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\markboth{Marco Lenci}{Ergodic Properties of the Quantum Ideal Gas ...}
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\newcommand{\td} {thermodynamical limit}
\newcommand{\psd} {pseudo-differential}
\newcommand{\tdlim} {\lim_{m,L\to\infty \atop m/L \to\rho}}
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\newcommand{\ra} {\rangle}
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\begin{document}
\title{
Ergodic Properties of the Quantum Ideal Gas in the
Maxwell-Boltzmann Statistics
}
\author{
Marco Lenci \thanks{On leave at: Mathematics Department, Princeton
University, Princeton, NJ 08544, U.S.A. E-mail:
marco@math.princeton.edu} \\
Dipartimento di Matematica \\
Universit\`a di Bologna \\
40127 Bologna \\
Italy \\
}
\date{April 1996}
\maketitle
\begin{abstract}
It is proved that the quantization of the Volkovyski-Sinai model
of ideal gas (in the Maxwell-Boltzmann statistics) enjoys at the
\td\ the property of quantum \mix\ in the following sense:
\begin{displaymath}
\lim_{|t|\to\infty} \tdlim \omega_{\beta,L}^{m}
(e^{iH_{m}t/\hbar} A e^{-iH_{m}t/\hbar} \, B) =
\tdlim \omega_{\beta,L}^{m}(A) \cdot
\tdlim \omega_{\beta,L}^{m}(B).
\end{displaymath}
Here $H_m$ is the Schr\"odinger operator of $m$ free
particles moving on a circle of length $L$; $A$ and $B$ are
the Weyl quantization of two
classical observables $a$ and $b$;
$\omega_{\beta,L}^{m}(A)$ is the corresponding quantum Gibbs state.
Moreover one has
\begin{displaymath}
\tdlim \omega_{\beta,m} (A) = P_{\rho,\beta}(a)
\end{displaymath}
where $P_{\rho,\beta}(a)$ is the classical Gibbs measure.
\par
The consequent notion of quantum \erg ity is also independently
proven.
\end{abstract}
\section{Introduction.} \label{sec-intro}
\sectcount
The purpose of the present paper is to analyze the quantum \erg\
properties of the Volkovyski-Sinai model of ideal gas, quantized
according to a Maxwell-Boltzmann statistics (i.e. all particles are
distinguishable). This paper represents the companion paper of
\cite{gm} where the same result is proved for a strongly analogous
system, namely the infinite harmonic chain with suitable restrictions
on the normal mode frequencies.
\par
These two systems provide examples of {\em kinematic quantum chaos}.
We borrow the expression {\em kinematic chaos} from the enlightening
paper by \cite{jlp}. By that we mean trivial motion whose chaotic
behavior is due to the randomness of the infinite-dimensional initial
condition (see, besides \cite{jlp} itself, the remark after
Theorem \ref{thm-erg}). For both the ideal gas and the
harmonic chain, this is a classical feature of the system, which
works on the quantum level as well because of the {\em exact Egorov
Theorem} (Lemma \ref{lemma-q-evol-c}). The latter is a fundamental
fact here, since it allows us to treat the quantum time evolution as the
classical one.
\par
A clear explanation of the motivations for the investigation of the
quantum behavior of infinite systems is given in \cite{gm}, as well
as further references. Here we just sketch what the immediate problems
are, concerning the search for {\em quantum chaos}.
\par
To fix the ideas, let $H$ be a self-adjoint operator acting on
$L^{2}(\Omega), \Omega\in \R^{m}$, resulting from the quantization of
some classical Hamiltonian \fn\ ${\cal H}$ over $\R^{m}\times\Omega$.
Suppose, as it happens in all interesting cases, that $\sigma(H)$ is
discrete. Consider two operators $A,B \in {\cal L}(L^{2}(\Omega))$,
which is regarded as our set of observables, and define $\Theta[t](A) :=
e^{iHt/\hbar} A e^{-iHt/\hbar}$, the Heisenberg evolution. All the
physical experiments one can do on such a system imply a certain
``measure'' on the observables is used. In quantum mechanics such
measures are {\em states} over the algebra of the operators. These
{\em quantum ensembles} (\cite{r}, \S1.3) are typically described as
\begin{equation}
\omega_{\varrho}(A) = \frac{{\rm Tr} \, A\, \varrho}
{{\rm Tr} \,\varrho},
\end{equation}
with $\varrho$ a suitable trace-class operator. A suitable definition of
\mix\ would be, then (\cite{br}, \cite{b}, \cite{gm})
\begin{equation}
\lim_{|t|\to\infty} \omega_{\varrho}(\Theta[t](A),\, B) =
\omega_{\varrho}(A) \: \omega_{\varrho}(B),\ \ \
\forall A,B \in {\cal L}(L^{2}(\Omega)).
\end{equation}
It is easily realized, writing $\omega_{\varrho}$ by means of the
matrix elements of the operators, w.r.t. the eigenbasis of $H$, that
such a property can never be verified, {\em for each underlying
classical dynamics}. The same is true for any reasonable definition
of \erg ity, as Von Neumann's formula (\cite{vn}) shows (see \cite{gm}
for details). This is a consequence of the quasi-periodicity of the
classical evolution -- as long as we have finite degrees of freedom --
and it is called ``quantum suppression of classical chaos''
(\cite{jlpc}, \cite{jlp}). Hence the idea of taking the number of
degrees of freedom to infinity.
\par
The system we consider is the quantization of the ideal gas in the
formulation given in \cite{s}: i.e. $m$ non-interacting particles moving
freely on a circle of length $L$, when $m$ and $L$ are taken to infinity,
subject to the finite density requirement $m/L \to\rho$.
\par
For an outline of the ``analytic approach'' we will follow in studying
the quantum infinite system in question, the reader is
definitely referred to the introduction of the companion paper
\cite{gm}, due to the similarities of the two works. There the
consequences of such a study are also properly emphasized.
In the next paragraph we just point out the structural differences,
between the two models, which require non obvious modifications of
the arguments of \cite{gm} valid for the harmonic chain.
Specifically
\begin{itemize}
\item[(i)] The other important classical mechanism which provides the
unpredictability (mixing) of the time evolution here, besides the
mentioned kinematic effect, is the {\em symmetry} of the
observables under particle exchange. This corresponds to the physical
fact that one is not able to distinguish between the particles in a gas.
Actually, such a restriction on the observables also has the noticeable
outcome to allow the interchange of the time average limit with the \td\
(see Theorem \ref{thm-erg}, Corollary \ref{cor-erg}, and relation
(\ref{thm-erg-1})). The remark after Theorem \ref{thm-erg} will
contain more comments. On the quantum side, the simmetry of the
observables would entail for a Bose-Einsten or a Fermi-Dirac statistics.
Nevertheless, we use Maxwell-Boltzmann for the sake of convenience: see
remark 2 in \S\ref{subs-quant}.
\item[(ii)] The phase space of any $m$-particle subsystem is
$\R^{m}\times(L\,S^{1})^{m}$, with $L\,S^{1}$ denoting henceforth the
circle of radius $L$. In other words the phase space has a cylindrical
structure. This has the effect of making necessary a Bloch
decomposition of $L^2(\R^{m})$, and consequently a direct fiber
decomposition of operators, if we want to consider functions on phase
space as symbols of operators on a one-to-one correspondence under
quantization. Equivalently, $\R^{m}\times(L\,S^{1})^{m}$ generates
a cylindrical Heisenberg group (see \cite{dbg} and \S\ref{subs-quant})
and its only faithful unitary representation is given by a fibered
$L^2$ space.
\item[(iii)] The coherent states we use are those
adapted to the cylindrical phase space, as presented in
\S\ref{appendix-cs} (refer also to \cite{p}, \cite{dbg}).
\item[(iv)] The infinite-particle limit is a true thermodynamic limit,
because here not only we have $m\to\infty$, but also $L\to\infty$ under
the constraint of finite density $\displaystyle \frac{m}{L}\to\rho,
\rho>0$.
\item[(v)] On the other hand in this model the classical dynamics is just
free motion. This entails a simplification: the Weyl symbol of the quantum
$m$-particle Schr\"odinger operator $H_m$ is obviously the classical
Hamiltonian $\Hm = (1/2) \sum_{i=1}^{m} p_{i}^{2}$, namely $H_{m} =
\Op(\Hm)$. Then the Weyl symbol of the (unnormalized here) quantum Gibbs
measure is
\begin{displaymath}
e^{-\beta H_{m}} = \Op (e^{-\beta\Hm}),
\end{displaymath}
where, here and above, $\Op := \Op^{W}$ denotes the operation of Weyl
quantization of a symbol, whose definition in the present context is
recalled in \S\ref{subs-quant} below.
\end{itemize}
Now, as it will be explained in sharper detail in \S\ref{subs-Gibbs},
consider the normalized quantum Gibbs state at inverse temperature
$\beta$ (for a system of $m$ free particles in $L\,S^{1}$), namely the
functional
\begin{equation}
\omega_{\beta,L}^{m}(A) := \frac{\mbox{Tr}\, A \, e^{-\beta H_{m}}}
{\mbox{Tr}\, e^{-\beta H_{m}}} ,
\end{equation}
defined, for instance, over the class of bounded operators $A$ which
are ``$L$-periodic in the coordinate variable'' (refer to
\S\ref{subs-quant}). The application Tr is practically the trace over
$L^{2}(\R^{m})$ (see \S\ref{subs-Gibbs} for details). Then the main
result of the present work, in analogy with \cite{gm}, can be stated as
\begin{equation}
\lim_{|t|\to\infty} \tdlim \omega_{\beta,L}^{m} \left(
\Theta^{m}[t] (A)\, B \right) = \tdlim \omega_{\beta,L}^{m}(A)
\cdot \tdlim \omega_{\beta,L}^{m}(B).
\label{intro-mix}
\end{equation}
(see also Theorem \ref{thm-mix} and (\ref{thm-mix-1})). Here $A$ and
$B$ are actually dependent on $L$ and $m$ and represent the operators
quantizing two classical observables, $a$ and $b$, over the Hilbert
spaces for $m$ particles in a $L$-circle (square integrable Bloch \fn s).
Also $\Theta^{m}[t] (A) := e^{iH_{m}t/\hbar} A e^{-iH_{m}t/\hbar}$.
Moreover
\begin{equation}
\tdlim \omega_{\beta,m}(A) = P_{\rho,\beta}(a).
\end{equation}
Formula (\ref{intro-mix}) is the quantum \mix\ property and induces a
consistent formulation of the quantum \erg ity: for instance
\begin{equation}
\lim_{T\to\infty} \tdlim \omega_{\beta,L}^{m} ( (\Xi^{m}[T]
(A))^{2}) = \tdlim ( \omega_{\beta,L}^{m}(A) )^{2},
\end{equation}
if $\Xi^{m}[T] (A) = (1/2T) \int_{-T}^{T} \Theta^{m}[t] (A) \,dt$.
\smallskip\par
The present model of quantum ideal gas is structurally different
from the free Bose gas or the free Fermi gas in the grandcanonical
ensemble, discussed e.g. in \cite{br}, \S5.2. The arguments
(\cite{br}, \cite{b}) yielding the mixing property with respect to the
KMS states through the asymptotic abelianess of the CCR (or CAR)
automorphisms generated by free dynamics do not apply in this context.
\par
The paper is organized as follows: in the next section we briefly recall
the ideal gas model and construct its quantization; in \S\ref{sec-statement}
we state the results, whose proofs are given in \S\ref{sec-proofs}.
Finally, in the appendix we recall the construction of the coherent
states on the cylinder (obtained e.g. in \cite{dbg}) with some
additional details; we collect two technical lemmas; and we exploit the
properties of the convolution over $L\,S^{1}$, as $L\to\infty$, which
are crucially used in the proofs.
\section{The Classical Ideal Gas and Its Quantization.}
\label{sec-ideal-quant}
\sectcount
\subsection{The Volkovyski-Sinai model.} \label{subs-classical}
To make the exposition self-contained, a brief reminder is here given of
the Volkovyski-Sinai model of ideal gas. The reader is referred to
\cite{s}, Lecture 8, for details.
\par
Consider a system of $m$ free particles of unit mass constrained to move
on a circle of length $L$. Its Hamiltonian \fn\ is
\begin{equation}
\Hm(p,q) = \frac{1}{2} \sum_{j=1}^{m} p_{j}^{2} = p^{2}/2
\end{equation}
defined on the phase space $\LLm := \R^{m} \times (L\,S^{1})^{m}$ which,
with a view to the limiting case $L\to\infty$ to be considered below, will
be identified with $\R^{m} \times \TLm,\ \TLm := [-L/2,L/2)^{m}$. The
physical intuition is to stretch the circle $L\,S^{1}$ more and more
towards a straight line, namely $\R$.
