# To unbundle under UNIX, shell this downloaded file.
# (type "sh filename" if the downloaded file
# is called "filename"). Then LaTeX the individual files.
echo unitary.tex 1>&2
cat >unitary.tex <<'End of unitary.tex'
\documentstyle[]{amsart}
\title{On the dynamics of a general unitary operator}
\author{Oliver Knill\thanks{Division of Physics, Mathematics and Astronomy,
Caltech, 91125 Pasadena, CA} }
\address{Division of Physics, Mathematics and Astronomy,
Caltech, 91125 Pasadena, CA, USA}
\date{11. March 1996}
\subjclass{Primary: 28D05, 28D20, 58F11}
\keywords{Ergodic theory, spectral theory, unitary dynamics}
\newcommand{\Acal}{\mbox{$\cal A$}} \newcommand{\Bcal}{\mbox{$\cal B$}}
\newcommand{\Gcal}{\mbox{$\cal G$}} \newcommand{\Hcal}{\mbox{$\cal H$}}
\newcommand{\Kcal}{\mbox{$\cal K$}} \newcommand{\Xcal}{\mbox{$\cal X$}}
\newcommand{\Ucal}{\mbox{$\cal U$}} \newcommand{\Vcal}{\mbox{$\cal V$}}
\newcommand{\RR}{{\bf R}} \newcommand{\ZZ}{{\bf Z}}
\newcommand{\NN}{{\bf N}} \newcommand{\CC}{{\bf C}}
\newcommand{\QQ}{{\bf Q}} \newcommand{\TT}{{\bf T}}
\newcommand{\tr}{{\rm tr}}
\newcommand{\im}{{\rm Im}}
\newcommand{\re}{{\rm Re}}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{propo}[thm]{Proposition}
\newtheorem{coro}[thm]{Corollary}
\newenvironment{proof}{\begin{trivlist}\item[]{\em Proof.\/\ }}%
{\hfill$\Box$\end{trivlist}}
\def\marnote#1{\marginpar{\scriptsize\raggedright #1}}
\theoremstyle{definition}
\newtheorem{definition}[thm]{Definition}
\newtheorem{example}[thm]{Example}
\theoremstyle{remark}
\newtheorem{remark}[thm]{Remark}
\newtheorem{discuss}[thm]{}
\setlength{\topmargin}{-1cm}
\setlength{\headheight}{1.5cm}
\setlength{\headsep}{0.3cm}
\setlength{\textheight}{24cm}
\setlength{\oddsidemargin}{0.5cm}
\setlength{\textwidth}{16.0cm}
\setlength{\parindent}{0cm}
\setlength{\parindent}{0cm}
\begin{document}
\bibliographystyle{plain}
\maketitle
\begin{abstract}
A unitary operator $U$ acting on the weakly compact unit ball $B$
of a Hilbert space $H$ gives a topological dynamical system $(B,U)$.
We prove that for every ergodic $U$-invariant measure $m$,
the dynamical system $(B,U,m)$ is conjugated to a strictly
ergodic group translation. Especially, the topological entropy of $(B,U)$
is zero. As an application, it follows that any flow on the unit ball
given by a Schr\"odinger equation $i h \psi = L \psi$
has zero topological entropy.
\end{abstract}
\pagestyle{myheadings}
\thispagestyle{plain}
\section{Introduction}
We consider here the problem to find the set of invariant measures of a
linear unitary flow $U_t$ on a Hilbert space $H$.
Important examples of such dynamical systems are the
Schr\"odinger flow $U_t \psi=e^{it L} \psi$ defined by $i \dot{\psi} = L \psi$
or the Koopman flow $f_t(x)=U_tf=f(T_tx)$ on $L^2(X,m)$
associated to an invariant measure $m$ of a
differential equation $\dot{x}=f(x)$ on a manifold $X$. As a motivation for
the problem one can say that the existence
of invariant measures for $U_t$ allows to make statements about
recurrence of a wave $\psi$ in quantum mechanics or a measurable set
$f=1_A$ in classical mechanics. Examples in both settings show that
a linear unitary dynamical system is in general non
trivial.
In quantum mechanics, an orbit $\psi_t$ of the time evolution
can be observed by measuring the
probabilities $|(\phi,U_t \psi)|^2$ that the wave $\psi$
is in the pure state $\phi$ at time $t$. For a differential equation with
invariant measure $m$, one can observe
$m(A \cap T_t(C))=(1_A, U_t 1_C)$, the probability with which an orbit $T_t(x)=x_t$
starting in the set $C$ hits an other set $A$ at time $t$.
It is therefore in both cases natural to look for invariant Borel
measures on $H$ with respect to the weak topology.
The set of invariant measures of the flow is contained
in the set of invariant measures of a time one map so that
one can consider the general problem, to determine
the set of invariant measures of the topological dynamical system $(B,U)$,
where $U$ is an arbitrary unitary operator and $B$ is the weakly compact
unit ball.
