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\begin{document}
\vspace*{2.5cm}
\noindent
{\bf CHAOS, QUANTIZATION AND THE CLASSICAL LIMIT }\\
{\bf ON THE TORUS\footnote{ To appear in: Proceedings of the XIVth Workshop on Geometrical Methods in
Physics\\ Bialowieza-Poland july 1995}}\vspace{1.3cm}\\
\noindent
\hspace*{1in}
\begin{minipage}{13cm}
S. De Bi\`evre$^{1}$ \vspace{0.3cm}\\
$^{1}$ UFR de Math\'ematiques et LPTM, Universit\'e Denis Diderot\\
\makebox[3mm]{ }Tour 45-55, 5\`eme \'etage \\
\makebox[3mm]{ }2 Place Jussieu \\
\makebox[3mm]{ }75251 Paris Cedex 05 \\
\makebox[3mm]{ }FRANCE \\
\makebox[3mm]{ }email: debievre@mathp7.jussieu.fr \\
\end{minipage}
\vspace*{0.5cm}
\begin{abstract}
\noindent
The algebraic and the canonical approaches to the quantization of a class of classical symplectic
dynamical systems on the two-torus are presented
in a simple unified framework. This allows for ready comparison between the two very different
approaches and is well adapted to the study of the semi-classical behaviour of the
resulting models. Ergodic translations and skew translations, as well as the hyperbolic toral
automorphisms and their Hamiltonian perturbations are treated. Ergodicity is
proved for the algebraic quantum model of the translations and skew-translations and exponential
mixing in the algebraic quantum model of the hyperbolic automorphisms. This latter
result is used to show the non-commutativity of the classical and large time limits. Turning to
the canonical model, recent results are reviewed on the behaviour in the classical limit of the
eigenvalues and eigenvectors of the quantum propagators; the
link with the ergodic or mixing properties of the underlying dynamics is explained. An example of
the non-commutativity of the classical and large-time limits is proven here as well.
\end{abstract}
% section 1
\section{\hspace{-4mm}.\hspace{2mm}INTRODUCTION}
%\subsection{\hspace{-5mm}.\hspace{2mm}First Subsection of the Current
%Section
%(main words capitalized)}
\hspace*{0.8cm}
Chaos in Hamiltonian dynamical systems is---to say the least---a much studied field. With the word
``chaos" interpreted sufficiently vaguely, it encompasses a vast number of subjects such as
(in)stability, periodic orbit theory, ergodicity, mixing, entropy, Lyapunov exponents,
and hyperbolicity. Two types of questions present themselves rather naturally.
The first one
is: ``How should one define and study the notions of classical chaos in quantum mechanics"? One can
for example legitimately ask what it means for a quantum dynamical system to be ergodic, mixing,
hyperbolic or what the dynamical entropy of a quantum dynamical system is. These questions were
originally addressed by the $C^*$-algebraic school in quantum statistical mechanics interested in
understanding---among other things---the approach or return to equilibrium in systems with an
infinite number of degrees of freedom, where such questions do indeed arise. A beautiful and recent
review of this field is to be found in Ref 1, where special attention is paid to entropic
questions.
Whereas this first
question is---as stated---purely a problem of quantum dynamics, the second question is
semi-classical in nature: ``If a quantum system has a classical limit that is ``chaotic", how does
this reflect on the properties of the quantum dynamics"? In this context, one looks at systems with
a finite (often small) number of degrees of freedom, having a quantum Hamiltonian with discrete
spectrum. In the energy domain, for example, one studies the eigenvalues and eigenfunctions of this
Hamiltonian in a small $\hbar$ dependent interval around a fixed energy value $E_0$. The question is
to understand how the behaviour of these quantities is affected by the nature of the Hamiltonian
dynamics of the classical Hamiltonian on the energy surface at energy $E_0$. In the time domain,
one is interested in detecting differences between the quantal and classical evolution of
observables, and in the (non-)commutativity of the $\hbar\to0$ and $t\to \infty$ limits. Generally,
one wishes to understand the signature of ergodic, mixing, or hyperbolic behaviour, the influence of
periodic orbits, etc.. The physical motivation for this set of problems comes from nuclear, atomic
and solid state physics. There is a vast literature on this subject (often referred to as
quantum chaology), of varying degrees of mathematical rigour\cite{gvz,o,s}.
Of course, both
questions are related. Somehow they refer both to the question of what ``quantum chaos" is, if it
exists at all. They nevertheless reflect different motivations and originate in completely
different physical problems, which explains why they have been studied by two different schools of
reseachers, studying different models. This situation seems to have generated some amount of
Babylonic confusion (and even some bad temper) since in some cases, the same terms are used to denote
entirely different notions: ``quantum ergodicity" is a good example of this. Indeed, precisely
because they have discrete spectrum, the Hamiltonians investigated by the second school can not
generate a quantum dynamics that displays ``chaotic" or even simply ergodic behaviour in the sense
in which this is understood by the first school\cite{be}. Nevertheless, they can ``recuperate"
ergodic (mixing, etc.) properties in the classical limit, and it is in this sense the term ``quantum
ergodic" is used by the second school\cite{s}. This perhaps somewhat unfortunate state of affairs
should however not be construed to mean the concepts themselves are unclear.
There is one class of (toy)
model classical systems, a quantum version of which has been introduced and studied by both schools.
These are the hyperbolic automorphisms of the torus. I will describe both quantum models below and
use them to illustrate the typical problems one addresses in each case. This will, I hope, allow the
reader to appreciate both the differences and the similarities between the two lines of thought. I
will profit from the occasion to discuss some semi-classical questions in the algebraic
model, a problem that seems not to have been addressed until now. I will also prove, in both
models, the non-commutativity of the large time and semi-classical limits (Theorem 5 and Theorem
\ref{nc2}), which constitute the only new results in this contribution.
Let me now first describe the toral automorphisms.
Let $\T^{(2)}=\R^2/\Z^2$ be the two-torus,with local coordinates $x=(q,p)\in
[0,1[\times [0,1[$ and symplectic two-form $\omega= dq\wedge dp$. Let
$A\in SL(2,\Z)$; then $A$ acts on $\T^{(2)}$ in the obvious way. Clearly $A$ is
a symplectic diffeomorphism of $\T^{(2)}$ referred to as an automorphism of the
torus. The case
$$A=\pmatrix{2&1\cr1&1}$$
is often referred to as ``Arnold's cat" . More generally, if $\mid Tr A
\mid > 2$,
then $A$ is an Anosov system. This means in particular that all orbits obtained
by iterating $A$ are exponentially unstable. This is therefore the prototypical example
of a chaotic Hamiltonian system, which is in particular mixing and ergodic, as explained in some
more detail below. Since it is essentially linear, it is also very simple. Hence it seems fair to
hope that one should be able to quantize it and that its quantum version should shed some light on
the two preceding questions.