\par
On $\LLm$ the motion is given by the flow map: $\fLmt(p,q) := (p,q+pt)$.
The dynamical system is completely integrable and $\LLm$ is decomposed in
a non-countable family of invariant tori: all motions are quasi-periodic.
Introducing any ``reasonable" measure on $\LLm$ (in \cite{s} the
microcanonical measure is used; for reasons which will become clear when
quantizing, the natural measure to introduce here is the canonical one)
the system is of course not even \erg.
\par
But a suitable \td\ of it might be. The construction of the infinite
dynamical system is based on the idea that the particles should be
``unlabeled'' the particles. More precisely, the phase space for the
infinite system is defined as follows:
\begin{equation}
\Linf := \{ (p,q);\ q \mbox{ countable subset of }\R,\: p:q \to\R \},
\label{Linf}
\end{equation}
with the interpretation that $q = \{x_{1},x_{2},\ldots,x_{j}, \ldots \}$
contains the positions of the particles, now undistinguishable, and
$p$ is a \fn\ such that $p(x_{j})$ gives the velocity of the particle
located at $x_{j}$. It may occur that more than one particle -- say
$n$ -- are located at $x_{j}$: in this case, with an abuse of notation with
respect to definition (\ref{Linf}), $p(x_{j})$ is an $n$-tuple of
velocities. The flow $\finft: \Linf\to\Linf$ is defined accordingly:
$\finft(p,q) = (p',q')$, where $q' := \{ x+p(x)t;\: x\in q\}$ and
$p':q'\to\R$ is $p'(x+p(x)t) := p(x)$.
\par
What we have introduced is the natural limiting object of the spaces
$\LLm/\Sm$, $\Sm$ being the group of permutations of $m$ coordinates
and momenta. Such spaces are expressly
defined in \cite{s} in a way completely analogous to (\ref{Linf}),
i.e. as the collection of all couples $(p,q)$ with $q\subset
L\,S^{1}, \, \# q\leq m$ and $p:q \to\R$ having $m$ values, in the sense
specified above. They can be regarded as belonging to $\Linf$, since
$q \subset L\,S^{1} \simeq [-L/2,L/2)$. This motivates the above
identification.
\par
This remark enables us to consider \fn s defined
on $\Linf$ as having a natural restriction on $\LLm/\Sm$. And a \fn\
$f$ on $\LLm/\Sm$ is simply a totally symmetric \fn\ on $\LLm$,
namely $f \circ \PLm$, where $\PLm$ is the natural immersion $\LLm
\to \Linf$:
\begin{equation}
\PLm(p_{1},\ldots,p_{m},q_{1},\ldots,q_{m}) := (p:q\to\R,q:=
\{ q_{1},\ldots,q_{m} \} ),
\end{equation}
$p$ taking values $p(q_{1}) = p_{1}, \ldots, p(q_{m}) = p_{m}$.
It will be useful in the remainder to notice that
\begin{equation}
\finft \circ \PLm = \PLm \circ \fLmt
\label{comm-of-flows}
\end{equation}
To complete the definition of our infinite dynamical system, we need
to specify the measurable \fn s, i.e. to fix a $\sigma$-algebra
${\cal A}$ and -- after that -- a probability measure on it. The
answer in \cite{vs} is ${\cal A} := \sigma(\gamma(\Delta)
)_{\Delta\in{\cal B}(\R)}$. Here $\Delta$ runs among the Borel sets
in $\R$ and $\gamma(\Delta)$ is the $\sigma$-algebra of all subsets of
$\Linf$ depending only on the positions and momenta of the {\em
unlabeled} particles in $\Delta$.
\footnote{More precisely, $\gamma(\Delta)$ is the
$\sigma$-algebra for which the measurable \fn s are all functions $f$
on $\Linf$ depending only on values taken by the particles in $\Delta$
{\em and} measurable when viewed as \fn s on $\LLm$:
$f \circ \PLm$.}
For the sake of simplicity, we just restrict ourselves to real \fn s.
\par
Examples of measurable \fn s are: $f_{\Delta}(p,q) := \# (q\cap\Delta)$,
the number of particles of the configuration $q$ located in $\Delta$; or
$g_{\Delta}(p,q) := \sum_{x\in\Delta} p(x)$, the total momentum of the
particles in $\Delta$.
\par
We endow ${\cal A}$ with the measure $P_{\rho,\beta}$ defined by the
following properties:
\begin{enumerate}
\item
The distribution of particles in the configuration space is
Poissonian with parameter $\rho$; that is
\begin{equation}
P_{\rho,\beta} (\{(p,q);\: f_{\Delta}(p,q) = n\}) = e^{-\rho|\Delta|}
\frac{(\rho|\Delta|)^{n}}{n!};
\label{Poisson}
\end{equation}
for every $\Delta\in{\cal B}(\R)$. This implies that the distributions
over two disjoint Borel sets $\Delta_{1}$ and $\Delta_{2}$ are independent.
\item
The momenta are independent centered Gaussian variables with variance
$1/\beta$. This means that, fixed $A\in {\cal B}(\R^{n})$ and a vector
$(x_{1},\ldots,x_{n}) \in\R^{n}$, we have
\begin{equation}
P_{\rho,\beta} ( \{ (p(x_{1}),\ldots,p(x_{n})) \in A \} \:\a\:
\{ \, \{x_{1},\ldots,x_{n}\}\subset q \} ) = \int_{A} \left[
e^{-\beta p^{2}/2}dp \right]_{N}
\label{Maxwell-distr}
\end{equation}
$[\cdots]_{N}$ being the normalized measure. This is a Maxwell
distribution with inverse temperature $\beta$.
\end{enumerate}
This measure has been chosen intentionally as the limit of the
canonical measures over the phase spaces of finite numbers of
particles.
\begin{prop}
Let $a$ measurable in ${\cal A}$. At the \td, i.e. $m,L \to\infty;
m/L \to\rho$,
\begin{displaymath}
\int_{\LLm} (a\circ\PLm)(p,q) \left[ e^{-\beta p^{2}/2}dpdq
\right]_{N} \longrightarrow P_{\rho,\beta}(a).
\end{displaymath}
\label{limit-of-meas}
\end{prop}
As regards the proof, the statement concerning the distribution
of the positions is clearly explained in \cite{s}, while
(\ref{Maxwell-distr}) trivially holds since the distributions over the
momentum spaces for a finite number of particles are given as Maxwellian
with inverse temperature $\beta$.
\par
We conclude this section formulating the key
\begin{thm}
{\rm\cite{vs}} The measure $P_{\rho,\beta}$ is invariant under $\finft$
and the dynamical system $(\Linf,{\cal A},P_{\rho,\beta},\finft)$ is
a K-flow.
\label{vs-thm}
\end{thm}
\smallskip
It may be worth noticing here that the infinite dynamical system just
recalled has a nice abstract construction, which is described in
\cite{gla} (and shortly in \cite{b}, Example 2.34). It is called the
{\em Poisson system} constructed for a one-dimensional free particle
with a Maxwell velocity distribution. Its ergodic properties follow
via a general technique, namely the {\em Bernoulli construction}.
This view-point shows clearly that ${\cal A}$ is generated by the
following sets:
\begin{equation}
B_{\Delta,\Gamma}^{(n)} := \big\{ (p,q)\in\Linf \:\a\: \# \{ x\in q
\,|\, x\in\Delta, p(x) \in\Gamma \} = n \big\},
\label{B-Delta,Gamma}
\end{equation}
where $\Delta,\Gamma \in {\cal B}(\R)$. Such a remark will we useful
while proving the statements.
\par
Nonetheless the approach we have chosen has the advantage of
constructing the infinite-particle dynamical system through a \td\
(see above, and in particular Proposition \ref{limit-of-meas}). This
fact will be crucial throughout this paper.
\subsection{The quantization} \label{subs-quant}
The Hilbert spaces associated to a quantum system of $m$ particles on
a circle of length $L$ are denoted by $\Ltwok, k\in [0,1/L)^{m}$. Each
of these is defined as
\begin{equation}
\Ltwok := \big\{ f \mbox{ on }\R^{m} \a \forall j\in\Z^{m}, f(q+Lj) =
\ei{Lk\cdot j} f(q), \int_{\TLm}\b |f|^{2}\b <\infty \big\},
\label{def-Ltwok}
\end{equation}
that is the space of the Bloch \fn s of parameter $k$ which are
square-integrable on a given fundamental domain. Concerning this
definition, we remark:
\begin{enumerate}
\item
The above family of spaces is the familiar one for
Schr\"odinger operators with periodic potential (see
\cite{rs}, \S XIII.16). They have to be simultaneously considered for
all values of $k\in [0,1/L)^{m}$, otherwise the quantization
application is not well defined since the Schr\"odinger
representation of the Heisenberg group is not faithful: e.g., in one
dimension, if we selected only $k=0$, then $T(L,0) = T(0,0) = \Id$
(see below).
\par
As a matter of fact, the whole Hilbert space $L^{2}(\R^{m})$ is
recovered through the standard direct integral formula (\cite{rs})
\begin{equation}
\int_{[0,1/L)^{m}}^{\oplus} \Ltwok dk \simeq L^{2}(\R^{m}).
\label{Bloch-dec}
\end{equation}
More details are given in \S\ref{appendix-cs} and in \cite{dbg}.
A basis for $\Ltwok$ is obviously $e_{\alpha}^{(k)}:= L^{-m/2}
\ei{(\alpha+k)\cdot x}$, with $\alpha\in (\Z/L)^{m}$.
\item
The choice of the Hilbert spaces in (\ref{def-Ltwok})
corresponds to a Maxwell-Boltzmann statistics, since we ask for no
wave \fn\ symmetry with respect to particle permutations. One could
also think of quantizing this system according to a Bose-Einstein or a
Fermi-Dirac statistics. To give a physical explanation, we are
considering particles which are in principle enumerable but our
observables do not see this enumeration. This approximation makes
sense in the semiclassical realm.
\footnote{On the other hand, we stress that this paper does not
involve any kind of semiclassical limit.}
\end{enumerate}
A further step towards the definition of the quantization
application is the introduction of the \Fou\ transform and
antitransform in $\LLm$. We define first the dual of the phase space:
$\LLms := \R^{m}\times\TLms$ where $\TLms := (\Z/L)^{m}$. Now, if
$b\in {\cal S}(\LLm)$, that is the Schwartz class of \fn s in
$\LLm$, then for $(\eta,\xi)\in\LLms$, the \Fou\ transform of
$b$ is
\begin{equation}
\hat{b}(\eta,\xi) := \int_{\R^{m}} \int_{\TLm} b(p,q)\,
\emi{(\eta\cdot p + \xi\cdot q)} dq dp.
\end{equation}
Correspondingly, the antitransformation is given by
\begin{equation}
b(p,q) := \frac{1}{L^{m}} \sum_{\xi\in\TLms} \int_{\R^{m}}
\hat{b}(\eta,\xi)\, \ei{(p\cdot\eta + q\cdot\xi)} d\eta.
\label{Fou-antitransf}
\end{equation}
The Heisenberg group to be considered in this situation is the
naturally induced {\em cylinder subgroup} of the Heisenberg group on
$\R^{2m}\times\R$, namely $\LLms\times\R$ endowed with the product law
\begin{equation}
(\eta,\xi,\tau)(\eta',\xi',\tau') = (\eta+\eta,\xi+\xi',
\tau+\tau' +\frac{1}{2}(\eta\cdot\xi' - \xi\cdot\eta')).
\end{equation}
Accordingly (see \cite{fo}), its unitary Schr\"odinger representation
in $L^{2}(\R^{m})$ is defined in the following way:
\begin{equation}
(T(\eta,\xi)f)(x) = \ei{\xi\cdot(\eta/2 + x)} f(x+\eta)
\end{equation}
This can be formally written as $T(\eta,\xi) = \ei{(\eta\cdot P +
\xi\cdot Q)}$, where $Q$ corresponds to the multiplication operator by
$q$, and $P = (2\pi i)^{-1} \nabla_{x}$. We have therefore taken
$\hbar = (2\pi)^{-1}$. It is evident that $T(\eta,\xi)$ preserves
$\Ltwok$: we denote $T^{(k)}(\eta,\xi)$ its restriction to that space.
\par
We are now in position to define the quantization application.
If $b$ is a \psd\ symbol of a given finite order (again \cite{fo} or
\cite{sh}) on $\LLm$, then
\begin{equation}
\Op(b) := \frac{1}{L^{m}} \sum_{\xi\in\TLms} \int_{\R^{m}}
\hat{b}(\eta,\xi)\, T(\eta,\xi) \,d\eta;
\label{Opb1}
\end{equation}
with $\hat{b}$ possibly interpreted in distributional sense.