We are not aware that this problem has been addressed somewhere already.
In the two cases, when $U$ has only absolutely continuous spectrum
(i.e. scattering states in quantum mechanics, the mixing property
in classical mechanics) or discrete spectrum (i.e. bound states in quantum
mechanics, group translation in classical mechanics), the answer to the
problem is quite easy. However, both discrete and absolutely continuous
spectrum are exceptional in a Baire category sense: a generic unitary
operator $U$ or a generic Koopman operator $U_T$ has purely singular
continuous spectrum \cite{Sim95,Sim95p,ChNa90,Kni96}.
Singular continuous spectrum appears in the quantum mechanics of
solid state physics \cite{Cycon}, especially when the potential
takes only finitely many values (see for example \cite{Hof+95}).
In classical mechanics, it appears, whenever a
topological shift is embedded because many shift invariant measures
have purely singular continuous spectrum \cite{Kni96}.
\section{Topological dynamics of a unitary operator}
Given a unitary operator $U$ on a separable Hilbert space $H$.
The unit ball $B \subset H$ is compact in the weak topology
and $U: B \rightarrow B$ is a homeomorphism.
For $\phi \in H$, let $H_{\phi}$ be the cyclic subspace of $\phi$ and let
$B_{\phi} = \{ ||\phi || \leq 1 \}$ be the unit ball in $H_{\phi}$.
For $\phi,\psi \in H$, the complex measure $\mu=\mu_{\psi,\phi}$
on $\TT=\{ |z|=1\}$ is defined by the functional
$f \mapsto (\psi, f(U) \phi)$ on $C(\TT)$.
\begin{lemma}
\label{Lemma}
The dynamics of $U$ on $B_{\phi}$ is topologically conjugated to a
shift on a compact, weakly closed subset in $l^{\infty}(\ZZ)$.
\end{lemma}
\begin{proof}
Consider
$$ F: B_{\phi} \rightarrow l^{\infty}(\ZZ,\CC), \; \psi
\mapsto \{ (\phi, U^n \psi) \}_{n \in \ZZ} \; , $$
which maps $\psi$ to the Fourier series of the complex
measure $\mu_{\phi,\psi}$ on $\TT$.
$F$ is injective, because if $(\phi, U^n \psi_1)=(\phi, U^n \psi_2)$ for
all $n$, then $(U^{-n} \phi, \psi_1) = (U^{-n} \phi,\psi_2)$ for all $n$ so that
$\psi_1=\psi_2$ in $H_{\phi}$.
If $l^{\infty}(\ZZ)$ has the weak topology, then $F$ is continuous
and $F$ is therefore a homeomorphism
onto its image $Y_{\phi}=F(B_{\phi})$. Moreover, $F$ conjugates
$U: B_{\phi} \rightarrow B_{\phi}$ to the shift on $Y_{\phi}$.
\end{proof}
For $\phi \in H$, denote by $\mu_{\phi} = \mu_{\phi,\phi}$ the spectral
measure of $\phi$. If $\mu_{\phi}$ has only atoms, then $\phi$ is called
discrete. If $\mu_{\phi}$ has no atoms, then $\phi$ is called continuous.
If $\mu_{\phi}$ is absolutely
continuous with respect to Lebesgue measure on $\TT$, then $\phi$ is called
absolutely continuous. If $\phi$ is both orthogonal to the discrete vectors
and all absolutely continuous vectors, it is called singular continuous.
(For refereneces on the spectral theory of unitary operators,
see (\cite{Queffelec} Chapter II) or \cite{CFS} Appendix), for the topological
dynamics and ergodic theory see \cite{DGS,Walters,Petersen}).
\begin{propo}
\label{Propo}
a) Assume $\phi$ is discrete. Then
$U$ restricted to the weak closure of the orbit of $\phi$ is a strictly
ergodic dynamical system, which is topologically conjugated to a group
translation on a compact topological group. \\
b) If $\phi$ is absolutely continuous.
Then $U^n\phi \rightarrow 0$. \\
c) Given $\phi \in H$. There exists a dense $G_{\delta}$ of unitary operators
for which $\phi$ is singular continuous and such that $\phi$ is
recurrent in the sense that there exists
a sequence $n_k \rightarrow \infty$ such that
$U^{n_k} \phi \rightarrow \phi$. \\
d) There is a dense set $\psi$ in the continuous subspace of $U$, such that
the orbit of $\psi$ has $0$ as an accumulation point.
\end{propo}
\begin{proof}
a) The Fourier transform of $\mu_{\phi}
= \sum_{j} a_j \delta(e^{i \lambda_j})$ is the Bohr almost periodic
sequence $F(\phi)=\hat{\mu}_n \sim \sum_{j} a_j e^{i \lambda_j n}$.
The translates of the sequence $F(\phi)$ have a compact
closure $X_{\phi} \subset Y_{\phi} \subset l^{\infty}(\ZZ)$ and
the shift $T$ acts as a group translation on the compact hull $X_{\phi}$.