Two completely different quantum analogs of this system have
been proposed\cite{hb,bns} in the literature, one by each school.
The oldest ``quantum Arnold cat" that roams the literature\cite{hb} is obtained by
a ``canonical quantization" that
sticks closely to the ``ordinary" quantum
mechanics of systems with a finite number of degrees of freedom (in this case one). Here the first
question is: ``What is the quantum Hilbert space of states for a system having the torus as phase
space?". The resulting space turns out to be finite dimensional, in agreement with the fact that
the torus has finite volume. The observables are then identified
with the self-adjoint operators on this Hilbert space. The classical limit corresponds to taking
the dimension of the space to infinity. This quantization starts from the torus viewed as phase space
of a classical system with one degree of freedom. It has an interpretation in terms of an optical
system\cite{hb} and is completely analogous to the one of spin. In that case, the classical phase
space is a sphere and the quantum Hilbert space is $\C^{2\ell +1}$ for spin $s=\hbar\ell$: the
classical limit corresponds to $\ell\to\infty, \hbar\to 0, \hbar\ell= s$. The quantized automorphisms
are realized as unitary operators on the finite dimensional Hilbert space of states and the interest
is in the behaviour of its eigenelements in the classical limit. This model is described in section
3.
The algebraic ``quantum Arnold cat" is the most recent of the two\cite{bns}. It
puts the emphasis on the algebra of observables of the model, defined as an abstract
von Neumann algebra, both in the classical and in the quantum model. The quantum algebra
is a deformation of the classical one (providing the link with non-commutative
geometry\cite{r}) and it is in this sense only that it is viewed as a ``quantization". Following the
usual algebraic formulation of (non-)commutative
dynamical systems, it considers ``states" to be functionals on this algebra (interpreted
as expectation values of observables). The dynamics is given by a discrete group of automorphisms
of the algebra. The ergodic, mixing and entropic properties of this model have been studied in much
detail\cite{bns,ents,nt,aaft,e} in the spirit of the algebraic school mentioned above. In those
papers the model is presented as a quantum analog of the classical automorphisms, rather than as a
quantization, and its classical limit does not seem to have been studied. I will present in section
2 a (slight variation on) the above quantum algebraic model. Rather than relying on $C^*$ and von
Neumann algebraic methods, I will use tools from ordinary quantum mechanics and Weyl quantization to
define and study it. The resulting formulation allows for easy proofs of the various ergodic
properties of the system (Theorems \ref{qmix} and \ref{expqmix}) and is well adapted to the study of
the classical limit, a fact I will illustrate by showing the non-commutativity of the
$\hbar\to0$ and
$t\to\infty$ limits (Theorem \ref{hbart}).
In standard quantum mechanics of a particle on the line,
wave functions are functions (or tempered distributions) $\psi(x)$ of one variable $x\in\R$.
Observables are self-adjoint operators on suitable domains of $L^2(\R)$. Weyl (or canonical)
quantization establishes a link between classical observables (i.e. functions on phase space) and
quantum ones. For $f\in C^\infty (\R^2)$, write
\begin{equation}
f(q,p) = \int\int \tilde f(a,b)
\exp{-\frac{i}{\hbar}[ap-bq]}\ \frac{dadb}{2\pi\hbar},
\end{equation}
and
\begin{equation}
Op^Wf= \int\int \tilde f(a,b)
U(a,b)\frac{dadb}{2\pi\hbar},
\end{equation}
where
\begin{equation}
U(a,b) = \exp -\frac{i}{\hbar}[aP-bQ]
\end{equation}
are the phase space translation operators in the usual notations (See Ref.12 for further
technical details).
If now a Hamiltonian $H(q,p)$ is given, yielding a classical dynamics, then the corresponding
Schr\"odinger equation is
\begin{equation}
i\hbar\partial_t\psi_t=Op^WH\psi_t
\end{equation}
which is solved by $\psi_t=U(t)\psi_0=\exp-\frac{i}{\hbar}tOp^WH\ \psi_0$.
Recalling that if $H$ is a quadratic function ($H=p^2/2$ or $H=p^2/2\pm x^2/2$, for example) then the
corresponding classical flow is linear, one sees that to any $A\in SL(2,\R)$ is in this manner
associated a unitary propagator
\begin{equation}
(U(A)\psi) (x) = \int e^{ \frac{i}{\hbar}S_A(x,y)}\psi(y)dy
\end{equation}
where $S_A$ is the classical action associated to $A$.
With these elements of basic quantum mechanics recalled, it will be easy to understand both the
algebraic and the canonical quantum ``Arnold's cats". The next section
is devoted to the definition and study of the algebraic model, the last one being reserved to the
canonical one.
\noindent{\bf Acknowledgements:} My understanding of the link of the approach presented in
section 2 with the algebraic approach has benefited from helpful conversations I had with G. Emch,
both ``live", in the marvelous setting of the Bialowieza workshop, as well as via email in the
following months.
\section{THE ALGEBRAIC QUANTUM ``ARNOLD'S CAT"}
In the algebraic approach to dynamical systems the starting point are the observables of the theory
and their evolution, as in the Heisenberg picture. The states of the system are defined as expectation
values. In the present case, this approach can be understood as follows. First note that the
classical observables for any system having the torus as phase space are real-valued functions
$f(q,p)$ on the torus, which can usefully be thought of as periodic functions on the plane $\R^2$.