\par
The restriction of this operator to the invariant space $\Ltwok$ will be
once more denoted by $\Op(b)^{(k)}$. The above definition is nothing
else than the standard Weyl quantization induced by the cylindrical
Heisenberg group and subject to our choice of inverse Fourier transform
(\ref{Fou-antitransf}). As a matter of fact, elementary algebraic
manipulations yield the following explicit formula: for
$f^{(k)}\in \Ltwok$,
\begin{equation}
\left( \Op(b)^{(k)}f^{(k)} \right)(x) = \int_{\R^{m}} \b
\int_{\R^{m}} b \left(\frac{x+y}{2},p\right) \ei{p \cdot (x-y)}
f^{(k)}(y) dy dp.
\label{Opb2}
\end{equation}
Remark that, since $f^{(k)} \not\in {\cal S}(\R^{2m})$, a priori
so this makes no sense even as an oscillatory integral. The sense we
can give it is, again, distributional, as we know (\cite{sh}) that for
\psd\ symbols, $\Op(b): {\cal S'} \to{\cal S'}$. Notice also
that, as usual, (\ref{Opb2}) implies that if $b$ depends
only on one canonical variable, for instance $p$, then $\Op(b) = b(P)$
in the spectral-theoretic sense. In particular, if the quantum Hamiltonian
is $H_{m} := -(\frac{1}{8\pi^{2}}) \Delta_{x} = \Op(\Hm)$, we have
$\Op(e^{-\beta\Hm}) = e^{-\beta H_{m}}$.
\par
We can now calculate the {\em Weyl composition} $a \c b$ of two
symbols $a$ and $b$, i.e. the (unique) symbol such that $\Op(a \c
b) = \Op(a)\Op(b)$. This is done by means of (\ref{Opb1}), remembering
that $T(\eta,\xi)$ obeys the multiplication law of the Heisenberg
group and that a symbol is obtained by its correspondent operator by
substituting $\ei{(p\cdot\eta + q\cdot\xi)}$ to $T(\eta,\xi)$ when
the operator is in the form (\ref{Opb1}) -- compare (\ref{Fou-antitransf})
to (\ref{Opb1}). The result, after some manipulations of the
integrals, is -- not surprisingly -- an adaptment of the
corresponding formula for the Euclidean space case (\cite{fo}, \S
2.1):
\begin{eqnarray}
&\pha& (a \c b)(p,q)\b = \label{Weyl-comp} \\
&=& \! \frac{1}{L^{2m}} \! \sum_{\xi_{1},\xi_{2}}
\! \int \! \hat{a}(\eta_{1},\xi_{1})
\hat{b}(\eta_{2},\xi_{2}) e^{\pi i(\eta_{1}\cdot\xi_{2} -
\xi_{1}\cdot\eta_{2})} \ei{[(\eta_{1}+\eta_{2})\cdot p +
(\xi_{1}+\xi_{2})\cdot q]} d\eta_{1} d\eta_{2} = \nonumber \\
&=& \! \frac{1}{L^{2m}} \! \sum_{\xi_{1},\xi_{2}} \! \int \!
a \left( p \!+\! \frac{\xi_{1}}{2},q \!+\! q_{1} \right)
b \left( p \!+\! \frac{\xi_{2}}{2},q \!+\! q_{2} \right)
\ei{(\xi_{2}\cdot q_{1} - \xi_{1}\cdot q_{2})} dq_{1}dq_{2}.
\nonumber
\end{eqnarray}
Notice that the sum is carried over $\xi_{1},\xi_{2} \in
\TLms$ and the integration over $q_{1},q_{2} \in \TLm$.
\par
By the above formula we deduce that the Weyl composition has a
property that may be called the {\em quasi-tracial property}:
\begin{equation}
\int_{\LLm} (a \c b)(p,q) \, dpdq = \int_{\LLm} a(p,q) b(p,q) \,
dpdq,
\label{quasi-tracial}
\end{equation}
even though we cannot hope to have the tracial property for some $\Ltwok$,
i.e. (\ref{quasi-tracial}) with ${\rm Tr}_{\Ltwok} \Op(a)\Op(b)$ on
the l.h.s., as explained in \cite{dbg}. Actually, as it will be clear
in \S\ref{subs-Gibbs}, what appears in the l.h.s. is something
resembling ${\rm Tr}_{L^{2}(\R^{m})}$.
\par
Given $f^{(k)},g^{(k)} \in\Ltwok$, we define the \Fou-Wigner \fn\
relative to those two vectors as $V_{f^{(k)},g^{(k)}} (\eta,\xi) :=
\la f^{(k)},T(\eta,\xi) g^{(k)} \ra$. This is completely
analogous to what is found in \cite{fo}. The Wigner \fn\
$W_{f^{(k)},g^{(k)}}$ is defined to be the (possibly distributional)
\Fou\ transform of $V_{f^{(k)},g^{(k)}}$ and thus, from (\ref{Opb1}),
\begin{equation}
\la f^{(k)},\Op(b) g^{(k)} \ra_{\Ltwok} = \int_{\LLm} b(p,q) \,
W_{f^{(k)},g^{(k)}}(p,q) \, dpdq;
\end{equation}
to be understood as $W_{f^{(k)},g^{(k)}}$ being the distribution
kernel of $b \mapsto \la f^{(k)},\Op(b) g^{(k)} \ra$. Again the
standard form for this ``\fn'' (\cite{fo}), calculated from (\ref{Opb2})
or from its very definition,
\begin{equation}
W_{f^{(k)},g^{(k)}}(p,q) := \int_{\R^{m}} \emi{p\cdot z}
\overline{f^{(k)} (q-z/2)} \, g^{(k)}(q+z/2) \, dz.
\label{Wigner-fn}
\end{equation}
has to interpreted in the weak sense.
\section{Statement of the Results} \label{sec-statement}
\sectcount
Suppose we have, $\forall L>0; m\in\Z, m\ge 1$ a measure space
$(X_{L}^{m},d\theta)$
\footnote{We are intentionally a little informal here, in
order to keep the notation not to cumbersome. A better label for
$d\theta$ would be $d\theta_{L}^{m}$. The same for $d\nu$ in the
following.}
and a family of states
\begin{equation}
\{ f_{\lambda} \}_{\lambda \in X_{L}^{m}} \subset \bigcup_{k \in
[0,1/L)^{m}} L_{(k)}^{2} (\TLm)
\end{equation}
labeled by the index $\lambda$ ranging in $X_{L}^{m}$. Call
$w_{\lambda}(p,q) := W_{f_{\lambda},f_{\lambda}}(p,q)$, the Wigner
\fn\ corresponding to $f_{\lambda}$.
\par
{\sc Hypothesis.} We suppose that
\begin{equation}
\int_{X_{L}^{m}} w_{\lambda}(p,q) d\theta(\lambda) \equiv 1,
\label{hyp}
\end{equation}
as a distribution on $\LLm$.
\par
{\sc Remark.} Such family of states represent in this context the
quantum substitute for the classical phase space. As a matter of
fact, (\ref{hyp}) says that the $\{ f_{\lambda} \}$ are evenly
distributed, as a whole, over $\LLm$. Hence they play the role
of ``points''. This is clearly seen in the case of the {\em \cs s},
perhaps the most remarkable example of states fulfilling the above
hypothesis. They are introduced in \S\ref{appendix-cs} of the appendix.
However, (\ref{hyp}) can be restated by saying that we are given a set
of states complete over all $\Ltwok, \, k\in [0,1/L)^{m}$ -- see
\S\ref{subs-Gibbs}.
\par\smallskip
Since $\| e^{-\beta H_{m}} f_{\lambda} \|^{2} = \la f_{\lambda}, e^{-2\beta
H_{m}} \, f_{\lambda} \ra$,
\footnote{Again a remark about the notation. A more formal symbol
for the scalar product would be $\la \cdot, \cdot \ra_{\Ltwok}$
where $k=k(\lambda)$ is the uniquely determined $k \in [0,1/L)^{m}$
such that the arguments are in $\Ltwok$. Since the scalar
products have the same structure on each $\Ltwok$ we will drop that
subscript.}
then (\ref{hyp}) immediately implies
\begin{equation}
\int_{X_{L}^{m}} \| e^{-\beta H_{m}} f_{\lambda} \|^{2}
d\theta(\lambda) = \int_{\LLm} e^{-2\beta\Hm(p)}\, dpdq = L^{m} \left(
\frac{\pi}{\beta} \right)^{m/2}.
\label{Gibbs}
\end{equation}
We define:
\begin{equation}
d\nu(\lambda) := \frac{\| e^{-\beta H_{m}} f_{\lambda} \|^{2}
d\theta(\lambda)} {\int_{X_{L}^{m}} \| e^{-\beta H_{m}}
f_{\lambda'} \|^{2} d\theta(\lambda')}
\label{dnu}
\end{equation}
and
\begin{equation}
g_{\lambda} := \frac{e^{-\beta H_{m}} f_{\lambda}} {\|e^{-\beta H_{m}}
f_{\lambda} \|}
\end{equation}
be the image under the quantum Gibbs measure of each of our states.
\begin{defn}
$a\in {\cal A}$ is said an {\em asymptotic symbol} if $\exists
m_{0} \in\N, L_{0} >0$ such that $\forall m \ge m_{0}, L \ge L_{0}$,
$a\circ\PLm$ is a \psd\ symbol over $\LLm$.
\label{def-asymp-sym}
\end{defn}
\par
{\sc Remark.} Notice that with such a definition, asymptotic symbols
are rather rigid objects. In fact, fixed an $m \ge m_{0}$, take
$L_{0} \le L \le L_{1}$. Now $\mbox{Im}(\PLm) \subseteq
\mbox{Im}(\Pi_{L_{1}}^{m}) \subseteq \Linf$. So $a\circ\PLm$ is just
the restriction of $a\circ \Pi_{L_{1}}^{m}$ to $\LLm$. But also
$a\circ\PLm$, in order to be a symbol must be $C^{\infty}$ and
$\TLm$-periodic. This means that $\forall i=1, \ldots, m$,
\begin{eqnarray}
(a\circ \Pi_{L_{1}}^{m})(\ldots,-L/2,\ldots) &=&
(a\circ \Pi_{L_{1}}^{m})(\ldots,+L/2,\ldots); \\
\frac{d}{dq_{i}} (a\circ \Pi_{L_{1}}^{m})(\ldots,-L/2,\ldots) &=&
\frac{d}{dq_{i}} (a\circ \Pi_{L_{1}}^{m})(\ldots,+L/2,\ldots),
\end{eqnarray}
where $(\ldots,\pm L/2,\ldots)$ stands for $(p_{1},\ldots,p_{m},q_{1},
\ldots,q_{i-1},\pm L/2, q_{i+1}, \ldots,q_{m})$.
Thus, parity arguments imply that for every $|q_{i}| \ge L_{0}/2$
\begin{equation}
\frac{d}{dq_{i}} (a\circ \Pi_{L_{1}}^{m})(\ldots,q_{i},\ldots) = 0.
\end{equation}
Hence, for a fixed large $m$, $(a\circ \Pi_{L_{1}}^{m})$ is a
constantwise continuation of $(a\circ \Pi_{L_{0}}^{m})$, and the
former is completely determined by the latter. This also explain we
had to ask for {\em asymptotic} symbols: one could not request
the above property to hold $\forall m \ge 0, L \ge L_{0}$.
Examples of such \fn s are to be found in the families ${\cal
B}^{(n)}$ defined after Lemma \ref{lemma1-mix} in \S\ref{sec-proofs}.
\smallskip\par
We are now ready to state the theorems. For any operator $A$ acting over
$L^{2}(\R^{m})$, define:
\begin{eqnarray}
\Theta^{m}[t]\, (A) &:=& \ei{t H_{m}} A \emi{t H_{m}};
\label{Theta-t} \\
\Xi^{m}[T]\, (A) &:=& \frac{1}{2T} \int_{-T}^{T} \Theta^{m}[t] (A) \,
dt. \label{Xi-T}
\end{eqnarray}
The quantum \mix\ property reads:
\begin{thm}
Suppose $a,b$ asymptotic symbols in $L^{2}(\Linf,P_{\rho,2\beta})$
and denote
\begin{displaymath}
I(t,L,m) := \int_{X_{L}^{m}}\la g_{\lambda}, \Theta^{m}[t]
(\Op(a\circ\PLm)) \, \Op(b\circ\PLm) \, g_{\lambda} \ra \,
d\nu(\lambda).
\end{displaymath}
Then:
\begin{displaymath}
\lim_{|t|\to\infty} \tdlim I(t,L,m) = P_{\rho,2\beta}(a)
P_{\rho,2\beta}(b).