The claim follows from Lemma~\ref{Lemma}. \\
An alternative proof is obtained by diagonalizing $U$ so that $U$ becomes
a direct sum of operators $U_i$ on the complex plane, where all $U_i$ are
acting by multiplication with some $\lambda_i$. \\
b) If $\mu_{\phi}$ is absolutely continuous
then $\mu_{\phi,\psi}$ is absolutely continuous (\cite{Queffelec}).
The Riemann-Lebesgue lemma assures that
$\hat{\mu}_n=(\phi,U^n\psi) \rightarrow 0$. Because this holds for all
$\phi$, we know that $U^n \psi \rightarrow 0$ in the weak topology. \\
c) A dense $G_{\delta}$ of measures is rigid, that is, it has the property that
for a subsequence $\hat{\mu}_{n_k} \rightarrow 1$ \cite{ChNa90}. A dense
$G_{\delta}$ set of operators has all vectors singular continuous \cite{Sim95}.
Let $\Ucal$ be the set of all unitary operators with the strong operator topology.
The map $\Ucal \to M(\TT)$ attaching to $U$ the spectral measure
$\mu_{\psi,U}$ is continuous and surjective.
A given vector $\psi$ is therefore rigid for
a generic $U$ in the sense that $(\psi, U^{n_k} \psi) \rightarrow 1$ for
some sequence $n_k$, which implies $U^{n_k} \psi$ converges to $\psi$. \\
d) A point $\psi$ is called weakly wandering, if there exists
a sequence $k_i$ such that all vectors $U^{k_i} \psi$ are orthogonal.
Weakly wandering points are dense in the subspace of all continuous vectors
by a theorem of Krengel (\cite{Krengel}).
\end{proof}
\begin{remark}
It would be interesting to know whether $(B,U)$ can be transitive if the Hilbert
space $H$ has
dimension $>1$. If $U$ has a two dimensional discrete subspace $H_p$,
then there can not exist a dense orbit because it would also give
a dense orbit on the unit sphere of the discrete subspace, where the dynamics is
minimal on the closure of an orbit. This contradicts that an initial point in
an eigenspace stays there and is not dense on the sphere.
If $U$ has some absolutely continuous spectrum,
then $(B,U)$ can not be transitive because it would also be transitive on
the absolutely continuous subspace which would contradict
Proposition~(\ref{Propo}) b).
For the transitivity of $(B,U)$, it is necessary that all spectral measures are
purely singular continuous.
\end{remark}
\section{Ergodic theory of a unitary operator}
We look now at ergodic properties of $(B,U)$, that is, we
determine the $U$-invariant measures $m$ on the compact space $B$.
\begin{thm}
\label{thm}
Given a unitary operator $U$ on a Hilbert space $H$ with unit ball $B$.
Every $U$-invariant measure $m$ defines a uniquely ergodic system
$({\rm supp}(m),U,m)$ which is topologically conjugated to a group translation.
\end{thm}
\begin{proof}
(i) Two vectors $\phi,\psi \in H$ define the complex measure $\mu=
\mu_{\phi,\psi}$. By Wiener's theorem,
$$ \lim_{n \to \infty} n^{-1} \sum_{k=1}^n |(\psi, U^k \phi)|^2
= \lim_{n \to \infty} n^{-1} \sum_{k=1}^n |\hat{\mu}_k|^2
= \sum_{x \in \TT} |\mu(\{x\})|^2 \; . $$
(Wiener's theorem is usually formulated for real probability measures on $\TT$
(see \cite{Katznelson}) but holds for any complex measures on $\TT$.)
Especially, the right hand side is zero, for $\phi,\psi$ in the continuous
subspace $H_c$
because then, $\mu$ is a continuous measure ($\mu=\mu_{\phi,\psi}$
is absolutely continuous
to both $\mu_{\psi}$ and $\mu_{\phi}$ \cite{Queffelec}). \\
(ii) For fixed $\psi$,
the map $f: B \rightarrow \RR$, $f(\phi)=|(\psi,\phi)|^2$ is continuous.
Let $m$ be a $U$-invariant measure on $B$. Then $f \in L^1(B,m)$.
By Birkhoff's ergodic theorem,
$$ \lim_{n \to \infty} n^{-1} \sum_{k=1}^n |(\psi, U^k \phi)|^2
= \lim_{n \to \infty} n^{-1} \sum_{k=1}^n f(U^k \phi)
\rightarrow \int_B f(\phi) \; dm(\phi)
= \int_B |(\psi,\phi)|^2 \; dm(\phi) \; . $$
(iii) Let $H=H_{d} \oplus H_{c}$ be the orthogonal
decomposition of $H$ into the discrete and continuous subspace with respect
to $U$.
Every $U$-invariant measure $m$ defines
a $U$-invariant measure $m_{c}$ on $H_c$ by push-forward
$m_c(A)=m(P_c^{-1}A)$, where $P_c: H \to H_c$ is the projection.
