\B(\Hi_1,\Hi_3))$ given through
\be \label{AACDEF}
C(\epsi,q,p) = \exp\left( \frac{\I\,\epsi}{2}\,(\nabla_p\cdot \nabla_x -
\nabla_\xi\cdot\nabla_q )\right)
A(\epsi,q,p)B(\epsi,x,\xi)\Big|_{x=q,\xi=p} \hspace{-1mm}=: A\,\widetilde\sharp  \,B\,.
\ee
\end{proposition}

The corresponding product on the level of the formal power series
is called Moyal product and denoted as
\[
\sharp  :M^{w_1} (\epsi, \B(\Hi_2,\Hi_3))\times
M^{w_2}(\epsi,\B(\Hi_1,\Hi_2)) \to M^{w_1 w_2}(\epsi
,\B(\Hi_1,\Hi_3))\,.
\]
The $\tau$-equivariance of symbols is preserved
under the pointwise product, the Weyl product and the Moyal
product.

\begin{proposition}\label{ABProSymbComp}
Let $A_\epsi \in  S^{w_1}_{\tau} (\epsi, \B(\Hi_2,\Hi_3))$ and
$B_\epsi \in S^{w_2}_{\tau} (\epsi, \B(\Hi_1,\Hi_2))$, then
$A_\epsi B_\epsi\in S^{w_1 w_2}_{\tau} (\epsi , \B(\Hi_1,\Hi_3))$
and $A_\epsi\,\widetilde{\sharp  }\,B_\epsi \in S^{w_1 w_2}_{\tau}
(\epsi, \B(\Hi_1,\Hi_3))$.
\end{proposition}

\begin{proof}
One has
\begin{eqnarray*}
A_\epsi(q-\gamma,p)B_\epsi(q-\gamma,p) &=&\tau_3(\gamma)
A_\epsi(q,p)\tau_2(\gamma)^{-1}\tau_2(\gamma)
B_\epsi(q,p)\tau_1(\gamma)^{-1}
\\&=&
\tau_3(\gamma) A_\epsi(q,p) B_\epsi(q,p)\tau_1(\gamma)^{-1}\,,
\end{eqnarray*}
which shows  $A_\epsi B_\epsi\in  S^{w_1 w_2}_{\tau} (\epsi,
\B(\Hi_1,\Hi_3))$ and inserted into (\ref{AACDEF}) yields
immediately also $A_\epsi\,\widetilde\sharp  \,B_\epsi \in S^{w_1
w_2}_{\tau} (\epsi, \B(\Hi_1,\Hi_3))$.
 \end{proof}
  An analogous
statement holds for the Moyal product of formal symbols.

A not completely obvious fact is the following variant of the
Calderon-Vaillancourt theorem.