\end{displaymath}
\label{thm-mix}
\end{thm}
\par
As regards the quantum \erg ity,
\begin{thm}
Let $a$ be an asymptotic symbol in
$L^{2}(\Linf,P_{\rho,2\beta})$. Let
\begin{displaymath}
J(T,L,m) := \int_{X_{L}^{m}} \| ( \Xi^{m}[T]
(\Op(a\circ\PLm)) - P_{\rho,2\beta}(a) ) \, g_{\lambda}
\|^{2}\, d\nu(\lambda).
\end{displaymath}
Then, for all $L,m$, the operator
\begin{displaymath}
\Xi^{m}[\infty] (\Op(a\circ\PLm)) := \lim_{T\to\infty} \Xi^{m}[T]
(\Op(a\circ\PLm))
\end{displaymath}
exists in the domain of $\Op(a\circ\PLm)$ and
\begin{displaymath}
\tdlim J(\infty,L,m) = 0.
\end{displaymath}
Furthermore, if $a$ is also bounded, then the two limits can be
inverted:
\begin{displaymath}
\lim_{T\to\infty} \tdlim J(T,L,m) = 0.
\end{displaymath}
\label{thm-erg}
\end{thm}
\medskip\par
{\sc Remark.} The interchange of the time limit with the \td, in the
above theorem, is remarkable. The fact that the time average can
be taken before the \td\ can be described saying that the
finite-particle system (the ``real'' one) is {\em quasi-ergodic}, for very
large $L$ and $m$; that is, the time average of any {\em decent}
\fn\ is close, in measure, to a constant. This is a feature of the
classical ideal gas which has nothing to do with quantum mechanics.
It is rather a consequence of the kinematic chaos and the restriction
to symmetric observables, as anticipated in \S\ref{sec-intro}, remark (i).
This can be seen quite easily, due to the integrability of the
motion: time averaging means almost everywhere averaging over a torus.
The invariant tori here are the sets $\{ p \} \times \TLm \in \LLm$,
and so the time average of $a(p,q)$ is simply $\int a(p,q) dq$; the
invariant \fn s would depend on $p$ only. Those \fn s, however, are
requested to be symmetric and thus they cannot concentrate around a
torus if they do not concentrate around all ``symmetric tori'' as well.
An example is
\begin{equation}
a(p_{1},\, \ldots\, ,p_{m}) = \left\{
\begin{array}{ll}
1 & \mbox{if } p_{i} \in \Gamma \ \forall i=1,\ldots,m; \\
0 & \mbox{otherwise},
\end{array}
\right.
\end{equation}
where $\Gamma$ is a Borel set of $\R$. Now the kinematic chaos effect
comes: at the \td, the support of this \fn, which is the probability
to find all the particles having momenta in $\Gamma$, is
exponentially small.
\par
We can compare this to the situation one has for the harmonic chain, as
shown in the companion paper \cite{gm}. In that case, there is no requirement
on the observables. The fact that they cannot concentrate over
invariant tori is instead due to the assumptions on the coupling matrix,
which shuffles the tori at the infinite-particle limit.
\medskip\par
An even clearer reason for referring to Theorem \ref{thm-erg} as quantum
\erg ity comes from the following:
\begin{cor}
Assume $a$ bounded asymptotic symbol. Then, $\forall \epsilon>0$,
set
\begin{displaymath}
K(\epsilon,T,L,m) := \nu \left( \{\lambda\in X_{L}^{m} \:\a\:
\left| \la g_{\lambda}, \Xi^{m}[T] (\Op(a\circ\PLm))\, g_{\lambda}
\ra - P_{\rho,2\beta}(a) \right| > \epsilon \} \right).
\end{displaymath}
Then
\begin{displaymath}
\lim_{T\to\infty} \tdlim K(\epsilon,T,L,m) =
\tdlim \lim_{T\to\infty} K(\epsilon,T,L,m) = 0.
\end{displaymath}
\label{cor-erg}
\end{cor}
This can be phrased as follows. Call {\em $(T,\epsilon)$-exceptional
initial states} those states $g_{\lambda}$ for which the quantum
expectation of the $T$-time average is greater than $\epsilon$. Then the
claim is that the measure of the $(T,\epsilon)$-exceptional initial
states vanishes when the \td\ and the time limit are achieved.
\par
{\sc Proof of Corollary \ref{cor-erg}.} Easy consequence of Theorem
\ref{thm-erg}, using a Cauchy-Schwartz inequality.
\qed
\subsection{The Quantum Gibbs State} \label{subs-Gibbs}
The results just formulated can be given a compact form, within the
realm of the $C^{*}$ dynamical systems theory. It is beyond the
purpose of this paper to go deep into that, so we do not outline the
main notions of such a theory, hoping that the statements in this
section are self-explanatory. However, a brief survey is given in
\cite{gm}, Appendix 2. Here we just observe that the relations we will
write are included in such a general frame. The interested reader is
referred to \cite{br} for complete details, and to \cite{b} for a
recent well-organized review.
\par
Consider ${\cal L}_{L}^{m}$, the space of all operators on
$L^{2}(\R^{m})$ which are invariant and bounded over all the fibers
$\Ltwok$. This is a $C^{*}$-algebra when endowed with the usual operator
norm. Associated to this algebra we define the Heisenberg dynamics
given by $\Theta^{m}[t]\, (A)$ as in (\ref{Theta-t}), and the quantum
Gibbs state expressed by
\begin{equation}
\omega_{\beta,L}^{m}(A) := \frac{\int_{[0,1/L)^{m}} \Tr (A \,
e^{-\beta H_{m}})\, dk}{\int_{[0,1/L)^{m}} \Tr (e^{-\beta H_{m}})
\, dk} ,
\label{omega}
\end{equation}
where $\Tr$ denotes the trace over $\Ltwok$. This functional is
clearly normalized
\footnote{Here normalized means $\omega_{\beta,L}^{m}(\Id) = 1$.}
and invariant for the $^{*}$-automorphism $\Theta^{m}[t]$.
Actually, it turns out to be a KMS state with parameter $\beta$ over
the $W^{*}$ dynamical system $({\cal L}_{L}^{m},\Theta^{m}[t],
\omega_{\beta,L}^{m})$.
\par
Now consider, for each value of $k \in [0,1/L)^{m}$, the standard
\Fou\ basis $\{ e_{\alpha}^{(k)} \}$ as defined in
\S\ref{subs-classical}. Next consider the family of all such vectors,
labeled by the index $\lambda := (\alpha,k) \in X_{L}^{m} =: \TLms
\times [0,1/L)^{m}$. Endow $X_{L}^{m}$ with the measure
\begin{equation}
d\theta(\alpha,k) := L^{m} \sum_{\xi\in\TLms} \delta(\alpha-\xi)
\, d\alpha\, dk.
\end{equation}
Such a family satisfies hypothesis (\ref{hyp}). In fact, calling
$w_{\alpha,k}$ the Wigner function relative to $e_{\alpha}^{(k)}$, a
straightforward computation from (\ref{Wigner-fn}) yields
\begin{equation}
w_{\alpha,k}(p,q) = \frac{1}{L^{m}} \: \delta(p-(\alpha + k)).
\label{w-alpha-k}
\end{equation}
Integrating this in $d\theta(\alpha,k)$ we obtain (\ref{hyp}). Thus,
define, as it is done in at the beginning of this section, $g_{\alpha,k}:=
e^{-\beta H_{m}} e_{\alpha}^{(k)} / \| e^{-\beta H_{m}} e_{\alpha}^{(k)} \|$.
By definition (\ref{omega}), if $A \in {\cal L}_{L}^{m}$, we have
\footnote{Notice that we are using $2\beta$ instead of $\beta$ in the
remainder.}
\begin{eqnarray}
\omega_{2\beta,L}^{m}(A) &=& \frac{\int dk \sum_{\alpha}
\la e^{-\beta H_{m}} e_{\alpha}^{(k)}, \, A\,e^{-\beta H_{m}}
e_{\alpha}^{(k)} \ra} {\int dk \sum_{\alpha} \| e^{-\beta H_{m}}
e_{\alpha}^{(k)} \|^{2} } = \nonumber \\
&=& \int_{X_{L}^{m}} \la g_{\alpha,k}, \, A\, g_{\alpha,k} \ra \:
d\nu(\alpha,k),
\end{eqnarray}
with $d\nu$ defined as in (\ref{dnu}). On the other hand, if one calls in
a natural way $A_{L}^{m} := \Op(a\circ\PLm)$, then a simple argument which
is better explained in the following (see formul\ae\ (\ref{mix-zero})
and (\ref{mix-first})) gives
\begin{equation}
\omega_{2\beta,L}^{m}(A_{L}^{m}) = \int_{\LLm} (a\circ\PLm) (p,q)
\left[ e^{-2\beta\Hm(p)}\, dpdq \right]_{N}.
\end{equation}
Proposition \ref{limit-of-meas} immediately yields
\begin{equation}
\tdlim \omega_{2\beta,L}^{m}(A_{L}^{m}) = P_{\rho,2\beta}(a).
\end{equation}
Hence, if $a,b$ are asymptotic symbols, we have just proved that Theorem
\ref{thm-mix} can be rewritten as
\begin{equation}
\lim_{|t|\to\infty} \tdlim \omega_{2\beta,L}^{m} \left(
\Theta^{m}[t] (A_{L}^{m}) \, B_{L}^{m} \right) =
\tdlim \omega_{2\beta,L}^{m}(A_{L}^{m})
\cdot \tdlim \omega_{2\beta,L}^{m}(B_{L}^{m}).
\label{thm-mix-1}
\end{equation}
In the same spirit, Theorem \ref{thm-erg} becomes
\begin{eqnarray}
&\pha& \lim_{T\to\infty} \tdlim \omega_{2\beta,L}^{m} ( (\Xi^{m}[T]
(A_{L}^{m}))^{2}) = \nonumber \\
&=& \tdlim \omega_{2\beta,L}^{m} ( (\Xi^{m}[\infty]
(A_{L}^{m}))^{2}) = \label{thm-erg-1} \\
&=& \tdlim ( \omega_{2\beta,L}^{m} (A_{L}^{m}) )^{2},
\nonumber
\end{eqnarray}
\par
{\sc Remark.} We have not defined an algebra of quantum observables for
the infinite-particle system, limiting ourselves to deal with finite
dimensions and to take a \td\ afterwards (see also comment 3 below). Had
we introduced such a mathematical framework, then relations
(\ref{thm-mix-1}) and (\ref{thm-erg-1}), for the state
$\tdlim \omega_{2\beta,L}^{m}$, would be contained in the general set of
chaoticity notions in $C^{*}$ dynamical system theory (see \cite{b},
Definitions 4.42, 4.43).
\footnote{We are not granted, in principle, all the equivalent \erg ity
and \mix\ properties recalled in that reference, since we have not
proved asymptotic commutativity.}
\par
Some comments concerning the above reformulation of the theorems as
compared to \cite{gm}:
\begin{enumerate}
\item
(\ref{thm-mix-1}) is completely analogous to statement (1.10) in
\cite{gm}. That is, the quantum \mix, forbidden in the finite-particle
frame by the quasi-periodicity of the Heisenberg evolution, regardless the
dynamics of the classical flow (see \S\ref{sec-intro}), is restored at
the \td.
\item
A formulation of the \erg ity similar to (1.9) in \cite{gm} has not
been chosen in this context because of the technicalities it would
require. The \cs s we have here (see appendix, \S\ref{appendix-cs})
are indexed by $\lambda \in \TLm \times \R^{m} \times [0,1/L)^{m} =:
X_{L}^{m}$, which is not exactly the classical phase space $\LLm$.
However, one could explicitly calculate $d\nu$ over $X_{L}^{m}$, for
particular choices of the \cs s, and find a limit measure space -- say
-- $(X,d\nu)$, but this turns out to be rather cumbersome and possibly
misleading. Understandably, though, (\ref{thm-erg-1}) and especially
Corollary \ref{cor-erg} carry the same physical meaning as the mentioned
result.
\item
As already exploited, we are able here to state the \erg ity results
with a commutation of the limits.
\item
As emphasized in \cite{gm}, \S1, Remark 2, the techniques we use
to prove the quantum \erg\ properties at the \td, have
the useful outcome to show that the r.h.sides of (\ref{thm-erg-1}) and
(\ref{thm-mix-1}) are the expected classical Gibbs averages. This
is why we have formulated Theorems \ref{thm-erg} and \ref{thm-mix}
in the first place.
\item
More importantly, here, results (\ref{thm-mix-1}) and (\ref{thm-erg-1})
were not known, at least to us.