Let $B_c$ be the unit ball $H_c$. By (i) and (ii),
$\int_{B_c} |(\psi,\phi)|^2 \; dm_c(\phi) = 0$ for all $\psi \in H_c$.
This implies that $m$-almost all vectors $\phi \in H_c$ are all orthogonal to
the given vector $\psi \in H_c$. Taking a countable dense set
$\{\psi_n\}_{n \in \NN}$ in $B_c$,
$m$ almost all vectors $\phi$ are orthogonal to all these vectors and so
to every vector in $B_c$. Therefore, the support of $m_c$ is contained in
$\{0\} \subset B_c$ or $m_c=0$.
Especially, every invariant measure $m$ has its support on
the subspace $H_d$ and the claim follows from Proposition~\ref{Propo} a).
\end{proof}
\begin{remark}
It follows that $U$ has continuous spectrum,
if and only if every $U$-invariant measure $m$ is located on $\{0\}$.
All nontrivial invariant measures have their support away from $\{0\}$.
Because a dense set of the support ${\rm supp}(\mu)$ of $m$
is on the sphere, every ergodic $U$-invariant measure has its support contained
in a sphere $S_r=\{ ||\phi||=r \}$.
\end{remark}
\begin{remark}
The mean ergodic theorem in Hilbert space deals with averages
$n^{-1} \sum_{k=1}^n U^k \psi$.
Invariant measures are constructed by taking any measure
$\rho$ and taking accumulation points of $n^{-1} \sum_{k=1}^n \rho(U^k)$.
Assume $U$ has some point spectrum not containing $1$.
Because $1$ is not an eigenvalue, there is no nonzero fixed point of $U$
and the mean ergodic theory gives
$n^{-1} \sum_{k=1}^n U^k \psi \rightarrow 0$ while
$n^{-1} \sum_{k=1}^n \delta(U^k \psi)$ converges to a nontrivial invariant
measure.
\end{remark}
\begin{coro}
The topological entropy of $U: B \rightarrow B$ is zero.
\end{coro}
\begin{proof}
The metric entropy of every invariant measure $m$ is zero. By Goodman's
variational theorem in ergodic theory, the topological entropy is the
supremum over all metric entropies and
is vanishing too.
\end{proof}
\begin{remark}
While $U$ is an isometry with respect to the norm (for which $B$
is not compact), it is not an isometry in the weak topology
and it is not apriory clear, that the topological entropy of $U$ is vanishing.
\end{remark}
\section{Additional remarks and questions}
\begin{discuss}
A measure on $B$ which is invariant under a $\RR$ group action
$t \mapsto U_t$, is also invariant under the $\ZZ$ action $n \mapsto U_1^n$
and Theorem~\ref{thm} shows that $U_t$
is measure theoretically conjugated to a flow on a compact topological
group. Especially: "there is no chaos for the Schr\"odinger flow":
for every measure $m$ on $B$ which is invariant under the
Schr\"odinger equation $i h \dot{\psi} = \Delta \psi + V \psi$, the
measure theoretical flow $(B,U_t)$ is measure theoretically
conjugated to a group translation and the flow has zero topological entropy.
\end{discuss}
\begin{discuss}
Let $G$ be a topological group and let $g \mapsto U_g$ be a unitary
representation of $G$ on a Hilbert space $H$. For a measure $m$ which
is invariant under all $U_g$, the dynamical system $(B,U_g)$ with time $G$
is topologically conjugated to a group translation.
If $U$ is only a contraction on a Hilbert space, then Wiener's theorem
still holds (compare \cite{Krengel}) and Theorem \ref{thm} extends
to all contraction semi-groups.
\end{discuss}
\begin{discuss}
While unitary operators are not interesting for the invariant subspace
problem because there are always plenty of invariant subspaces given as images
of spectral projections, it is interesting to ask
for the structure of invariant measures of $(H,A)$, if $A$ is a general
linear operator and $H$ is equipped with the weak topology. In the
compactification $H \cup \{\infty\}$, there must be some
invariant measures. Are still all nontrivial invariant measures having their
support on eigenspaces belonging to eigenvalues on the unit circle?
The nontriviality of such dynamical systems is illustrated by
the open question, whether there exists a system $(H,A)$, where every
nonzero vector $\psi$ has a dense orbit. (A positive answer
would of course solve the invariant subspace problem).
Rolevicz constructed a system $(H,A)$, which is
transitive in the norm topology of $H$ (\cite{Beauzamy}).
\end{discuss}
\begin{discuss}
A unitary operator is called reductive, if the lattice of invariant
subspaces of $U$ and $U^*$ coincide. By a result of Wermer (\cite{Wer52}),
the reductive unitary operators are exactly those
which do not contain a subspace on which $U$ is the shift. If follows that
a unitary operator is reductive if it has no Lebesgue spectrum, so that
a generic unitary operator is reductive.