The complex-valued $C^k$ ($0\leq k\leq\infty)$ functions form a $^*$-algebra under complex
conjugation and multiplication. If
$k=\infty$, it is also a Lie-algebra under the Poisson bracket. As in statistical mechanics, states
of the system are described by probability measures that allow one to associate to each observable
$f$ its mean value (or expectation value), given by $\int_{\To} f \ d\mu$. For absolutely continuous
measures with respect to Lebesgue measure, this becomes $\int_{\To} f(x) \rho(x) \ dx$ for some $L^1$-
density $\rho$. The translations $x=(q,p)\in\To\to (q+a, p+b)\in\To$ act by automorphisms on the
function algebras via $f(x)\to \tau_{(a,b)}f(x) = f(q+a, p+b)$. Similarly, for any $H\in
C^\infty(\To)$, the Hamiltonian flow $\Phi^H_t$ generates an automorphism $f\to f\circ
\Phi^H_{t}$. Finally, the toral automorphisms $A$ also define automorphisms $f\to \tau_Af = f\circ
A$. Note that, by concentrating on the automorphisms of the function algebras, the emphasis is
shifted away from the motion of individual points of the torus (i.e. away from individual
trajectories). Lebesgue measure defines the only state invariant under all the above
automorphisms. Nevertheless, for fixed
$\Phi^H_{t}$ or $A$, there are many other invariant states, in particular the ones concentrated on
one periodic orbit. They play an important role in the theory of classical dynamical systems and
I will come back to them below.
The only algebras I will be needing for what follows are the above algebras of smooth functions, but
in order to make the link with the algebraic literature, I will at each stage of the discussion
briefly sketch how $C^*$ and von Neumann algebras show up in this model. These remarks can be safely
ignored by the uninterested as well as by the uninitiated reader.
The uniform closure of any of the above function algebras
is the $C^*$-algebra $C^0(\To)$. The states on $C^0(\To)$ are in one to one correspondence
with the probability measures $\mu$ on $\To$, associating to each observable $f$ its mean value
$\int_\To f\ d\mu$. For some purposes, the $C^*$-algebra of continuous functions is still too
small. If one is convinced of the importance of Lebesgue measure by the above remarks, one might
want to construct the GNS representation of $C^0(\To)$ associated to $dx$. This realizes $C^0(\To)$
as a norm closed algebra of multiplication operators on $L^2(\To, dx)$. Its weak closure is the von
Neumann algebra $L^\infty(\To,dx)$. Normal states on $L^\infty(\To, dx)$ correspond to measures
that are absolutely continuous with respect to Lebesgue measure. Note that measures supported on
periodic orbits are not of this type. Actually, they do not define states on $L^\infty(\To, dx)$ at
all. So where they are deemed important, it is likely that the the use of this von Neumann algebra
should be avoided. At any rate, the von Neumann algebraic dynamical system associated to the classical
dynamical system $(\To, A)$ is the triple $(L^\infty(\To, dx), dx, \tau_A)$. This is the algebraic
version of the classical ``Arnold's cat"\cite{bns,be}.
Let me briefly recall the basic
dynamic properties of the classical hyperbolic toral automorphisms that will interest me here.
\begin{theorem}\label{classmix}Let $A$ be a hyperbolic toral automorphism and let $\mu$ be an
absolutely continuous probability measure on $\To$ (with respect to $dx$). Then, for all
$f\in L^\infty(\To, dx)$ \begin{equation}
\lim_{k\to\infty}\int_{\To} f\circ A^{k} \ d\mu = \int_{\To} f \ dx.
\end{equation}
\end{theorem}
This is a classical result, the proof of which I omit. It can at any rate also be obtained as a
corollary of Theorem \ref{expclassmix} below using a density argument. Theorem \ref{classmix} is
equivalent to saying that $A$ is mixing. It can be stated in ``physical terms" as follows. Thinking
of $dx$ as the ``equilibrium state" of the system, any deviation from equilibrium, of the type $d\mu
= \rho(x) \ dx$, with $\rho\in L^1(\To,dx)$, is washed away in the course of time. It shows,
incidentally, that the only $A$-invariant measure, absolutely continuous with respect to $dx$, is
$dx$ itself. Note that (2.1) fails to hold in general for measures that are not absolutely continuous
with respect to Lebesgue measure. In particular, it does of course not hold for the invariant
measures supported by the periodic orbits of $A$. The return to equilibrium of Theorem \ref{classmix}
is actually exponential, in the following sense. \begin{theorem}\label{expclassmix} Let $A$ be a
hyperbolic toral automorphism $({\rm Tr\ }A>2)$ and $0<\lambda_-<1$ its smallest eigenvalue. Then
there exists a constant $C_A$ so that, for all $f, g\in C^1(\To)$ and for all $k\in\N$,
\begin{equation}
\mid\int_{\To}f(A^{k}x)g(x) \ dx - \int_{\To}f(x) \ dx\int_{\To}g(x) \ dx\mid\leq
\lambda_{-}^k \parallel g\parallel_{2,1} \parallel f\parallel_{2,1}.
\end{equation}
\end{theorem}
This is considered to be an important signature of
chaos, since it can be interpreted as an exponential return to equilibrium at
least for sufficiently smooth observables $f$ and for measures $\mu$ with a $C^1$-density.
The above result is a well-known property of Anosov systems in general, but rather than relying on
general results\cite{b,l} I prefer to give a simple proof of this particular case which has the added
benefit of passing immediately to the quantum version, discussed below (Theorem \ref{expqmix}).
\noindent{\bf Proof:} The essence of the proof is contained in its last few lines, which you might
wish to read first. I start however with some preliminaries. Let $Au_{\pm}=\lambda_{\pm}u_{\pm},
\lambda_+\lambda_- = 1, \lambda_+>1$. Write $u_\pm = (\cos\theta_{\pm}, \sin\theta_{\pm})$ and
$s_{\pm}=\tan\theta_{\pm}$. Note for later purposes that the slopes $s_{\pm}$ are quadratic
irrationals.
It will obviously enough to show (2.2) for $f, g$ of mean zero, so from now on I work in the
subspace of $L^2(\To)$ orthogonal to the constant functions. Define there, on the obvious domain,
the self-adjoint operators $D_{\pm} = \frac{1}{2\pi i}(u_{\pm}\cdot\nabla)$ with eigenfunctions
$\chi_n = \exp i2\pi(n_2q-n_1p)$: $D_{\pm}\chi_n = (u_{\pm}\cdot n)\chi_n$. Also
\begin{equation}
D_{\pm}(f\circ A^k) = \lambda_{\pm}^k (D_\pm f)\circ A^k,
\end{equation}
and
\begin{equation}
D_{\pm}^{-1}g=\sum_{n\in\Z^{2*}}g_n(u_\pm\cdot n)^{-1}\chi_n.
\end{equation}
Note that here ``$\cdot$" denotes the symplectic form: $x\cdot y= x_1y_2-x_2y_1$, for $x,y\in\R^2$.
The only technical ingredient of the proof is the following estimate:
\begin{equation}
\parallel D_{\pm}^{-1}g\parallel_2\leq C_A \parallel g\parallel_{2,1}.