\end{enumerate}
\section{The Proofs.} \label{sec-proofs}
\sectcount
The first key fact is the following
\begin{lemma} \label{lemma-q-evol-c}
For every symbol $c$ defined on $\LLm$,
\begin{displaymath}
\ei{t H_{m}} \Op(c) \emi{t H_{m}} = \Op(c\circ\fLmt)
\end{displaymath}
\end{lemma}
This is true since we are dealing with a linear flow. This property
of linear flows -- sometimes referred to as the {\em exact Egorov
Theorem} for the evolution canonical transformation -- dates back at
least to Van Hove and is valid only for Weyl quantization, whose
restriction to $L$-periodic symbols we are
now using. Anyway, for the sake of
convenience, a direct proof is found in
\S\ref{appendix-pf-quant-evol} of the appendix.
\smallskip\par
In our case, applying this lemma to $a \circ \PLm$ and using the
remark in formula (\ref{comm-of-flows}), we have
\begin{equation}
\ei{t H_{m}} \Op(a\circ \PLm) \emi{t H_{m}} = \Op(a \circ \finft
\circ \PLm).
\label{quant-evol-comm}
\end{equation}
\subsection{Proof of Theorem 3.2.} \label{subs-pf-thm-mix}
In view of the above relation we call $a_{t} := a \circ \finft$. In
the rest of this proof, whenever there is no confusion, we denote
by a quote the immersion application from $\LLm$ to $\Linf$. Hence $a'
:= a \circ \PLm$ and so on. In other words, $a'$ is just our
observable $a$ looked at in the finite dimensional
phase space $\LLm$. By (\ref{hyp}) and (\ref{dnu}), the
definition of $I$, in the statement of the theorem, yields
\begin{equation}
I(t,L,m) = \frac{\int_{\LLm} (e^{-\beta\Hm} \c a'_{t} \c
b' \c e^{-\beta\Hm}) dpdq} {\int_{\LLm} e^{-2\beta\Hm}
dpdq}.
\label{mix-zero}
\end{equation}
Using twice (\ref{quasi-tracial}) -- once to permutate cyclically
the factors in (\ref{mix-zero}) and once to remove one of the $\c$
signs -- leads to
\begin{equation}
I(t,L,m) = \int_{\LLm} (a'_{t} \c b') (p,q)
\left[ e^{-2\beta\Hm(p)}\, dpdq \right]_{N},
\label{mix-first}
\end{equation}
since obviously $e^{-\beta\Hm} \c e^{-\beta\Hm} = e^{-2\beta\Hm}$.
We further denote by $\mLm$ the classical
Gibbs measure (at inverse temperature $2\beta$) over $\LLm$:
$\mLm(p,q) := L^{-m}(\pi/\beta)^{-m/2} e^{-\beta p^{2}}$, and by
$\mbm$ its component in the $p$-space: $\mbm(p) := (\pi/\beta)^{-m/2}
e^{-\beta p^{2}}$.
\par
In view of (\ref{Weyl-comp}), (\ref{mix-first}) becomes, after
some elementary but tedious rearrangements of the nested integrals,
\begin{equation}
I(t,L,m) = \int_{\LLm} [a'_{t}\, (b' *_{L} \Phi_{L}^{m})]
(p,q) \: d\mLm(p,q),
\label{mix-second}
\end{equation}
where
\begin{equation}
\Phi_{L}^{m}(p,q) = e^{\beta p^{2}} \frac{1}{L^{m}}
\sum_{\xi\in\TLms} e^{-\beta(p-\xi/2)^{2}} \, \ei{\xi\cdot q},
\label{phiLm-1}
\end{equation}
and $*_{L}$ means convolution in the $q$-variable on $\TLm$. A
particular care must be taken here about this convolution {\em on a
torus}, in order to prevent mistakes: see \S\ref{appendix-conv}.
\par
Thus $\Phi_{L}^{m}$ is completely factorizable, with $\Phi_{L}$ being
a natural symbol for each of his factor: if
$f(p_{1}, \ldots,p_{m}, q_{1}, \ldots, q_{m})
= f_{1}(p_{1}, q_{1}) \cdots f_{m}(p_{m}, q_{m})$, then
\begin{equation}
(f *_{L} \Phi_{L}^{m})(p,q) = (f_{1} *_{L} \Phi_{L})(p_{1},q_{1})
\cdots (f_{m} *_{L} \Phi_{L})(p_{m},q_{m}).
\label{phiLm-2}
\end{equation}
This property will be useful in the following.
Also, if $1$ is the \fn\ on $\LLm$ identically equal to 1,
\begin{equation}
1 *_{L} \Phi_{L}^{m} = 1.
\label{phiLm-3}
\end{equation}
We are going to prove the statement of the theorem, starting from
(\ref{mix-second}), for $b$ in a dense subspace of
$L^{2}(\Linf,P_{\rho,2\beta})$. To accomplish that we need to make
the following construction.
\par
Fix a positive integer $n$ and consider the \fn\ $\beta(p,q) \in
C_{0}^{\infty}(\R^{2n})$
\footnote{Not to be confused with the inverse
temperature $\beta$, a fixed parameter throughout this paper.}
, that is, infinitely differentiable \fn s with compact
support. For $m > n$ define the application
\begin{equation}
\NLmn(\beta)(p_{1}, \ldots, p_{m}, q_{1}, \ldots, q_{m}) :=
\sum_{j_{1}=1}^{m} \cdots \sum_{j_{n}=1}^{m} \beta (p_{j_{1}},
\ldots, p_{j_{n}}, q_{j_{1}}, \ldots, q_{j_{n}}).
\label{NLm}
\end{equation}
So $\NLmn(\beta)$ is a \fn\ defined on $\R^{2m}$, and thus in particular
on $\LLm$. The use of this application is explained by next
\begin{lemma} \label{lemma1-mix}
If $\beta(p,q) \in {\cal S} (\R^{2n})$
\footnote{This denotes the Schwartz class.}
then there exists a \fn\ $b\in L^{2}
(\Linf,P_{\rho,2\beta})$ s.t. $\NLmn(\beta) = b \circ \PLm$.
Plus, the following properties hold:
\begin{equation}
\int_{\LLm} |(b\circ\PLm) (p,q)|^{2} \: d\mLm(p,q)
\le m^{2n} \int_{\R^{2n}} |\beta(p',q')|^{2} \: d\mu_{L}^{n}
(p',q') ;
\label{lemma1-one}
\end{equation}
\begin{equation}
\int_{\LLm} (b\circ\PLm) (p,q) \: d\mLm(p,q) =
\frac{m!}{L^{m} (m-n)!} \int_{\R^{2n}} \beta(p',q')
\: d\breve{\mu}^{n} (p')\, dq'.
\label{lemma1-two}
\end{equation}
\end{lemma}
\par
{\sc Proof of Lemma \ref{lemma1-mix}.} The first inequality simply
follows by definition (\ref{NLm}): we have $m^{2n}$ integrals over
$\LLm$. To the cross-term integrals we apply Cauchy-Schwartz in order
to obtain $m^{2n}$ terms equal to $\| \beta \|^{2}_{L^{2}(\LLm)}$.
These get reduced to integrals over $\LLn$, since the
measure is decomposable and the integrand \fn s depend only on $2n$
variables; finally they are extended to all of $\R^{2n}$.
\par
To prove the rest we approximate $\beta$ with suitably
chosen indicator \fn s over $\R^{2n}$. More precisely take two
sufficiently fine partitions of $\R$, $\{ \Gamma_{j} \}, \{
\Delta_{\ell} \}\subset {\cal B}(\R)$ with $\sup_{j,\ell} \{
\mbone(\beta_{j}), |\Delta_{\ell}| \}$ small.
\footnote{$| \cdot |$ denotes the Lebesgue measure.}
Let $\chi_{j_{1},\ldots,j_{n}, \ell_{1},\ldots,\ell_{n}}
(p',q')$ be the indicator \fn\ of the set $\Gamma_{j_{1}} \times \cdots
\times \Gamma_{j_{n}} \times \Delta_{\ell_{1}} \times \cdots \times
\Delta_{\ell_{1}}$ and approximate $\beta$ with $\beta_{a} :=
\sum c_{j_{1},\ldots,\ell_{n}}\, \chi_{j_{1},\ldots,\ell_{n}}$. So
\begin{equation}
\NLmn(\beta_{a})(p,q) = \sum_{j_{1},\ldots,\ell_{n}}
c_{j_{1},\ldots,\ell_{n}} \NLmn(\chi_{j_{1},\ldots,\ell_{n}})(p,q).
\label{step-in-lemma1}
\end{equation}
Since $\chi_{j_{1},\ldots,\ell_{n}}$ is completely factorizable,
then it is easy to realize, by definition (\ref{NLm}),
that $\NLmn(\chi_{j_{1},\ldots,\ell_{n}}) (p,q) = N_{L,m}^{(1)}
(\chi_{\Gamma_{j_{1}} \times \Delta_{\ell_{1}}}) \cdots N_{L,m}^{(1)}
(\chi_{\Gamma_{j_{n}} \times \Delta_{\ell_{n}}})$. We see that
$N_{L,m}^{(1)}(\chi_{\Gamma \times \Delta}) (p,q)$ takes integer
values between $0$ and $m$. Specifically if counts the number of
particles in the configuration $(p,q) \in\LLm$ whose momentum is
contained in $\Gamma$ and whose coordinate in $\Delta$.
\footnote{This explains why we have chosen such a notation for
$\NLmn$.}
So $N_{L,m}^{(1)}(\chi_{\Gamma \times \Delta}) = N_{\Gamma \times
\Delta} \circ \PLm$, where $N_{\Gamma \times \Delta}: \Linf \to \N$
is defined by
\begin{equation}
N_{\Gamma \times \Delta} (p,q) := \# \{ x \in q \cap \Delta
\,|\, p(x) \in \Gamma \},
\end{equation}
where, with sloppy notation, $(p,q)$ denotes a point in $\Linf$.
Recalling what we said in \S\ref{subs-classical}, this \fn\
obviously belongs to ${\cal A}$: see in particular
(\ref{B-Delta,Gamma}) and comments thereby. Therefore so does every finite
product of similar \fn s. Looking at (\ref{step-in-lemma1}), and
subsequent comments, this proves that there exists a $b_{a} \in
{\cal A}$ such that $\beta_{a} = b_{a}\circ\PLm$. The analogous
statement holds for $\beta$ as well, by density.
\par
Let us go over to the proof of (\ref{lemma1-two}). Fix a sequence
$(j,\ell):=(j_{1},\ldots,\ell_{n})$ like those we have in formula
(\ref{step-in-lemma1}) and fix $n$ integers $k := (k_{1}, \ldots,
k_{n})$ such that $k_{1} + \ldots + k_{n} \le m$. Now consider the
set $A_{j,\ell}^{(k)} := \{ N_{L,m}^{(1)} (\chi_{\Gamma_{j_{i}} \times
\Delta_{\ell_{i}}}) = k_{i} \, ;\, \forall i=1, \ldots, n \} \in \LLm$,
i.e. the set of the configurations having $k_{1}$ particles in
$\Gamma_{j_{1}} \times \Delta_{\ell_{1}}$, $k_{2}$ particles in
$\Gamma_{j_{2}} \times \Delta_{\ell_{2}}$, and so on. Notice that
(use some combinatorics and, anyway, refer to \cite{s})
\begin{equation}
\mLm( A_{j,\ell}^{(k)} ) = \frac{m!} {(m -\sum k_{i})!}
\left[ \prod_{i=1}^{n} \frac{1}{k_{i}!}
\left( \frac{ |\Delta_{\ell_{i}}| }{L} \mbone(\Gamma_{j_{i}})
\right)^{k_{i}} \right] \left( 1 - \sum_{i=1}^{n}
\frac{ |\Delta_{\ell_{i}}| }{L} \mbone(\Gamma_{j_{i}})
\right)^{m -\sum k_{i}}.
\end{equation}
We have seen that $\NLmn(\chi_{j_{1},\ldots,\ell_{n}}) =
\sum_{k} k_{1} \cdots k_{n} \, A_{j,\ell}^{(k)}$. So, when the
initially chosen partition is fine, $\mLm(\chi_{j_{1},\ldots,
\ell_{n}}) = m! /(L^{m} (m-n)!) \prod_{i} |\Delta_{\ell_{i}}|
\mbone(\Gamma_{j_{i}}) + o(|\Delta_{\ell_{i}}|, \mbone(
\Gamma_{j_{i}}) )$. Looking back at (\ref{step-in-lemma1}) this proves
that (\ref{lemma1-two}) holds with negligible errors for $\beta_{a}$
and thus is exact for $\beta$.