A unitary Koopman operator $U_T$ is reductive
if and only $T$ has positive metric entropy because a transformation with
positive entropy contains a Bernoulli shift as a factor \cite{Sinai}.
\end{discuss}
\begin{discuss}
In quantum mechanics, one is also interested in the dynamics
$A \mapsto U A U^*$, induced on the space of bounded operators.
Because $V: A \mapsto U A U^*$ is a unitary operator on the Hilbert space of
Hilbert-Schmidt operators $\Bcal_2$
(see \cite{Amrein}), Theorem~\ref{thm} implies
that for every invariant measure $m$
on $\Bcal_2$, the dynamical system $(\sup(m),V)$ is topologically
conjugated to a group translation. If $U$ has continuous spectrum,
then $V$ has also continuous spectrum (\cite{Amrein}).
It is also evident that if $U$ has some point spectrum, then $V$ contains the same
point spectrum. We don't know, whether
the inner automorphism $V: A \mapsto U A U^*$ acting on the weakly compact
unit ball of the algebra of {\it all} operators has zero topological entropy.
\end{discuss}
\begin{discuss}
Point spectrum is unstable,
because operators with singular continuous spectrum form a dense $G_{\delta}$
in the group of all unitaries \cite{Sim95}.
The theory of rank one perturbations also
illustrates, how point spectrum can be distroyed \cite{Sim95p}.
This should be compared to the situation in
Hamiltonian mechanics, where quasi-periodic motion disappears
by small perturbations because of the generic ergodicity results of
Oxtoby-Ulam or Katok-Stepin. In ergodic theory, a small perturbation of
an automorphism with discrete spectrum renders it continuous by Halmos
weak mixing theorem. However, these are quite general perturbations.
Quasi-periodic motion is often observed in experiments both in
Hamiltonian mechanics as well as in quantum mechanics. KAM theory
gives an explanation of that fact in some circumstances.
It would be interesting to know of criteriums for
a topological space of selfadjoint Schr\"odinger operators to contain
some open set of operators with point spectrum.
\end{discuss}
\begin{thebibliography}{10}
\bibitem{Amrein}
W.O. Amrein.
\newblock {\em Non-Relativistic Quantum Dynamics}, volume~2 of {\em
Mathematical Physics Studies}.
\newblock D.Reidel Publishing Company, Dordrecht, 1981.
\bibitem{Beauzamy}
B.~Beauzamy.
\newblock {\em Introduction to Operator theory and invariant subspaces},
volume~42 of {\em North-Holland Mathematical Library}.
\newblock North-Holland, Amsterdam, 1988.
\bibitem{ChNa90}
J.R. Choksi and M.G.Nadkarni.
\newblock Baire category in spaces of measures, unitary operators and
transformations.
\newblock In {\em Invariant Subspaces and Allied Topics}, pages 147--163.
Narosa Publ. Co., New Delhi, 1990.
\bibitem{Sinai}
I.P. Cornfeld and Ya. G.Sinai.
\newblock Dynamical systems {II}.
\newblock In Ya.G. Sinai, editor, {\em Encyclopaedia of Mathematical Sciences,
{Volume 2}}. Springer-Verlag, Berlin Heidelberg, 1989.
\bibitem{CFS}
I.P. Cornfeld, S.V.Fomin, and Ya.G.Sinai.
\newblock {\em Ergodic Theory}, volume 115 of {\em {Grundlehren} der
mathematischen {Wissenschaften} in {Einzeldarstellungen}}.
\newblock Springer Verlag, 1982.
\bibitem{Cycon}
H.L. Cycon, R.G.Froese, W.Kirsch, and B.Simon.
\newblock {\em {Schr\"odinger} Operators---with Application to Quantum
Mechanics and Global Geometry}.
\newblock Springer-Verlag, 1987.
\bibitem{DGS}
M.~Denker, C.~Grillenberger, and K.~Sigmund.
\newblock {\em Ergodic Theory on Compact Spaces}.
\newblock Lecture Notes in Mathematics 527. Springer, 1976.
\bibitem{Hof+95}
A.~Hof, O.~Knill, and B.~Simon.
\newblock {Singular continuous spectrum for palindromic Schr\"odinger
operators}.
\newblock {\em Commun. Math. Phys.}, 174:149--159, 1995.
\bibitem{Katznelson}
Y.~Katznelson.
\newblock {\em An introduction to harmonic analysis}.
\newblock John Wiley and Sons, Inc, New York, 1968.
\bibitem{Kni96}
O.~Knill.
\newblock Singular continuous spectrum and quantitative rates of weakly mixing.
\newblock Caltech 1996, submitted, 1996.
\bibitem{Krengel}
U.~Krengel.
\newblock {\em Ergodic Theorems}, volume~6 of {\em De Gruyter Studies in
Mathematics}.
\newblock Walter de Gruyter, Berlin, 1985.
\bibitem{Petersen}
K.~Petersen.
\newblock {\em Ergodic theory}.
\newblock Cambridge University Press, Cambridge, 1983.