\end{equation}
It is enough to write the proof of (2.5) for $D_-$. Note that this operator has dense pure point
spectrum covering the entire axis, but that $0$ is not an eigenvalue since $s_-$ is irrational. As a
result,
$D_-^{-1}$ is well-defined, but unbounded. The estimate (2.5) shows that any $g\in C^1$ of mean zero
is in its domain. This follows from a simple calculation, using the observation that, since $s_-$ is
a quadratic irrational, basic properties of its continued fraction expansion imply that
$$
\exists C>0, {\rm \ so \ that\ }\quad\forall m, n>0, \ \frac{1}{2Cm}<\mid n-ms_-\mid.
$$
The proof of (2.2) is now immediate:
\begin{eqnarray*}
\int_{\To} (f\circ A^k)(x) g(x) \ dx &=& \\
&=&\\
&\leq&\lambda_-^k\parallel D_-^{-1}g\parallel_2\parallel D_-f\parallel_2,
\end{eqnarray*}
from which (2.2) follows.\endproof
Recall finally that the translation $x\to x+\alpha$ is ergodic
(but not mixing) iff the slope of $\alpha$ is irrational and $\alpha_1, \alpha_2\in\R\setminus\Q$.
Also, the skew translation $x=(q,p)\to(q+\alpha, p+kq)$ is ergodic for $\alpha\in\Q, k\in\Z^*$.
We are now ready to turn to the quantized version of the above model. I will follow ordinary
quantum mechanics to obtain the quantum observables by Weyl quantization of the
classical observables in the standard manner.
For $f\in C^\infty(\To)$, with
\begin{equation}
f(q,p) =
\sum_{n} f_{n} \exp 2\pi i(n_2q-n_1p),
\end{equation}
(1.1)--(1.3) yield
\begin{equation}
Op^Wf = \sum_{n\in\Z^2} f_{n} U(2\pi\hbar n_1, 2\pi\hbar n_2).
\end{equation}
These operators form a $^*$-algebra\cite{f}, that I will denote by $C^\infty_\hbar(\To)$ and which is
the quantization of the classical $^*$-algebra of periodic $C^\infty$ functions. Note that
(2.6)--(2.7) continues to make sense provided
\begin{equation}
\sum_{n\in\Z^2} \vert f_n\vert< \infty,
\end{equation}
so that the sum in (2.6) is uniformly convergent (guaranteeing that $f\in C^0(\To)$) and that the one
in (2.7) is norm-convergent. In the following, whenever I write $Op^Wf$, (2.8) will be assumed to
hold.
Whereas $C^\infty_\hbar(\To)$ provides an obvious quantum analog of $C^\infty(\To)$, I now make
contact with the algebraic approach, by claiming a good quantum analog of the classical von Neumann
algebra
$L^\infty(\To, dx)$ is obtained by taking the weak closure of the above algebra
$C^\infty_\hbar(\To)$, which I shall denote by
$L^\infty_\hbar(\To)$. Similarly, one could define $C^0_\hbar(\To)$ as the $C^*$-algebra obtained
by taking the uniform closure of $C^\infty_\hbar(\To)$. The elements of $L^\infty_\hbar(\To)$
are
operators of the form (2.7), but with coefficients possibly not satisfying (2.8). Note furthermore
that the algebra
$L^\infty_\hbar(\To)$ is the commutant\cite{e} of the algebra generated by $U(1,0)$ and $U(0,1)$, in
accordance with the interpretation of these operators as phase space translation operators. It is
indeed easy to check that $$ [U(a,b),U(k,\ell)]=0,\quad \forall k,\ell\in\Z\iff
\exists n\in\Z^2, \ a=2\pi\hbar n_1,\quad b=2\pi\hbar n_2,
$$
from which one concludes that the ($\hbar$ dependent) algebra $L^\infty_\hbar(\To)$ of quantum
observables is generated by the $U(2\pi\hbar n, 2\pi\hbar m)$.
In other words, and to summarize, the
classical observables of a system on the torus are the subalgebra of all observables $f(q,p)$ for a
system on the full plane having the periodicity property
$f(q+k,p+\ell)=f(q,p),\ \forall k,\ell\in\Z$. In the same way, the quantum observables are those
operators
$F$ on
$L^2(\R)$ satisfying
$U(k, \ell)FU(-k,-\ell)=F$.
Turning now to the states of the system, each density matrix $\rho$ on
$L^2(\R)$ ($0\leq \rho\leq 1,\ {\rm Tr\ }\rho = 1$) defines a (normal) state on $L^\infty_\hbar(\To)$ via
$$
F\in L^\infty_\hbar(\To)\to {\rm Tr\ } F\rho\in\C.
$$
I will argue below (Theorem \ref{qmix}) that these play the role played by the absolutely continuous
measures in the classical model. This ends my discussion of the cinematical setup.
As for the dynamics, standard quantum mechanics, as embodied in equation (1.5), tells us how
observables evolve: for $A\in SL(2,\Z)$,
\begin{equation}
Op^Wf\to \hat\tau_A(Op^Wf)=U(A)^{-1}\ Op^Wf\ U(A).
\end{equation}
Similarly, translations act
via
\begin{equation}
Op^Wf\to \hat\tau_{(a,b)}(Op^Wf) = U(a,b)^{-1}\ Op^Wf\ U(a,b).
\end{equation}
Combining (2.9)-(2.10), one can also quantize skew-translations. Recall\cite{f} that
\begin{equation}
U(A)^{-1}\ Op^Wf\ U(A)=Op^W(\tau_A f),\qquad U(a,b)^{-1}\ Op^Wf\ U(a,b)=Op^W(\tau_{(a,b)} f),
\end{equation}
which is a basic property of Weyl quantization related to the fact that when the
dynamics is affine, ``quantization and evolution commute" (as in a harmonic
oscillator). As a result $\hat\tau_A$ and $\hat\tau_{(a,b)}$ define automorphisms of
$C^\infty_\hbar(\To)$ and extend to automorphisms of $L^\infty_\hbar(\To)$. Note that, for $H\in
C^\infty(\To)$, one similarly defines a one-parameter group of automorphisms by
\begin{equation}
Op^Wf\to e^{\frac{i}{\hbar}t Op_W H}\ Op^Wf\ e^{-\frac{i}{\hbar}t Op_W H},
\end{equation}
but now
\begin{equation}
e^{\frac{i}{\hbar}t Op_W H}\ Op^Wf\ e^{-\frac{i}{\hbar}t Op_W H} \not=Op^W(f\circ \Phi_t^H),
\end{equation}
a fact I shall come back to below.