\qed
\par\medskip
Let us call ${\cal B}^{(n)} \in {\cal A}$ the space of \fn s $b$
granted by Lemma \ref{lemma1-mix} when $\beta \in C_{0}^{\infty}
(\R^{2n})$. From now on we will suppose $b\in {\cal B}^{(n)}$,
so that $b' := b\circ\PLm = \NLmn(\beta)$. In so doing we
will be proving Theorem \ref{thm-mix} for $b \in \oplus_{finite} {\cal
B}^{(n)}$. But this is dense in $L^{2}(\Linf,P_{\rho,2\beta})$ since,
looking at the proof of Lemma \ref{lemma1-mix}, the closure of
${\cal B}^{(n)}$ contains the product of $n$ \fn s like $N_{\Gamma
\times \Delta}$. This means that in the algebra $\overline{(\oplus\,
{\cal B}^{(n)})}$ we are able to find the indicator \fn s of the sets
$N_{\Gamma \times \Delta}^{-1} (n),\, \forall n\in\N$. But these
generate ${\cal A}$ (look at \ref{B-Delta,Gamma} and refer to
\cite{gla} and \cite{b}).
\par
Under the above assumption, we go back to (\ref{mix-second}): since $b' =
\NLmn(\beta)$, then $b' *_{L} \Phi_{L}^{m} = \NLmn(\beta *_{L}
\Phi_{L}^{n})$
\footnote{Where this time $*_{L}$ means convolution in the
$q$-variable over $\TLn$. We warn the reader again about the possible
confusion arising from the fact that $\beta(p,q)$ is defined on $\R^{2n}$.
When in a $*_{L}$-convolution, it has to be considered as restricted
to $[-L/2,L/2)^{n}$ and periodic according to the identification
$[-L/2,L/2)^{n} \simeq \TLn$. See again \S\ref{appendix-conv}.}
because of the mentioned properties of $\Phi_{L}^{m}$
(see (\ref{phiLm-1}) to (\ref{phiLm-3})). If we denote by
$\gamma_{L} := \beta *_{L} \Phi_{L}^{n}$, it is obvious that
$\gamma_{L} \in {\cal S}(\R^{2n})$ and so $\NLmn(\gamma_{L}) = c'_{L}$
for some $c_{L} \in L^{2}(\Linf,P_{\rho,2\beta})$, by Lemma
\ref{lemma1-mix}. This allows us to rewrite (\ref{mix-second}) as
\begin{equation}
I(t,L,m) = \la a'_{t}, c'_{L} \ra_{L^{2}(\LLm,\mLm)}.
\label{mix-third}
\end{equation}
If we are able to find a limit for $c_{L}$ then we are done with the
cumbersome part of this proof. To this goal, we formulate the following
\begin{lemma} \label{lemma-mix-tech}
There exists a $\gamma_{\infty} \in {\cal S} (\R^{2n})$ such that
\begin{equation}
\| \gamma_{\infty} - \gamma_{L} \|^{2}_{L^{2} (\LLn,
\mu_{L}^{n})} = \OL.
\label{lemma-t-one}
\end{equation}
Furthermore
\begin{equation}
\int_{\R^{2n}} \gamma_{\infty}(p',q') \, d\breve{\mu}^{n}
(p')\, dq' = \int_{\R^{2n}} \beta(p',q') \, d\breve{\mu}^{n}
(p')\, dq'.
\label{lemma-t-two}
\end{equation}
\end{lemma}
The proof of this lemma is found in \S\ref{appendix-mix-tech}.
\medskip\par
In analogy with the above notations we call $c_{\infty}$ the
observable in \linebreak[3]
$L^{2}(\Linf,P_{\rho,2\beta})$ obtained applying Lemma
\ref{lemma1-mix} to $\gamma_{\infty}$. Comparing now
(\ref{lemma-t-one}) in Lemma \ref{lemma-mix-tech} with
(\ref{lemma1-one}) we deduce that
\begin{equation}
\tdlim \| c'_{\infty} - c'_{L} \|^{2}_{L^{2} (\LLm,\mLm)} = 0.
\label{mix-fourth}
\end{equation}
Dropping for the sake of simplicity the subscript in the scalar
product notation, this means that, when $m,L \to\infty,\, m/L \to\rho$,
\begin{eqnarray}
&\pha& | \la a'_{t}, c'_{L} \ra - P_{\rho,2\beta}(a_{t}
c_{\infty}) | \le \\
&\le& \| a'_{t} \|^{2} \| c'_{L} - c'_{\infty} \|^{2} +
| \mLm (a'_{t} c'_{\infty}) - P_{\rho,2\beta}(a_{t} c_{\infty}) |
\to 0, \nonumber
\end{eqnarray}
because of Proposition \ref{limit-of-meas}. Now we use the other main
ingredient of this proof, i.e. the classical result, Theorem \ref{vs-thm}.
We obtain
\begin{equation}
\lim_{|t|\to\infty} \tdlim I(t,L,m) = P_{\rho,2\beta}(a)
P_{\rho,2\beta}(c_{\infty}).
\end{equation}
Now, using the integrals of $\gamma_{\infty}$ and $\beta$ to compare
the integrals of $c_{\infty}$ and $b$ (apply (\ref{lemma-t-two}) into
(\ref{lemma1-two})), we see that $\mLm(c'_{\infty}) = \mLm(b')$.
Taking the limits, $P_{\rho,2\beta}(c_{\infty}) = P_{\rho,2\beta}(b)$,
which, together with the last relation, completes the proof.
\qed
\subsection{Proof of Theorem 3.3.} \label{subs-pf-thm-erg}
First of all it has to be noticed that both statements of Theorem
\ref{thm-erg} (respectively relation (\ref{thm-erg-1})) cannot be
derived so trivially from Theorem \ref{thm-mix} (resp.
(\ref{thm-mix-1})). This will be seen below in each case.
\par
We borrow the notation from the previous proof: so, for
example, $a'_{t} := a \circ \finft \circ \PLm$. Also, let $a'_{T} :=
(1/2T) \int_{-T}^{T} a'_{t} dt$. Formula (\ref{quant-evol-comm}) proves
that
\begin{equation}
\Xi^{m}[T] \, (A_{L}^{m}) = \Op(a'_{T}),
\label{erg-00}
\end{equation}
where, as in \S\ref{subs-Gibbs}, we call $A_{L}^{m} := \Op(a')$.
\par
The existence of $\Xi^{m}[\infty] (A_{L}^{m})$ is a trivial
consequence of the Heisenberg evolution: we can easily figure it out
looking at its matrix elements w.r.t. the
bases $\{ e_{\alpha}^{(k)} \} \subset \Ltwok$. These bases
diagonalize the Hamiltonian $H_{m}$, as well as any operator \fn\
of $P$ only. We call such eigenvalues
\begin{equation}
E_{\alpha}^{(k)} = \frac{1}{2} (\alpha + k)^{2} = \frac{1}{2}
\sum_{i=1}^{n} (\alpha_{i} + k_{i})^{2}.
\label{eigenv}
\end{equation}
Now it is easy to see that, $\forall\, k\in [0,1/L)^{m},\,
\alpha,\gamma \in \TLms$,
\begin{equation}
\la e_{\alpha}^{(k)},\, \Xi^{m}[\infty] (A_{L}^{m})\, e_{\gamma}^{(k)}
\ra = \la e_{\alpha}^{(k)},\, A_{L}^{m}\, e_{\gamma}^{(k)} \ra \:
\delta_{E_{\alpha}^{(k)},E_{\gamma}^{(k)}},
\label{erg-0}
\end{equation}
where $\delta$ is the Kronecker $\delta$-\fn. This formula
shows that $\Xi^{m}[\infty] (A_{L}^{m})$ is well defined on all vectors
in $D(A_{L}^{m})$.
\par
One might think to prove now the statement regarding $J(\infty,L,m)$
by simply substituting the Heisenberg invariant operator
$\Xi^{m}[\infty] (A_{L}^{m})$ to $\Op(a')$ and $\Op(b')$ in Theorem
\ref{thm-mix}. We cannot quite do this, since such operator is not
in general \psd. It is obvious, though, that it can be
approximated to any extent by \psd\ operators, and the result
would follow by density. However, as remarked in \S\ref{subs-Gibbs},
we have not defined a proper $C^{*}$-algebra for the infinite-particle
system. Thus, we cannot talk of any density and have to prove the
theorem directly.
\par\noindent
$\Xi^{m}[\infty] (A_{L}^{m})$,
roughly speaking, represents the
quantization of $\displaystyle a'_{\infty} :=
\lim_{T\to\infty} a'_{T}$, which is not in
general a symbol, being possibly not even
continuous. But simple considerations based
upon the trivial dynamics over $\LLm$ (see also
the remark after the statement of this theorem)
show that it is almost everywhere (namely for
$p=(p_{1},
\ldots,p_{m})$ having rationally independent components) equal to
\begin{equation}
c'(p,q) = c'(p) := \frac{1}{L^{m}} \int_{\TLm}
a'(p,q) \, dq
\label{erg-c-prime}
\end{equation}
which is a symbol. Moreover we denote it $c'$
since one can straightforwardly find a $c\in
{\cal A}$ such that $c' = c\circ\PLm$. The
whole idea of this proof is exactly to show
that, in some sense, $\Op(c')$ is a.e. equal to
$\Xi^{m}[\infty] (A_{L}^{m})$, so that the
former can be substituted to the latter in the
definition of
$J(\infty,L,m)$ in order to apply Theorem \ref{thm-mix} (also compare
(\ref{thm-erg-1}) and (\ref{thm-mix-1})).
\par
We first remark some basic properties of $\Op(c')$. Since $c'(p,q) =
c'(p)$ then $\Op(c')$ is diagonal w.r.t. $\{ e_{\alpha}^{(k)} \}$.
Its diagonal matrix elements, using (\ref{w-alpha-k}), are found to be
\begin{equation}
\la e_{\alpha}^{(k)},\, \Op(c')\, e_{\alpha}^{(k)} \ra =
\frac{c'(\alpha+k)} {L^{m}} = \frac{1} {L^{m}} \int_{\TLm}
a'(\alpha+k,q) dq = \la e_{\alpha}^{(k)},\, A_{L}^{m}\,
e_{\alpha}^{(k)} \ra,
\label{erg-1}
\end{equation}
showing incidentally, as it ought to be, that $\Op(c')$ is
invariant for time evolution. More importantly, (\ref{erg-1}),
together with (\ref{erg-0}), implies that $\Op(c')^{(k)} =
(\Xi^{m}[\infty] (A_{L}^{m}))^{(k)}$
\footnote{Remember that with $A^{(k)}$ we denote $A_{|\Ltwok}$ as
explained in \S\ref{subs-quant}.}
for those $k\in [0,1/L)^{m}$ for which $H_{m}^{(k)}$ is diagonal.
\par
Using (\ref{Wigner-fn}) over a generic $f_{\lambda}$ picked up from
the set of states satisfying (\ref{hyp}), one can see that
$w_{\lambda}(p,q)$ contains a (possibly countable) sum of $\delta$-\fn
s in $p$. This simple argument shows that, in order for $\{
f_{\lambda} \}$ to verify (\ref{hyp}), a factor of the measure space
$(X_{L}^{m}, d\theta)$ must be $([0,1/L)^{m},d\tau(k))$, with $d\tau$
absolutely continuous w.r.t. the Lebesgue measure.
\footnote{This is a manifestation of the fact that all fibers
$\Ltwok$ need to be taken into account, as mentioned in
\S\ref{subs-quant}, Remark 1. We can convince ourselves of this also
looking at the two examples of $\{ f_{\lambda} \}$ we have explicitly
written: the \Fou\ basis in \S\ref{subs-Gibbs}
and the \cs s in the appendix,
\S\ref{appendix-cs}. In both cases $(X_{L}^{m},
d\theta) = ([0,1/L)^{m},dk)) \times$ some
measure.} So if we prove that
$\sigma(H_{m}^{(k)})$ is simple for
Lebesgue-almost all $k$'s, then
\begin{equation}
J(\infty,L,m) := \int_{X_{L}^{m}} \| (\Op(c') - P_{\rho,2\beta}(a) )
\, g_{\lambda} \|^{2}\, d\nu(\lambda)
\end{equation}
and we can apply Theorem \ref{thm-mix} with $a = b = c - P_{\rho,2\beta}(a)$,
which is time invariant. This would complete the proof of the first
claim.
\par
Rescaling (\ref{eigenv}) by a factor $L^{m}$, what we need is
equivalent to the following
\begin{lemma}
\begin{displaymath}
\a \{ k\in [0,1)^{m} \,|\, \exists j,n \in \Z^{m}\:
s.t. \: (j+k)^{2} = (n+k)^{2} \} \a = 0.