\bibitem{Queffelec}
M.~Queff\'elec.
\newblock {\em Substitution Dynamical Systems---Spectral Analysis}, volume 1294
of {\em {Lecture Notes in Mathematics}}.
\newblock Springer, 1987.
\bibitem{Sim95}
B.~Simon.
\newblock Operators with singular continuous spectrum: {I}. {General}
operators.
\newblock {\em Annals of Mathematics}, 141:131--145, 1995.
\bibitem{Sim95p}
B.~Simon.
\newblock Spectral analysis of rank one perturbations and applications.
\newblock volume~8 of {\em CRM Proceedings and Lecutre Notes}, pages 109--149,
1995.
\bibitem{Walters}
P.~Walters.
\newblock {\em An introduction to ergodic theory}.
\newblock Graduate texts in mathematics 79. Springer-Verlag, New York, 1982.
\bibitem{Wer52}
J.~Wermer.
\newblock On invariant subspaces of normal operators.
\newblock {\em Proc. Amer. Math. Soc.}, 3:270--277, 1952.
\end{thebibliography}
\end{document}
End of unitary.tex
echo discrete.tex 1>&2
cat > discrete.tex <<'End of discrete.tex'
\documentstyle[]{amsart}
\newcommand{\RR}{{\bf R}} \newcommand{\ZZ}{{\bf Z}}
\newcommand{\NN}{{\bf N}} \newcommand{\CC}{{\bf C}}
\newcommand{\QQ}{{\bf Q}} \newcommand{\TT}{{\bf T}}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{propo}[thm]{Proposition}
\newtheorem{coro}[thm]{Corollary}
\newenvironment{proof}{\begin{trivlist}\item[]{\em Proof.\/\ }}%
{\hfill$\Box$\end{trivlist}}
\setlength{\topmargin}{-1cm}
\setlength{\headheight}{1.5cm}
\setlength{\headsep}{0.3cm}
\setlength{\textheight}{24cm}
\setlength{\oddsidemargin}{0.5cm}
\setlength{\textwidth}{16.0cm}
\setlength{\parindent}{0cm}
\setlength{\parindent}{0cm}
\title{A remark on quantum dynamics}
\author{Oliver Knill \thanks{Division of Physics, Mathematics and Astronomy,
California Institute of Technology, 253-37,
Pasadena, CA, 91125 USA. } }
\date{March 30, 1996}
\begin{document}
\bibliographystyle{plain}
\maketitle
\begin{abstract}
For the study of the long time behavior of the
quantum dynamics $\psi \mapsto \exp(itL) \psi$,
where $L$ is a bounded self adjoint operator on a Hilbert space $H$,
we propose to replace $L$ by an operator $\tilde{L}$ solving
$\cos(\tilde{L})=a L$ with $||a L||<1$. Like this,
the time evolution can be computed by iterating
$ A: (\psi,\phi) \mapsto (2 a L \psi - \phi, \psi)$ on $H \oplus H$
which is the time one map for the unitary evolution of $\tilde{L}$.
This is efficient for determining the Fourier coefficients
$\hat{\mu}_n=(\psi,\psi(n))$ with $A^n(\psi,0)=(\psi(n),\phi(n))$
of measures $\mu_{\psi}$ on the circle
which determine the spectral measures
$\nu_{\psi}=(\mu_{\psi}+\overline{\mu}_{\psi})/2$ of $L$. Furthermore,
many properties of the quantum dynamics are not affected by the change
$L \mapsto \tilde{L}$ because the spectral measures are only distorted.
Numerical computations of the Wiener averages
$\lim_{n \rightarrow \infty} n^{-1} \sum_{k=1}^n |\hat{\mu}_k|^2$
becomes so a convenient tool for testing the existence of eigenvalues for $L$.
We illustrate the time discretisation for a tight binding model of an
electron in a constant or random magnetic field in the plane. For random
magnetic fields, where the existence of discrete spectrum is not known, our
experiments suggest that there is no.
\end{abstract}
\section{Introduction}
For simulating a classical quantum mechanical system with Hamiltonian $L$,
one has to determine $\psi_t=\exp(i t L) \psi$, where $\psi=\psi_0$
is the initial wave in the Hilbert space.
The problem is, how to compute $\psi_t$ efficiently. Both technical and
conceptual difficulties can be avoided by working with a tight binding
approximation for $L$, a space discretisation which renders $L$ to be a bounded
operator. Still, with a bounded operator $L$, computing $\exp(i t L) \psi$
is numerically is so convenient and is a time-consuming task especially in higher
dimensions (see \cite{Ric91} for examples of calculations using super-computers).
Many properties of the orbit $\psi_t$ are determined by the spectral measure
$\mu_{\psi}$. It is in many respects irrelevant, whether we
evolve with the Hamiltonian $L$
or use the Hamiltonian $f(L)$, where $f$ is a smooth real function. The reason
is that the spectral measures of $f(L)$ are only
distorted versions of the spectral measures of $L$.