To further support my claim that the algebras $C^\infty_\hbar(\To)$ and $L^\infty_\hbar(\To)$ are good
quantum analogs of $C^\infty(\To)$ and $L^\infty(\To, dx)$ and that the above automorphisms are
the analogs of their classical counterparts, I now prove the following theorem.
\begin{theorem}\label{qmix} \begin{itemize}
\item[(i)] Let $\rho$ be a density matrix on $L^2(\R)$. Then
$$
{\rm Tr\ }Op^Wf\ \rho = {\rm Tr\ }U(a,b)^{-1}Op^Wf\ U(a,b)\ \rho,\ \forall (a,b)\in\R^2,\ \forall
f\in C^\infty(\To),
$$
iff
$$
{\rm Tr\ }Op^Wf\ \rho = f_0 = \int_{\To} f(x) \ dx, \ \forall f\in
C^\infty(\To).
$$
\item[(ii)] There exists a density matrix $\rho_0$ so that, $\forall f\in
C^\infty(\To)$,
$$
{\rm Tr\ }Op^Wf\ \rho_0 = \int_{\To} f(x) \ dx.
$$
Moreover, $\rho_0$ can be chosen of the form $\rho_0=Op^W\sigma_0$, with $\sigma_0\in {\cal
S}(\R^2)$.
\item[(iii)] In addition, for all $f,g\in C^\infty(\To)$,
$$
{\rm Tr\ }Op^Wf\ Op^Wg\ \rho_0 = {\rm Tr\ }Op^Wg Op^Wf\ \rho_0 = \int_{\To} f(x) g(x) \ dx.
$$
\item[(iv)] For any density matrix $\rho$, and $\forall f\in
C^\infty(\To)$,
$$
\lim_{K\to\infty} {\rm Tr\ }\bigl[\frac{1}{K}\sum_{k=0}^{K-1}U(a,b)^{-k}Op^Wf\ U(a,b)^k\bigr]\rho=
{\rm Tr\ }Op^W f\ \rho_0,
$$
provided $x\to (q+a, p+b)$ is an ergodic translation.
\item[(v)] If $\rho$ is a density matrix and $A$ a hyperbolic toral automorphism, then
$$
\lim_{k\to \infty} {\rm Tr\ }U(A)^{-k}Op^Wf\ U(A)^k\ \rho = {\rm Tr\ }Op^W f\ \rho_0.
$$
\end{itemize}
Statements (iii), (iv) and (v) remain true with $Op^Wf, Op^Wg$ replaced by any $F,G\in
L^\infty_\hbar(\To)$.
\end{theorem}
Theorem \ref{qmix} (v) was proven in Ref.6. Actually,
the von Neumann algebra proposed in Ref.6 as the quantum analog (deformation) of the classical one
differs from the one constructed here, but it is algebraically and topologically isomorphic to
it\cite{e}. In fact, it is just the von Neumann algebra obtained from the GNS representation of the
faithful normal tracial state $\rho_0$.
The interpretation of the above results is rather clear. The density matrix $\rho_0$ plays the role
played by Lebesgue measure in the classical model. Note that, although $\rho_0$ is certainly not
unique as a density matrix, (i) says that there is a unique translationally invariant (normal) state
on $L^\infty_\hbar(\To)$. According to (iv), quantum translations are ergodic with respect to
$\rho_0$; the same is true for the quantized ergodic skew-translations, as is easily checked. The
quantized hyperbolic automorphisms are mixing, according to (v) (Compare to Theorem
\ref{classmix}). In algebraic terms,
$(L^\infty_\hbar(\To),
\rho_0, \hat
\tau_{(a,b)})$ is a quantum ergodic system and $(L^\infty_\hbar(\To), \rho_0, \hat \tau_A)$ is a
quantum mixing system as defined in Ref.1. So it is clear that the ergodic properties of the
classical model carry over unaltered and effortlessly to the algebraic quantum model: the reason for
this is (2.11). While finishing these notes, I noticed that an equivalent of Theorem \ref{qmix} is
proven in Ref.16, but using anti-Wick quantization, working in the Bargmann representation.
\noindent{\bf Proof:} Once (ii) is proven, (i), (iii) and (iv) follow immediately from (2.11), upon
taking $f=\chi_n, g=\chi_{n'}$. To prove (ii), take, in the notations of (1.1)--(1.3),
$\tilde\sigma_0\in C_0^\infty(\R^2),\ \tilde\sigma_0(0,0)=1$, and such that ${\rm
supp\ }\tilde\sigma_0$ is contained in the ball of radius $\pi\hbar$. Since
$$
{\rm Tr\ }Op^W\sigma_0\ U(a,b)^{-1} = \tilde\sigma_0(a,b)
$$
it follows that, for all $f\in C^\infty(\To)$ (see (2.7))
$$
{\rm Tr\ }Op^W\sigma_0\ Op^Wf = f_0.
$$
Note that $Op^W\sigma_0$ is a Hilbert-Schmidt operator. It remains to show it can be chosen
positive:
$Op^W\sigma_0\geq0$. Let $\eta_0(y)=\bigl(\frac\omega{\pi\hbar}\bigr)^{1/4}\exp -\omega y^2/2\hbar$
and $W\eta_0$ its Wigner function. Lemma 2.85 in Ref.15 (or a direct computation) shows that,
if $\tau\in{\cal S}(\R^2),\ \tau\geq0$, then $Op^W(\tau * W\eta_0)\geq0$, where $*$ stands for
convolution. Hence we need to solve $\sigma_0=\tau*W\eta_0$ or $\tilde\sigma_0=\tilde\tau
\widetilde{W\eta_0}$ for $\tau\geq0$. Taking $\tilde\tau =\tilde {\bar k}*\tilde k,\ \tilde k\in
C_0^\infty(\R^2)$, with the support of $\tilde k$ sufficiently small around the origin, (ii)
follows. To prove (v), note that it is enough to show that
$$
\lim_{k\to\infty} <\psi, U(A)^{-k} Op^W\chi_n\ U(A)^k\psi>=0,
$$
for all $\psi\in L^2(\R), n\in\Z^{*2}$. Using (2.11), this follows readily from the Riemann-Lebesgue
lemma together with the observation that, since $A\in SL(2, \Z)$ is hyperbolic, both components of
the vector $A^kn$ tend to infinity with $k$. \endproof
I
now state and prove, without further comment, the quantum equivalent of Theorem \ref{expclassmix}, a
version of which is proven in Ref.8.