\end{displaymath}
\label{lemma-lattice}
\end{lemma}
\par
{\sc Proof of Lemma \ref{lemma-lattice}.}
\footnote{I thank D.Dolgopyat for this simple proof.}
Thinking of it as a geometric problem in $\R^{m}$, when such $j,n$ exist,
then $-k$ lies in the axial hyperplane of the segment joining $j$ to
$n$, i.e. the set of points in the space equally distant from $j$ and $n$.
By very construction there is only a countable number of such hyperplanes.
\qed
\medskip\par
As far as the last statement of Theorem \ref{thm-erg} is concerned,
we see again that it cannot be derived as a corollary of the mixing
theorem since we are taking time limits of both operators. But we can
give a direct proof using the techniques of \S\ref{subs-pf-thm-mix}
and the classical \erg ity result contained in Theorem \ref{vs-thm}.
\par
Exactly as in (\ref{mix-zero}) and (\ref{mix-first}) we can write
\begin{equation}
J(T,L,m) = \int_{\LLm} \left[ (a'_{T} - P_{\rho,2\beta}(a)) \c
(a'_{T} - P_{\rho,2\beta}(a)) \right] (p,q)
\left[ e^{-2\beta\Hm(p)}\, dpdq \right]_{N},
\label{erg-2}
\end{equation}
having used (\ref{erg-00}). We have become familiar with this object
in \S\ref{subs-pf-thm-mix}, and we have seen that integrating -- w.r.t.
the Gibbs measure -- the Weyl composition of two \fn s means
integrating the product of the two \fn s, one of which scrambled by a
convolution (see (\ref{mix-first}), (\ref{mix-second}) and
(\ref{mix-third})). At the thermodynamic limit, this amounts to say
that
\begin{equation}
\tdlim J(T,L,m) = P_{\rho,2\beta} ((a_{T} - P_{\rho,2\beta}(a))
\, c^{(T)}),
\label{erg-3}
\end{equation}
where $c^{(T)}$ is the limit of the ``scrambled \fn s'' constructed
upon $(a_{T} - P_{\rho,2\beta}(a))$. Its existence is granted by Lemmas
\ref{lemma-mix-tech} and \ref{lemma1-mix}, which yielded
(\ref{mix-fourth}). From the construction we have just recalled it
can be seen that if $(a_{T} - P_{\rho,2\beta}(a))$ is bounded then
$c^{(T)}$ is as well.
\par
A remark is in order here: in \S\ref{subs-pf-thm-mix} we have worked
with symbols belonging to ${\cal B}^{(n)}$, and those are unbounded
by definition, being the limits of \fn s like $\NLmn(\beta)$ defined
in (\ref{NLm}). But a simple argument shows that a bounded $a \in
{\cal A} = \sigma (\oplus \, {\cal B}^{(n)})$ remains bounded after
the above procedure, since, roughly speaking, it gets deformed in the
same way in each of its ${\cal B}^{(n)}$-components.
\par
Finally, we can apply Lebesgue dominated convergence in (\ref{erg-3})
since the integrand \fn\ is bounded and tends pointwise
to zero as $T\to\infty$. Thus, the $T$-limit of (\ref{erg-3}) gives
the last statement in Theorem \ref{thm-erg}, whence the end of the proof.
\qed
\section{Acknowledgments}
I would like to thank S.Graffi and Ya.G.Sinai for addressing this
problem to me, giving stimulation as well as useful advices. I also
wish to thank A.Parmeggiani and A.Martinez for interesting
discussions on this subject. A grant from I.N.F.N. is highly
acknowledged.
\appendix
\section{Appendix}
\sectcount
\subsection{Coherent states for the cylinder.} \label{appendix-cs}
We begin this section by recalling some notions about the Bloch
decomposition (\ref{Bloch-dec}), following \cite{rs} and \cite{dbg}.
The idea is very simple: given a \fn\ $f\in L^{2}(\R^{m})$, and
therefore its \Fou\ transform $\hat{f}(p)$, we pick up from the
latter only the terms at $p=\xi+k,\ (\xi\in\TLms)$ to construct
$f^{(k)}$ which clearly lies in $\Ltwok$.
\par
In formula
\begin{equation}
f^{(k)}(x) := \frac{1}{L^{m}} \sum_{\xi\in\TLms} \hat{f}(\xi+k)
\ei{(\xi+k)\cdot x}.
\label{def-fk}
\end{equation}
Considering the scalar products in the dual spaces (respectively
$L^{2}(\R^{m})$ and $\ell^{2}((\Z/L)^{m}+k)$), it is easy to see the
decomposition property which justifies (\ref{Bloch-dec}):
\begin{equation}
\la f,g \ra_{L^{2}(\R^{m})} = \int_{[0,1/L)^{m}} \la f^{(k)},
g^{(k)}\ra_{\Ltwok} dk
\label{Bloch-dec2}
\end{equation}
An explicit formula for $f^{(k)}$, more direct than
(\ref{def-fk}), is also computable with the aid of the Poisson
summation formula:
\begin{equation}
f^{(k)}(x) = \sum_{n\in\Z^{m}} \emi{Ln\cdot k} f(x+Ln)
\end{equation}
\smallskip
We can now proceed to the construction of a remarkable example of states
satisfying the assumptions of the theorems.
Let a family of generalized \cs s for the Euclidean $2m$-dimensional
phase space be given
\begin{equation}
f_{(u,v)} := T(-u,v) f_{0}
\label{cs-on-Rm}
\end{equation}
as constructed in \cite{p}, where $(u,v)\in\R^{2m}$ and $f_{0}\in
L^{2}(\R^{m})$, usually a Gaussian centered at the origin.
According to our preparatory remark, we give the following
\begin{defn}
{\rm\cite{dbg}} The family $f_{(u,v)}^{(k)}$ where $(u,v,k)\in X_{L}^{m}
:= \TLm\times\R^{m} \times [0,1/L)^{m}$ constructed as above is called
a set of {\em \cs s on} $\LLm$.
\label{cs-on-torus}
\end{defn}
We endow $X_{L}^{m}$ with the measure $d\theta (u,v,k) := du\,dv\,dk$ and
check that they verify the hypothesis of the theorem.
\par
We shall work on the \Fou\ antitransform of $w_{\lambda}$, i.e. on the
\Fou-Wigner \fn\ relative to the state $f_{\lambda}$.
\begin{eqnarray}
&\pha& \int_{\LLm}dudv \int_{[0,1/L)^{m}}dk \left\la
f_{(u,v)}^{(k)}, \Tkex f_{(u,v)}^{(k)} \right\ra_{\Ltwok} =
\nonumber \\
&=& \int_{\LLm}dudv \la f_{(u,v)},\Tex f_{(u,v)}
\ra_{L^{2}(\R^{m})} = \nonumber \\
&=& \int_{\LLm}dudv \la T(-u,v)f_{0}, T(-u,v)\Tex f_{0}
\ra_{L^{2}(\R^{m})}\,\ei{(\eta\cdot v + \xi\cdot u)} = \\
&=& \int_{\LLm}dudv \la f_{0}, \Tex f_{0} \ra_{L^{2}(\R^{m})}\,
\ei{(\eta\cdot v + \xi\cdot u)} = \nonumber \\
&=& \la f_{0}, \Tex f_{0} \ra_{L^{2}(\R^{m})} \,\delta_{\xi}
\delta(\eta) = \delta_{\xi}\delta(\eta), \nonumber
\end{eqnarray}
which is another way to state (\ref{hyp}). The first step is justified
by (\ref{Bloch-dec2}) and the third by the commutation relations in the
Heisenberg group.
\par
Finally, this set of \cs s is perhaps the most important among the
possible collections one could choose. As a matter of fact, such
states are indeed introduced to be as {\em localized} as the Heisenberg
principle permits, as clearly explained in \cite{dbg}. Had we
performed a limit $\hbar\to 0$, exploiting the Wigner \fn\ as a measure
of the degree of localization of a state, we would have seen that
\begin{equation}
W_{f_{(u,v)}^{(k)},f_{(u,v)}^{(k)}}(p,q) \longrightarrow
\delta(p-u)\delta(q-v) \mbox{ \ \ as } \hbar\to 0;
\end{equation}
where the dependence on $\hbar$ is implicit in the construction of
$f_{(u,v)}^{(k)}$. This is why one can say that such a state is a good
analogue of a point in the phase space. Therefore we see that the
physical meaning of our quantum \erg\ properties gets clearer and more
classical, of course, in the semiclassical regime.
\subsection{Proof of Lemma 4.1.}
\label{appendix-pf-quant-evol}
Let us check that equality on all the matrix elements with
respect to the standard basis of $\Ltwok$, $\{ e_{\alpha}^{(k)}
\}_{\alpha\in\TLms}$ introduced in \S\ref{subs-quant}. It is
easily computed that
\begin{equation}
T^{(k)}(\eta,\xi) e_{\alpha}^{(k)} = \ei{\eta\cdot (\xi/2 + \alpha +
k)} e_{\alpha+\xi}^{(k)}.
\label{Tealpha}
\end{equation}
If we now denote $c^{t}(p,q) := (c\circ\fLmt) = c(p,q+pt)$, we can
compute its \Fou\ transform which turns out to be $\hat{c^{t}}
(\eta,\xi) = \hat{c}(\eta - \xi t,\xi)$. Thus, substituting into
(\ref{Opb1}) and changing variable,
\begin{equation}
\Op(c^{t}) := \frac{1}{L^{m}} \sum_{\xi\in\TLms} \int_{\R^{m}}
\hat{c}(\eta,\xi)\, T(\eta + \xi t,\xi) \,d\eta.
\label{Opbt}
\end{equation}
In order for the statement to hold for every $c$, it is a
necessary and sufficient condition that $\forall\alpha,\gamma \in
\TLms$
\begin{equation}
\la e_{\alpha}^{(k)}, T(\eta + \xi t,\xi) e_{\gamma}^{(k)} \ra =
\la e_{\alpha}^{(k)}, \ei{t H_{m}} T(\eta,\xi) \emi{t H_{m}}
e_{\gamma}^{(k)} \ra.
\label{to-verify}
\end{equation}
Using (\ref{Tealpha}) we find on the r.h.s.
\begin{equation}
\la e_{\alpha}^{(k)}, T(\eta + \xi t,\xi) e_{\gamma}^{(k)} \ra =
\ei{(\eta + \xi t) \cdot (\xi/2 + \gamma + k)} \, \delta_{\alpha,
\gamma+\xi}.
\end{equation}
Since $P^{(k)} e_{\alpha}^{(k)} = (\alpha + k) e_{\alpha}^{(k)}$, on
the l.h.s. we have
\begin{eqnarray}
&\pha& \la e_{\alpha}^{(k)}, \ei{t H_{m}} T(\eta,\xi) \emi{t H_{m}}
e_{\gamma}^{(k)} \ra = \nonumber \\
&=& e^{\pi i (\alpha + k)^{2} t} \, e^{-\pi i (\gamma + k)^{2} t} \,
\ei{\eta\cdot (\xi/2 + \gamma + k)}\, \delta_{\alpha,\gamma+\xi} \\
&=& \ei{((\xi/2 + \gamma + k) \xi t + (\xi/2 + \gamma + k)\eta)} \,
\, \delta_{\alpha,\gamma+\xi}, \nonumber
\end{eqnarray}
where we have substituted for $\alpha$ its value $\gamma+\xi$. This
relation finally verifies (\ref{to-verify}).
\qed
\subsection{Convolutions over tori and over Euclidean spaces.}
\label{appendix-conv}
The purpose of this section is to clarify the meaning of the symbol
$*_{L}$, indicating $q$-convolution over $\TLm$, when applied to \fn s
that are in principle defined over larger sets, like the \fn s
$b\circ\PLm$, for instance. Also, we want to understand how this is
related to $*_{\infty}$, the ordinary convolution on $\R^{m}$, when
$L \to\infty$.
\par
Since the arguments here are essentially descriptive, we specialize
to one dimension, without loss of generality. If $f$ is a nice \fn\
defined on $\R$, denote by $f^{(L)}$, its {\em periodic restriction},
i.e. the \fn, {\em defined again on $\R$}, which is $L$-periodic and
coincides with $f$ on $[-L/2,L/2)$. Then if $g$ is also a nice \fn\
on $\R$, we define
\begin{eqnarray}
(f *_{L} g)(x) &:=& (f^{(L)} * g^{(L)})(x) = \int_{-L/2}^{L/2}
f^{(L)}(y) g^{(L)}(x-y)\, dy = \nonumber \\
&=& \int_{-L/2}^{L/2} f(y) g^{(L)}(x-y)\, dy.
\label{conv-1}
\end{eqnarray}
Thus, for instance, $(f *_{L} g)(x) \ne \int_{-L/2}^{L/2} f(y)
g(x-y) dy$. As $L \to\infty$, however, we expect this to be
approximately true, at least for a fixed $x\in\R$. As a matter of fact,
requiring some properties of $f$ and $g$, one can prove a useful lemma.