It is therefore natural, to look for a function $f$ such that the discrete
time step $\exp(i f(L))$ can be computed
easily. The purpose of this letter is to point out this possibility and to
illustrate in two examples, how
properties of the spectrum which can be determined experimentally
from the system, which are unchanged by the replacement $L \mapsto f(L)$.
\section{A unitary discretisation}
Given a bounded selfadjoint operator $L$ on a separable Hilbert space $H$.
After rescaling $L \mapsto a L$, with $a \in \RR$, we can assume $||L|| < 1$.
The equation $L=(U+U^*)/2$ has (among other solutions) the two unitary
solutions $U_{\pm} = L \pm i \sqrt{1-L^2}$. One gets
$$ U_{\pm}^n = \exp(in \arccos(L)) = T_n(L) \pm i R_n(L) \; , $$
where $T_n(x) = \cos(n \arccos(x))$ be the $n$'th Chebychev polynomial of
the first kind and let $R_n(x) = \sin(n \arccos(x))$ be
the $n$'th Chebychev function of the second kind. The arithmetic average
of the two solutions $\psi_{\pm}(n) =U^n_{\pm} \psi$
is the wave $\psi(n) = (\psi^+ + \psi^-)/2=T_n(L) \psi(0)$
which satisfies like $\psi_{\pm}$ the recurrence equation
$$ \psi(n+1) = 2L \psi(n) - \psi(n-1) \; . $$
This linear second order difference equation leads to the first order
evolution
\begin{equation}
\label{1}
A: \left( \begin{array}{c} \psi \\ \phi \\ \end{array} \right) \mapsto
\left( \begin{array}{cc} 2 L & -1 \\ 1 & 0 \\ \end{array} \right)
\left( \begin{array}{c} \psi \\ \phi \\ \end{array} \right) \;
\end{equation}
an infinite dimensional symplectic map on $H \oplus H$. \\
Write $(\psi,\phi)(n)=(\psi(n),\psi(n-1))=A^n (\psi(0),\psi(-1))$.
The solutions $\psi_{\pm}(n)$ are obtained when starting with
$(\psi(0),U_{\pm}^* \psi(0))$.
Important for us is the case, when taking the initial condition $(\psi(0),0)$.
We get then the wave $\psi = (\psi_+ + \psi_-)/2)$
which is real if $L$ and if $\psi(0)$ are real.
These observations prove the following proposition.
\begin{propo}
Given $\psi \in H$. Then, $\psi(n)$, obtained from
$A^n (\psi,0)=(\psi(n),\psi(n-1))$
solves $\psi(n+1) + \psi(n-1) = 2L \psi(n)$ and
is the superposition of the two solutions
$\psi_{\pm}(n) = U_{\pm}^n \psi$, where $U_{\pm}$ are two unitary operators
which both solve $U+U^*=2L$ and which have their spectrum in
$\{\pm \; {\rm Im}(z)>0\}$.
\end{propo}
An alternative derivation of (\ref{1}) can be obtained by observing that
the two operators
$$ A= \left( \begin{array}{cc} 2L & -1 \\ 1 & 0 \\ \end{array} \right), \;
B= \left( \begin{array}{cc} \exp(i \arccos(L)) & 0 \\
0 & \exp(-i \arccos(L)) \\ \end{array} \right) \; . $$
on $H \otimes H$ are conjugated if $||L||<1$, because
$A=C^{-1} B C$ with
$ C= \left( \begin{array}{cc}
L-i \sqrt{1-L^2} & L+i \sqrt{1-L^2} \\
1 & 1 \\ \end{array} \right) \;$
and because
$\exp( \pm i \arccos(L)) = L \pm i \sqrt{1-L^2} = U_{\pm}$ is
independent of the choice of $\tilde{L}=\arccos(L)$.
\section{Fourier coefficients of spectral measures}
For $\psi \in H$, the functional $f \mapsto (\psi,f(L) \psi)$
defines by Riesz representation theorem
a measure $\nu_{\psi}$ on $\RR$ called the spectral measure of
$\psi$. On the circle $\TT$, we are interested in the spectral measures
$\mu_{\psi,\pm}$ of the unitary operators $U_{\pm}$ which
are determined by their Fourier coefficients
$(\hat{\mu}_{\psi,\pm})_n=(\psi,U^n_{\pm} \psi)$.
\begin{propo}
Let $\mu_{\psi,\pm}$ be the spectral measures of $\psi$ with respect to
$U_{\pm}$. The measure $\mu_{\psi}=(\mu_{\psi,+} + \mu_{\psi,-})/2$
on the circle $\TT$ is related to the spectral
measure $\nu_{\psi}$ of $L$ on the line $\RR$ by
$\nu_{\psi} = (\mu_{\psi} + \overline{\mu_{\psi}})/2)$.