\begin{theorem} \label{expqmix} Let $\rho$ be a trace-class operator on $L^2(\R)$,
such that $0\leq\rho\leq1,\ {\rm Tr\ }\rho =1$ and suppose $\rho$ is of the
form $\rho=Op^W\sigma_\rho$ with $\sigma_\rho, D_+\sigma_\rho\in L^1(\R^2)$. Suppose $A$ is a
hyperbolic toral automorphism, as in Theorem \ref{expclassmix}. Let $f\in C^0(\To)$ so that
$\sum \vert n\vert\vert f_n\vert<\infty$. Then
there exists a constant $C$, depending on $A, f$ and $\rho$, so that, for all $k\in\N$,
\begin{equation}
\mid{\rm Tr\ } [ U(A)^{-k}Op^Wf\ U(A)^k\ \rho] - \int_{\To} f(x) \ dx\mid \leq C \lambda_-^k.
\end{equation}
\end{theorem}
\noindent{\bf Proof:} As before, we can suppose $f$ is of mean zero and compute
\begin{eqnarray*}
{\rm Tr\ } [ Op^W(f\circ A^k)\rho]&=&\int_{\R^2} (f\circ A^k)(x) \sigma_\rho(x)
\ \frac{dx}{2\pi\hbar}\\
&=& \int_{\R^2} D_+^{-1}(f\circ A^k)(x) (D_+\sigma_\rho)(x)
\ \frac{dx}{2\pi\hbar}
\end{eqnarray*}
>From (2.3), one infers
$$
D_+^{-1}(f\circ A^k)=\lambda_-^k((D_+^{-1}f)\circ A^k)
$$
so that
$$
\mid {\rm Tr\ } [ Op^W(f\circ A^k)\rho]\mid \leq \lambda_-^k\parallel D_+\sigma_\rho\parallel_1
\parallel D_+^{-1}f\parallel_\infty
$$
whence the result follows with an estimate as in the proof of Theorem 2.
\endproof
This ends my description of the basic ergodic properties of the quantized translations, skew
translations and toral automorphisms in their algebraic version. They mimic the classical properties
almost perfectly and are easy to prove thanks essentially to (2.11) which, one way or another,
reduces all quantum properties to classical ones. The situation would be less trivial and more
interesting should we study the quantization of skew translations of the form $(q,p)\to(q+\alpha, p+
h(q))$, which are easily quantized, or, better yet, Hamiltonian perturbed toral automorphisms,
which are known to still be Anosov, and hence exponentially mixing. They can be constructed as
follows\cite{bdb}. Let $H\in C^\infty(\To)$, write $\Phi^H_t$ for the corresponding Hamiltonian flow,
and consider $$
B_\epsilon = A\circ \Phi_\epsilon^H.
$$
The structural stability of Anosov systems guarantees that
this is still an Anosov symplectomorphism provided $\epsilon$ is taken small enough. If I now define
the unitary operator
\begin{equation}
U(B_\epsilon)=U(A)\exp -i\frac{\epsilon}{\hbar}Op^W H
\end{equation}
on $L^2(\R)$, then, as in (2.9), it defines an automorphism of the algebras $C^\infty_\hbar(\To)$ and
$L^\infty_\hbar(\To)$:
$$
Op^Wf\to \hat\tau_{B_\epsilon}(Op^Wf) = U(B_\epsilon)^{-1}Op^Wf\ U(B_\epsilon).
$$
Note that in this case the equivalent of (2.11) does no longer hold in view of (2.13).
At any rate, it would perhaps be rather more fun to find ways in which the quantized systems differ
radically from the classical ones. The obvious place to look for such phenomena is in
manifestations of the uncertainty principle. As is shown in Theorem \ref{hbart} below, the latter,
together with the mixing property of the quantized hyperbolic automorphisms is indeed at the origin
of the non-commutatitivity of the $\hbar\to 0$ and the $k\to\infty$ limits, which shows that at
sufficiently long times, quantum mechanics ``divorces" irremediably from classical mechanics.
Virtually all
important notions in the study of classical dynamical systems involve the limit as time goes to
infinity. On the other hand, the way in which quantum dynamical systems approach their classical
limit is governed by the limit $\hbar\to 0$. Consequently, the commutativity or absence thereof of
these two limiting procedures has been at the center of many problems in the study of ``quantum
chaos". One expects the two limits not to commute and this can be shown very easily in the present
model, using Theorem \ref{qmix}. A more detailed analysis (in particular of the time scales
involved) will be given elsewhere\cite{bodb}. Recall first the definition of the standard
Weyl-Heisenberg coherent states. For
$z\in\C$, and $Imz>0$, define the gaussian
$$
\eta_{0,z}(y)=(\frac{Imz}{\pi\hbar})^\frac{1}{4}e^{\frac{i}{2\hbar}zy^{2}},\quad y\in\R.
$$
The coherent states are then defined, for all $x=(q,p)\in\R^2$, by
$$
\eta_{x,z}(y)=(U(q,p)\eta_{0,z})(y)
$$
in the standard manner. They are, as usual, interpreted as states, ``optimally localized" at
$x\in\R^2$. It will be useful on occasion to use the bra-ket notation of Dirac:
$$
\eta_{x,z}(y)=.
$$
Note that two different coherent states $\mid x,z>$ and $\mid x',z>$ define different states (i.e.
different functionals) on $L^\infty_\hbar(\To)$ iff $x' \not= x+n,\ \forall n\in\Z^2$! It is
furthermore well known that $$
U(A)\mid x,z> = \mid Ax, A\cdot z>
$$
where $A\cdot z$ denotes the homographic action of $A$ on the upper half plane.
Let us write
$$
\rho(x_0)=\mid x_0,z> \cong
\int_0^{2\pi}\int_0^{2\pi}\frac{d\kappa}{(2\pi)^{2}} \mid x, z, \kappa>
$$
and one shows\cite{bdb}
$$
\int_0^1\int_0^1 \frac{dx}{2\pi\hbar} \mid x, z, \kappa>< x, z, \kappa\mid = {\rm Id}_{\hh}
$$
so that the map
$$
\psi\in\hh\to\in L^2(\To, dx)
$$
is an isometry. The image of $\hh$ under this map is a reproducing kernel subspace of
$L^2(\To)$. It can be seen as a space of complex polarized sections of a prequantum bundle over the
torus, determined by $N$ and $\kappa$.