\begin{theorem} \label{ABThCaldVailltorus}
Let $A\in S^{1}_{\tau}(\B(\Hi))$ and $\tau_1, \tau_2$   unitary
representations of $\Gamma$ in $\B(\Hi)$, then $\widehat A\in
\B(\Hi_{\tau_1},\Hi_{\tau_2})$ and for $A_\epsi \in
S^{1}_{\tau}(\epsi, \B(\Hi))$ we have that
\[
\sup_{\epsi\in[0,\epsi_0)} \|\widehat
A_\epsi\|_{\B(\Hi_{\tau_1},\Hi_{\tau_2} )}<\infty\,.
\]
\end{theorem}
\begin{proof}
Fix $n>d/2$ and let $w(x) = \langle x\rangle^{-n}$. We consider
the weighted $L^2$-space
\[
L^2_w = \left\{ \psi\in L^2_{\rm loc}(\R^d,\Hi):\,\,\int_{\R^d}\D
x\, w(x)^2 |\psi(x)|^2 <\infty\right\}\,.
\]
Let $j=1,2$, then $\Hi_{\tau_j}\subset  L^2_w$ and for any
$\psi\in \Hi_{\tau_j}$ one has the norm equivalence
\be\label{ABNE}
C_1\, \|\psi\|_{\Hi_{\tau_j}} \leq \|\psi\|_{L^2_w} \leq C_2 \,
\|\psi\|_{\Hi_{\tau_j}}
\ee
for appropriate constants $0<C_1,C_2<\infty$. The first inequality
in (\ref{ABNE}) is obvious and the second one follows by
exploiting $\tau_j$-equivariance of $\psi$ and unitarity of
$\tau_j$:
\begin{eqnarray*}
\|\psi\|_{L^2_w}^2 &=& \sum_{\gamma\in\Gamma}\int_{B+\gamma}\D x\,
w(x)^2\,  \|\tau_j(\gamma)^{-1}\psi(x)\|^2_\Hi =
\sum_{\gamma\in\Gamma}\int_{B+\gamma}\D x\,
w(x)^2 \, \|\psi(x)\|^2_\Hi\\
&\leq& \sum_{\gamma\in\Gamma} \sup_{x\in B+\gamma}\left\{
w(x)^2\right\}\,\int_B\D x\,\|\psi(x)\|^2_\Hi\leq
C_2\,\|\psi\|_{\Hi_{\tau_j}}\,.
\end{eqnarray*}
According to (\ref{ABNE}) it suffices to show that $\widehat
A\in\B(L^2_w)$ and to estimate the norm of $\widehat A_\epsi$ in
this space.

Let $\psi\in C^\infty_{\tau_1}(\R^d,\Hi)$, then by the general
theory $\widehat A \psi$ is smooth as well (cf.\ \cite{Fo},
Corollary 2.62) and thus, according to Proposition
\ref{ABInvProp}, $\widehat A \psi\in C^\infty_{\tau_2}(\R^d,\Hi)$.
Hence we can use (\ref{ABNE}) and find
\[
\big\| \widehat A\psi \big\|_{L^2_w} = \big\| w \widehat A\psi
\big\|_{L^2} \leq
 \big\| w \widehat A w^{-1} \big\|_{\B(L^2)} \big\|w\psi\big\|_{L^2} =
\big\| w \widehat A w^{-1} \big\|_{\B(L^2)}
\big\|\psi\big\|_{L^2_w}\,.
\]
However,  by Proposition \ref{AAPropMoyalproduct}, we have that
$w\,\widetilde\sharp  \,A\,\widetilde\sharp  \,w^{-1}\in S^{1}(\epsi,
\B(\Hi))$. Thus from the usual Calderon-Vaillancourt theorem it
follows that
\[
 \big\| w \ \widehat A \ w^{-1} \big\|_{\B(L^2)} \leq C_d\, \big\|
\, w\,\widetilde\sharp  \,A\,\widetilde\sharp  \,w^{-1} \big\|_{C^{2d+1}_{\rm
b}(\R^{2d})}\,.
\]
This shows that for $A\in S_{\tau}^{1}(\B(\Hi))$ we have
$\widehat A\in \B(\Hi_{\tau_1},\Hi_{\tau_2})$. Since also
$w\,\widetilde\sharp  \,A_\epsi\,\widetilde\sharp  \,w^{-1}\in S^{1}(\epsi,
\B(\Hi))$ for $A_\epsi \in S^{1}_{\tau}(\epsi,\B(\Hi))$, we
conclude that
\[
\sup_{\epsi\in[0,\epsi_0)} \|\widehat
A_\epsi\|_{\B(\Hi_{\tau_1},\Hi_{\tau_2})}<\infty
\]
by the same argument.
\end{proof}



\begin{remark} It is clear from the proof that   the previous
result still holds true under the weaker assumption that $\tau_1$
and $\tau_2$ are uniformly bounded, i.e. that
\[ \sup_{\gamma \in \Gamma} \|
\tau_j(\gamma) \|_{\B(\Hi)} \leq C\,,\quad j=1,2\,.
\]\er
\end{remark}