Call $S_{R} := [-R,R]$.
\begin{lemma}
Suppose $f \in C_{0}^{\infty}(\R)$ and ${\rm supp}\, f \subseteq
S_{R}$. Assume also that $|g|$ vanishes monotonically at infinity.
Defining
\begin{displaymath}
h(x) = (f *_{L} g - f *_{\infty} g)(x),
\end{displaymath}
then one has, for $L$ sufficiently large,
\begin{displaymath}
h(x) \: \left\{
\begin{array}{ll}
\le M (\, |g(-L/2)| + |g(L/2-R)| \,) & {\rm for}\ x\in
[-L/2,-L/2+R) \\
= 0 & {\rm for}\ x\in [-L/2+R,L/2-R] \\
\le M (\, |g(L/2)| + |g(-L/2+R)| \,) & {\rm for}\ x\in (L/2-R,L/2]
\end{array}
\right.
\end{displaymath}
where $M := R \max |f|$.
\label{lemma-conv-app}
\end{lemma}
\par
{\sc Proof of Lemma \ref{lemma-conv-app}.} Take $L$ so large that
$L/2 > R$ and $|g|$ is increasing in $(-\infty,-L/2+R]$ and decreasing in
$[L/2-R,+\infty)$.
\par
Looking at (\ref{conv-1}) and recalling the hypothesis on $f$, we can write
\begin{equation}
(f *_{L} g)(x) = \int_{S_{R}} f(y) g^{(L)}(x-y)\, dy.
\end{equation}
Hence
\begin{equation}
h(x) = \int_{S_{R}} f(y) (g^{(L)} - g)(x-y)\, dy.
\label{conv-2}
\end{equation}
Now, $g^{(L)}(x-y)$ coincides with $g^{(L)}(x-y)$ when $x-y \in
S_{L/2}$, that is, when $y \in [x-L/2,x+L/2]$. So (\ref{conv-2}) is
rewritten as
\begin{equation}
h(x) = \int_{S_{R} \setminus (x+S_{L/2})} f(y) (g^{(L)} - g)(x-y)
\, dy.
\end{equation}
It is easily seen that if $x \in [-L/2+R,L/2-R]$, then $S_{R} \subseteq
(x+S_{L/2})$ such that $h(x) = 0$ and part of the claim is proved.
If $x \in [-L/2,-L/2+R)$, making the change of variable $z=x-y$,
(\ref{conv-2}) gives
\begin{eqnarray}
|h(x)| &=& \left| \int_{x-R}^{-L/2} f(x-z) (g^{(L)} - g)(z) dz \right|
\le \nonumber \\
&\le& \max|f| \int_{-L/2-R}^{-L/2} (|g^{(L)}(z)| + |g(z)|) dz.
\end{eqnarray}
Since by definition, for $z< -L/2$, $g^{(L)}(z) = g(z+L)$, the
monotonicity property of $g$ gives the first case in the statement of
the lemma. The third case is of course analogous.
\qed
\subsection{Proof of Lemma 4.3.}
\label{appendix-mix-tech}
Before even getting started, let us agree upon denoting,
throughout this proof, by $(p,q)$ all momentum-coordinate variables,
be they defined on $\Lambda_{L}^{m}$ or on $\R^{2n}$ or on
$\Lambda_{L}^{1}$. Notice that, in the proof of Theorem \ref{thm-mix},
we referred to $n$-dimensional variables as $(p',q')$.
\par
The whole idea here is to realize that, if $n$ is fixed, the function
$\Phi_{L}^{n}$ defined as in (\ref{phiLm-1}), gets closer and closer,
in $\LLn$, to
\begin{eqnarray}
\Phi_{\infty}^{n}(p,q) &=& e^{\beta p^{2}} \int_{\R^{n}} d\xi
\: e^{-\beta(p-\xi/2)^{2}} \, \ei{\xi\cdot q} = \nonumber \\
&=& e^{\beta p^{2}} \, e^{4\pi i p\cdot q}
\left( \frac{4\pi}{\beta} \right)^{n/2}
\, e^{-(4\pi^{2}/\beta) q^{2}} = \label{def-phi-inf} \\
&=& e^{\beta p^{2}} \, e^{4\pi i p\cdot q} \upsilon^{n}(q),
\nonumber
\end{eqnarray}
where $\upsilon(q) := \sqrt{4\pi/\beta}\: e^{-(4\pi^{2}/\beta) q^{2}}$.
In the following we will use repeatedly the asymptotic estimate
$\upsilon(L/2) = \OL$.
\par
Anyway, $\Phi_{\infty}^{n}$ is defined by the fact that it has the same
\Fou\ spectrum of $\Phi_{L}^{n}$, suitably extended to all of $\R^{n}$.
If we denote by $\tilde{\cdot}$ the $q$-\Fou\ transform over $\TLn$, then
this amounts to say that, for $\xi\in (\TLn)^{*} = (\Z/L)^{n}$,
\begin{equation}
\int_{\R^{n}} \Phi_{\infty}^{n}(p,q) \, \emi{\xi\cdot q} \,dq =
e^{\beta p^{2}}\, e^{-\beta(p-\xi/2)^{2}} =: \tilde{\Phi}_{L}^{n}
(p,\xi).
\label{tilde-phi-inf}
\end{equation}
So the best candidate
for $\gamma_{\infty}$ is $\beta *_{\infty} \Phi_{\infty}^{n}$. The
symbol $*_{\infty}$ designates the $q$-convolution over
$\R^{n}$, as explained in \S\ref{appendix-conv}. First of all,
such a $\gamma_{\infty}$ verifies (\ref{lemma-t-two}):
this is a consequence of the fact that $\int_{\R^{n}}
\Phi_{\infty}^{n} = 1$, which is easily verified. Let us proceed to the
proof of (\ref{lemma-t-one}).
\smallskip\par
Recalling the warning in \S\ref{appendix-conv}, the main
inequality will be
\begin{eqnarray}
&\pha& \| \gamma_{L} - \gamma_{\infty} \|_{L^{2} (\LLn,\mu_{L}^{n})}
= \| \beta *_{L} \Phi_{L}^{n} - \beta *_{\infty} \Phi_{\infty}^{n}
\|_{L^{2} (\LLn,\mu_{L}^{n})} \le \label{tech-fund} \\
&\le& \| \beta *_{L} (\Phi_{L}^{n} - \Phi_{\infty}^{n}) \|_{L^{2}
(\LLn,\mu_{L}^{n})} + \| \beta *_{L} \Phi_{\infty}^{n} -
\beta *_{\infty} \Phi_{\infty}^{n} \|_{L^{2}(\LLn,\mu_{L}^{n})}.
\nonumber
\end{eqnarray}
The leftmost term can be worked out pointwise, using
(\ref{def-phi-inf}) and Lemma \ref{lemma-conv-app} of
\S\ref{appendix-conv}. If $(p,q) \in \LLn$,
\begin{equation}
|\, ( \beta *_{L} \Phi_{\infty}^{n} - \beta *_{\infty}
\Phi_{\infty}^{n}) (p,q) \,| \le M \upsilon^{n}
\left( \frac{L}{2} -R \right),
\label{tech-00}
\end{equation}
where $M \simeq \max (|\beta(p,q)| e^{\beta p^{2}})$ and $R$ is the
radius of the ball containing supp$\,\beta$. Now, since
$\mu_{L}^{n}$ is a probability measure,
\begin{equation}
\| \beta *_{L} \Phi_{\infty} - \beta *_{\infty} \Phi_{\infty}
\|_{L^{2}(\LLn,\mu_{L}^{n})} = \OL.
\label{tech-01}
\end{equation}
To work out the other term in (\ref{tech-fund}) we employ the
ideas stated at the beginning of this section about $\Phi_{\infty}^{n}$.
We start by applying the well-known convolution inequality to our case.
We have
\begin{equation}
\int_{\TLn} | \beta *_{L} (\Phi_{L}^{n} - \Phi_{\infty}^{n})
(p,q)|^{2} \, dq \le \| \beta(p,\cdot) \|_{L^{1}(\TLn,dq)}^{2}
\: \| (\Phi_{L}^{n} - \Phi_{\infty}^{n}) (p,\cdot)
\|_{L^{2}(\TLn,dq)}^{2}.
\label{tech-1}
\end{equation}
Notice that $\| \beta(p,\cdot) \|_{L^{1}(\TLn)} = \| \beta(p,\cdot)
\|_{L^{1}(\R^{n})}$, since $\beta$ is compactly supported. We also
see that
\begin{equation}
\| (\Phi_{L}^{n} - \Phi_{\infty}^{n}) (p_{1}, \ldots, p_{n},\cdot)
\|_{L^{2}(\TLn,dq)}^{2} = \prod_{i=1}^{n} \| (\Phi_{L} -
\Phi_{\infty}) (p_{i},\cdot) \|_{L^{2}(T_{L},dq_{i})}^{2},
\label{tech-2}
\end{equation}
since $\Phi_{L}^{n}$ and $\Phi_{\infty}^{n}$ are completely
factorizable: we call, obviously, $\Phi_{L}$ and $\Phi_{\infty}$ their
one-dimensional versions, on which we are immediately going to work.
As anticipated at the beginning of this section, we will be a little
imprecise and use again the label $(p,q)$ for $(p_{i},q_{i})$.
Now
\begin{equation}
\| (\Phi_{L} - \Phi_{\infty}) (p,\cdot) \|_{L^{2}(T_{L})}^{2} =
\frac{1}{L} \sum_{\xi\in(\Z/L)} | \tilde{\Phi}_{L} (p,\xi) -
\tilde{\Phi}_{\infty} (p,\xi) |^{2},
\label{tech-3}
\end{equation}
with $\tilde{\Phi}_{\infty} (p,\xi) = \int_{-L/2}^{L/2}
\Phi_{\infty} (p,q) \emi{\xi q} dq$. Looking back at
(\ref{tilde-phi-inf}) and using definition (\ref{def-phi-inf}),
we can write
\begin{eqnarray}
& \pha & | \tilde{\Phi}_{L} (p,\xi) - \tilde{\Phi}_{\infty} (p,\xi) |
= \nonumber \\
& = & \left| \int_{\R \setminus T_{L}} \Phi_{\infty}(p,q) \, \emi{\xi q}
\,dq \right| = \nonumber \\
& = & 2 e^{\beta p^{2}} \left| \int_{L/2}^{+\infty} \upsilon(q) \,
\cos(2\pi(\xi -2p) q)\, dq \right| = \label{tech-4} \\
& = & 2 e^{\beta p^{2}} \left| \left[ \frac{\upsilon(q) \, \sin(2\pi(\xi
-2p) q)} {2\pi (\xi -2p)} \right]_{L/2}^{+\infty} -
\int_{L/2}^{+\infty} \frac{\upsilon'(q) \, \sin(2\pi(\xi -2p) q)}
{2\pi (\xi -2p)}\, dq \right| \le \nonumber \\
& \le & e^{\beta p^{2}} g(\xi -2p) \upsilon(L/2), \nonumber
\end{eqnarray}
where $g(x)$ is a continuous \fn\ defined on $\R$ behaving like
$|x|^{-1}$ for large values of $x$. Note that, as $L\to\infty$,
$(1/L) \sum_{\xi \in (\Z/L)} g^{2}(\xi -2p) \to \int_{\R}
g^{2}(\xi) d\xi =: K$, {\em uniformly} for $p$ in a compact set.
So, looking at (\ref{tech-1}), and merging up (\ref{tech-2}),
(\ref{tech-3}) and (\ref{tech-4}), we have
\begin{eqnarray}
&\pha& \| \beta *_{L} (\Phi_{L}^{n} - \Phi_{\infty}^{n}) \|_{L^{2}
(\LLn,\mu_{L}^{n})}^{2} = \nonumber \\
&=& \frac{1}{L^{n}} \int_{\R^{n}} \| \beta *_{L} (\Phi_{L}^{n} -
\Phi_{\infty}^{n} (p,\cdot) \|_{L^{2}(\TLn,dq)}^{2} \:
d\breve{\mu}^{n} (p) \le \label{tech-5} \\
&\le& \left( \frac{(K+1) \upsilon^{2}(L/2)}{L^{n}} \right)^{n}
\int_{\R^{n}} \| \beta(p,\cdot) \|_{L^{1}(\R^{n})}^{2} e^{2\beta p^{2}}
\: d\breve{\mu}^{n} (p) = \OL, \nonumber
\end{eqnarray}
since $\beta$ has compact support. Inserting (\ref{tech-01}) and
(\ref{tech-5}) into the fundamental inequality
(\ref{tech-fund}) the proof is completed.
\qed
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\end{document}