For fixed $\hbar$ (i.e. for fixed $N$), $\hh$ is a finite dimensional Hilbert space, on
which the dynamics is generated by an $N$ by $N$ matrix, which can not display chaos in any sense of
the word. This is a simple manifestation of the so-called ``quantal suppression of classical chaos".
Indeed, if $\rho$ is a density matrix on $\hh$, any dynamical quantity of the type ${\rm Tr\
}U_\kappa(B_\epsilon)^{-k} Op_\kappa^Wf \ U_\kappa(B_\epsilon)^k \rho$,
or of the type
$
{\rm Tr\ } U_\kappa(B_\epsilon)^{-k} Op_\kappa^Wf\ U_\kappa(B_\epsilon)^kOp^W_\kappa g,
$
is necessarily a quasi-periodic function and in particular no ergodic or mixing property as in
Theorem \ref{qmix}(v) can ever hold for such a system. Actually, the limit as $k$ goes to infinity
simply does not exist for those quantities. As a result, if your main concern is to shed light on
the first question raised in the introduction, the canonically quantized toral automorphisms are
unlikely to be of any help to you, whereas their algebraic cousins might be. They do however form an
interesting testing ground in connection with the second question, as I now explain.
In particular, writing
$$
U_\kappa(B_\epsilon)\psi_j^{(N)} = \exp i\theta_j(N) \psi_j^{(N)},
$$
for the eigenvalues and eigenvectors of $U_\kappa(B_\epsilon)$, one is interested in finding out how
their behaviour in the semi-classical limit $N\to\infty$ is affected by the chaotic nature of the
underlying dynamics.
Before addressing this question, let me point out one might expect this to be a relatively simple
matter in the case $\epsilon = 0$ since the underlying classical system is linear and for such systems
we expect to be able to find the answer to any question via explicit computations, as in the proofs of
section 2. This expectation is not borne out for the case at hand. As an example, the
eigenstates of $U_\kappa(A)$ have only been computed for $A$ of the form (3.5), and even then it
is quite an arduous task\cite{degi}. In addition, it turns out to be hard to study the asymptotic
behaviour of these explicit expressions when they can be obtained. Nevertheless, some results such
as trace formulas relating the trace of the quantum propagator $U_\kappa(A)$ to a sum over periodic
orbits of the classical map can be obtained in closed form for these particular systems, because of
their strong number theoretic properties, in particular when $N$ is a prime number\cite{ke}. In the
end, these systems are surprisingly complicated. Nevertheless, they are often criticized for being too
special, and hence incapable of capturing the essence of the behaviour of quantum systems having a
classical limit that is chaotic. One therefore wishes to study the properties of the quantized
versions of larger classes of classically chaotic systems, for which closed form expressions can no
longer be expected to exist. The easiest candidates in this perspective are perhaps the Hamiltonian
perturbations described above. As already remarked, the failure of the equivalent of (2.11) in this
case makes them more truely quantum mechanical.
I now turn to some of
the semi-classical properties of $U_\kappa(B_\epsilon)$ and their link with the underlying dynamics.
\begin{theorem}\label{schn} Let $U(B_\epsilon)$ be as above and $f\in C^\infty(\To)$. Then
\begin{equation}
V_N(f)=\frac{1}{N}\sum_{j=1}^{N}\mid<\psi_j^{(N)}, Op^W[f - \int_{\To} f(x)
\ dx]\psi_j^{(N)}>\mid^2 \stackrel{N\to\infty}{\to} 0.
\end{equation}
For all $N$, there exists $E_N\subset\{1,\ldots,N\}$ with $\sharp E_N/N\to 1$ so
that $\forall j_N\in E_N$, $\forall f\in C^\infty(\To)$
\begin{equation}
<\psi^{(N)}_{j_N},\,Op^W_\kappa f\ \psi^{(N)}_{j_N}>\stackrel{N\to\infty}{\to}\int_{\To} f(x)\ dx.
\end{equation}
Also,
\begin{equation}
\frac{\mid\mid^2}{2\pi\hbar}\to 1
\end{equation}
in the sense of distributions.
\end{theorem}
\noindent{\bf Proof:} Write $\bar f = \int_{\To} f(x) \ dx$ and $f_K = \frac{1}{K}\sum_{k=1}^K
f\circ A^k$. Then, using the obvious inequality $\mid<\psi, A\psi>\mid^2\leq<\psi,A^*A\psi>$,
together with the basic semi-classical estimates\cite{bdb}
\begin{eqnarray}
U(B_\epsilon)^{-k}Op^Wf\ U(B_\epsilon)^k &=& Op^W(f\circ B_\epsilon^k) + {\cal O}_k(\hbar),\\
Op^Wf \ Op^Wg &=& Op^Wfg + {\cal O}(\hbar),\nonumber\\
\frac{1}{N}{\rm Tr\ } Op^Wg &=& \int_{\To} g(x) \ dx + {\cal O}(\hbar).\nonumber
\end{eqnarray}
one easily sees that
\begin{eqnarray*}
V_N(f)&=&\frac{1}{N}\sum_{j=1}^{N}\mid<\psi_j^{(N)}, Op^W[f - \bar f]_K\psi_j^{(N)}>\mid^2 +
{\cal O}_K(\hbar)\\
&\leq&\frac{1}{N}\sum_{j=1}^{N}<\psi_j^{(N)}, Op^W\bigl[\mid (f -
\bar f)_K\mid^2\bigr]\psi_j^{(N)}> + {\cal O}_K(\hbar)\\
&\leq&\int_{\To} \mid f_K -\bar f\mid^2 \ dx +{\cal O}_K(\hbar).