Finally we would also like to show that for $A\in S_{\tau}^1
(\B(\Hi))$ the adjoint of $\widehat A$ as an operator in
$\B(\Hi_\tau)$, denoted by $\widehat A^\dagger$, is  given through
the quantization of the pointwise adjoint, i.e.\ through
$\widehat{A^*}$. Here it is crucial that $\tau$ is a unitary
representation.

\begin{proposition}
Let  $S_{\tau}^1( \B(\Hi))$ with a unitary representation $\tau$
(with $\tau_1=\tau_2=\tau$) and let
 $\widehat A^\dagger$ be the adjoint of $\widehat A\in\B(\Hi_\tau)$,
 then $\widehat A^\dagger = \widehat{A^*}$.
\end{proposition}
\begin{proof}
Let $\psi\in \Hi_\tau$ and $\ph \in C^\infty_\tau$ such that
$\widetilde \ph := {\bf 1}_B\,\ph \in C^\infty_0(\R^d,\Hi)$, where
${\bf 1}_B$ denotes the characteristic function of the set $B$.
Such $\ph$ are dense in $\Hi_\tau$ and the corresponding
$\widetilde \ph$ can be used as a test function:
\begin{eqnarray*}
\big\langle\ph, \widehat A\psi\big\rangle_{\Hi_\tau} &=& \int_B\D
x\,\big\langle \ph(x),\, (\widehat A\psi)(x)\big\rangle_{\Hi} =
\int_{\R^d} \D x\, \big\langle \widetilde \ph(x),\,  (\widehat
A\psi)(x)\big\rangle_{\Hi}\\
&=& \int_{\R^d} \D x\, \big\langle (\widehat{A^*} \widetilde
\ph)(x),\,
\psi(x)\big\rangle_{\Hi}\\
&=& \int_{\R^d} \D x\, \Big\langle \int_{\R^d}\D y\, K_{A^*}(x,y)
\widetilde
\ph(y),\,\psi(x)\Big\rangle_{\Hi}\\
&=& \int_{\R^d} \D x\, \Big\langle \int_B\D y\, K_A^*(x,y)
\widetilde
\ph(y),\,\psi(x)\Big\rangle_{\Hi}\\
&=& \int_B \D x\,\sum_{\gamma\in\Gamma}
 \Big\langle \int_B\D y\, K_A^*(x+\gamma,y) \widetilde
\ph(y),\,\psi(x+\gamma)\Big\rangle_{\Hi}\\
&=& \int_B \hspace{-1pt}\D x\,\sum_{\gamma\in\Gamma}
 \Big\langle \hspace{-1pt}\int_B\hspace{-1pt}\D y\,\tau^{-1}\hspace{-1mm}(\gamma)
K_A^*(x,y-\gamma)\tau(\gamma)
\widetilde \ph(y),\,\tau^{-1}(\gamma)\psi(x)\Big\rangle_{\Hi}\\
&=& \int_B \D x\,\sum_{\gamma\in\Gamma}
 \Big\langle \int_B\D y\, K_A^*(x,y-\gamma)
 \ph(y-\gamma),\,\psi(x)\Big\rangle_{\Hi}\\
&=& \int_B \D x\,
 \Big\langle \int_{\R^d}\D y\, K_A^*(x,y)
 \ph(y),\,\psi(x)\Big\rangle_{\Hi}\\
&=& \int_B \D x\,
 \big\langle (\widehat{A^*}\ph)(x),\,\psi(x)\big\rangle_{\Hi}
= \big\langle\widehat{ A^*}\ph, \psi\big\rangle_{\Hi_\tau}
\end{eqnarray*}
In particular, we used the $\tau$-equivariance of the kernel
(\ref{ABKperiodic}) and of the functions in $\Hi_\tau$ and the
unitarity of $\tau$. By density we have $\widehat{A^*} = \widehat
A^\dagger$.
 \end{proof}

\noindent{\bf Acknowledgements.} G.\ P.\  is grateful for financial support
by the Research Training Network HYKE of the European Union and
by the  Priority Program  ``Analysis, Modeling and Simulation of Multiscale Problems''
of the Deutsche  Forschungsgemeinschaft.
 




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