\end{eqnarray*}
Hence, taking $N\to\infty$
$$
\lim_{N\to\infty}V_N(f) \leq \int_{\To} \mid f_K -\bar f\mid^2 \ dx
$$
whence the result follows upon invoking the ergodicity of the classical map. The rest of
the theorem then follows from standard arguments that I will not repeat here.\endproof
Roughly speaking, the theorem says that, in the classical limit, most of the invariant states of the
quantum system (i.e. its eigenstates) converge to Lebesgue measure. It is an open question if some
eigenstates could converge to other invariant measures of the classical dynamics, in particular
those concentrated on classical trajectories. The results of Theorem \ref{schn} also hold for
quantized ergodic translations and skew-translations\cite{bdb} and in fact for all quantized
ergodic symplectic transformations for which an estimate of the type (3.9) holds. The theorem can be
generalized in various ways. First of all, one might wonder what happens if one restricted the sum in
(3.6) to those eigenvalues lying in a fixed interval
$I$ on the circle. In that case one shows\cite{z,bo} that
$$
\frac{\sharp \{j\mid \theta_j(N)\in I\}}{N}\stackrel{N\to\infty}{\to} \mid I\mid,
$$
where $\mid I\mid$ denotes the Lebesgue measure of $I$. This is a kind of Weyl law, saying that the
eigenvalues equidistribute over the circle. Furthermore\cite{bo},
$$
V_N^I(f)=\frac{1}{\mid I\mid N}\sum_{j,\theta_j(N)\in I}\mid<\psi_j^{(N)}, Op^W[f - \int_{\To} f(x)
\ dx]\psi_j^{(N)}>\mid^2 \to 0.
$$
Since even perturbations of toral automorphisms are still expected to be atypical among chaotic
systems, other models have been quantized and studied, most particularly the Baker map
and the sawtooth maps\cite{sv,la,dbde}. These are discontinuous, uniformly hyperbolic systems, known
to be exponentially mixing\cite{l}, and hence ergodic. In Ref.25 an equipartition result of
the type of Theorem \ref{schn} is proven for those maps. Tracing through the proof of Theorem
\ref{schn}, it is clear that this amounts to showing an estimate of the type (3.9) for those maps.
The attentive reader will have noticed that Theorem \ref{schn} only uses the ergodicity of the
underlying classical system. Mixing has an effect on the off-diagonal matrix elements
$<\psi^{N}_{i},\,Op^W_\kappa f \psi^{N}_{j}>,\ i\not= j$ in the
classical limit\cite{z2,cr,bo}. Roughly speaking
$$
<\psi^{N}_{i_N},\,Op^W_\kappa f \psi^{N}_{j_N}>\to 0,\ i_N\not= j_N.
$$
A result of this type is used in the following theorem, which is an analog of Theorem 5.
\begin{theorem} \label{nc2} Let $A$ be as in (3.5). Then there exists a sequence of integers
$N_\ell\to\infty$ so that, $\forall\psi_{N_\ell}\in\hh, \parallel\psi_{N_\ell}\parallel=1$
\begin{equation}
\lim_{N_\ell\to\infty}\lim_{K\to\infty}\frac{1}{K}\sum_{k=0}^{K-1}<\psi_{N_\ell}, U(A)^{-k}
Op^W_\kappa f\ U(A)^k\psi_{N_\ell}>=\int_{\To} f(x) \ dx.
\end{equation}
On the other hand, for almost all $x_0\in\To$
\begin{equation}
\lim_{K\to\infty}\lim_{N\to\infty}\frac{1}{K}\sum_{k=0}^{K-1}=\int_{\To} f(x) \ dx,\label{irrational}
\end{equation}
whereas for all $x_0\in \To\cap\Q\times\Q$
\begin{equation}
\lim_{K\to\infty}\lim_{N\to\infty}\frac{1}{K}\sum_{k=0}^{K-1}= \frac{1}{T}\sum_{k=0}^{T-1}
f(A^kx_0), \label{rational}
\end{equation}
where $T$ is the period of the orbit of $A$ through $x_0$.
\end{theorem}
\noindent{\bf Remark:} This result, in particular (\ref{rational}) corrects an erroneous statement in
the Main Corollary of Ref.20, where the commutativity of both limits is claimed. In addition,
equation (\ref{irrational}) seems to contradict a claim made in Ref.28 about the supposed failure of
the correspondence principle in the quantized Arnold cat: it is suggested there that only the
behaviour of (\ref{rational}) could occur, (3.11) being excluded.
\noindent{\bf Proof:} Decomposing $\psi_N\in\hh$ on the basis of eigenvectors of $U(A)$, we have
$$
\psi_N = \sum_{r=1}^N c_r^{(N)} \psi_r^{(N)}.
$$
It is easy to see that
\begin{eqnarray*}
\lefteqn{\lim_{K\to\infty}\frac{1}{K}\sum_{k=0}^{K-1}<\psi_{N}, U(A)^{-k}
Op^W_\kappa f\ U(A)^k\psi_{N}>=}\hspace{2cm}\\
& & \sum_{\stackrel{r, s=1}{\theta_r(N)=\theta_s(N)}}^N\bar c_r^{(N)}
c_s^{(N)} <\psi_r^{(N)}, Op^W_\kappa f\ \psi_s^{(N)}>.
\end{eqnarray*}
It is proven in Ref.20 that there exists a sequence of integers $N_\ell\to\infty$ with the
following properties: there exists a constant $C$ so that
\begin{itemize}\item[(i)]$\sup_\ell \max\{{\rm deg\ }\theta_r(N_\ell)\mid r= 1, 2\dots , N\} - \delta_{r s}\int_{\To}f(x)
\ dx\mid <\frac{C}{\sqrt{N_\ell}}.$
\end{itemize}
Hence
\begin{eqnarray*}
\mid\lim_{K\to\infty}\frac{1}{K}\sum_{k=0}^{K-1}<\psi_{N}, U(A)^{-k}
Op^W_\kappa f\ U(A)^k\psi_{N}>&-&\int_{\To}f(x) \ dx\mid\\
&\leq& \frac{C}{\sqrt{N_\ell}}[1+
\sum_{\stackrel{r\not= s}{\theta_r(N)=\theta_s(N)}}\bar c_r^{(N)}c_s^{(N)}].
\end{eqnarray*}
The cross terms are controlled by remarking that, in an $L$-dimensional Euclidean space with
orthonormal basis $e_\ell\ ,\ell =1,\dots,L$, one has, for any $\psi=\sum_{\ell=1}^L c_\ell e_\ell$
$$
\sum_{\stackrel{\ell,\ell'=1}{\ell\not=\ell'}}^L \vert c_\ell\vert\vert c_{\ell'}\vert\leq
(L-1)\parallel\psi\parallel^2.
$$
Applying this argument to the sum over $r\not=s$ and recalling that (i) above shows each eigenspace
is at most $C$ dimensional, independently of $N$, the result follows. The two other
statements of the theorem follow from\cite{bo}
$$
\lim_{N\to\infty} = f(x).
$$
\endproof
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