\end{propo}
\begin{proof}
The two equations $\nu_{\psi} = (\mu_{\psi,\pm} + \overline{\mu_{\psi,\pm}})/2$,
follow from $U_{\pm}+U_{\pm}^* = 2L$. Adding them up gives
$2 \nu_{\psi} = \mu_{\psi} + \overline{\mu_{\psi}}$.
\end{proof}
Remarks. \\
1) The measure $\mu_{\psi} = \overline{\mu_{\psi}}$ on $\TT$
is the spectral measure of
the unitary operator $U=U_+ \oplus U_-$ on $H \oplus H$ which is
the simultaneous evolution of $\psi_+,\psi_-$ on two copies of
the Hilbert space $H$. The measure $\mu_{\psi}$ is symmetric with
respect to complex conjugation.
An unitary operator $V=iU$ solving $i (V-V^*) = 2(L \oplus L)$
has part of the spectrum in the positive real plane and half of the spectrum
in the negative real plane. \\
3) The idea, to study orthogonal
polynomials on $[-1,1]$ by lifting them
onto the circle goes back to Szeg\"o \cite{Szego}. More than one decade ago,
in \cite{GaLa84}, it was suggested to replace ordinary moments
$\int x^n \; d\mu$ by other moments $\int p_n(x) \; d\mu$
in order to get better information on the
spectral measures of operators. The case treated here,
when $p_n(x)=T_n(x)$ is the Chebychev polynomial, has
been left out in \cite{GaLa84}. Chebychev polynomials are also useful in
similar contexts like polynomial expansion of the Green functions
(see \cite{Man96}). (I thank V. Mandelshtam for kindly
sending me this preprint). \\
4) One checks that the involution $S(\phi, \psi) = (-\psi,\phi- L \psi)$
conjugates $A$ to its inverse:
$$ A^{-1} = S A S :
\left( \begin{array}{c} \psi \\ \phi \\ \end{array} \right) \mapsto
\left( \begin{array}{cc} 0 & 1 \\ -1 & 2 L \\ \end{array} \right)
\left( \begin{array}{c} \psi \\ \phi \\ \end{array} \right) \; \; $$
so that $A$ is reversible. This is also obvious from the fact that $A$ and $B$
are conjugated and that $R(\phi,\psi)=(\psi,\phi)$ satisfies $R B R = B^{-1}$.
\begin{propo}
The Fourier coefficients of the spectral measure
$\mu=\mu_{\psi}=(\mu_{\psi,+} + \mu_{\psi,-})/2$ satisfy
$$ (\hat{\mu}_{\psi})_n=(\psi,{\rm Re}(U_{\pm}^n) \psi) = (\psi,T_n(L) \psi) =
(\psi,\psi(n)) \; . $$
\end{propo}
\begin{proof}
If $L=(U_{\pm}+U_{\pm}^*)/2$, we write $U_{\pm}=\exp(i\Phi_{\pm})$,
so that $L=\cos(\Phi_{\pm})$.
Because the measure $\mu=\mu_{\psi}$ satisfies $\mu=\overline{\mu}$,
the Fourier coefficients $\hat{\mu}_n$ are real.
With $\cos(n \Phi_{\pm})=(U_{\pm}^n+(U_{\pm}^n)^*/2$, one obtains
$$ \hat{\mu}_n = (\psi,\frac{U_-^n+(U_+^n)^*}{2} \psi)
= (\psi,\cos(n \Phi_{\pm}) \psi)
= (\psi, \cos(n \arccos(L)) \psi) = (\psi,T_n(L) \psi) \; . $$
Because $2 \psi(n)=\psi(n)_+ + \psi(n)_- = U_+^n \psi(0) + U_-^n \psi(0)$,
we have also $\hat{\mu}_n = (\psi,\psi(n))$.
\end{proof}
\section{Advantages of Evolution~(\ref{1})}
1) {\bf Simplicity} .
The usual Schr\"odinger evolution $\exp(itL)$ needs a numerical integration like
$\exp(itL) \psi \sim \prod_{j=1}^n (1 + \frac{i t}{n} L) \psi$,
where $n$ is so large that $t^n/n!$ is smaller than the
accuracy $\epsilon$.
One problem with this is that the truncation produces
high-frequency noise after a relatively small number of time steps.
On the other hand,
${\rm Re}(\exp(i n \arccos(L))) \psi=T_n(L) \psi$ can be determined exactly
if $L$ is a discrete Schr\"odinger operator on a graph and $\psi$ has compact
support. An other method used in quantum dynamics is to diagonalize
a finite dimensional approximation $L_N$ of $L$ and
evolving its eigenfunctions (see for example in \cite{Ke+92}).
However it is not quantitatively controlled, how well a finite dimensional
Galerkin cut-off respects the actual dynamics. Moreover,
in dimension $d$, the size $N$ has to be so small that
an eigensystem of a $N^{d}$ matrix can be found. Evolution
(\ref{1}) is especially of advantage, if one wishes to
determine all the intermediate steps $\exp(i k \arccos(L))$. This is necessary
when finding averages like
$ n^{-1} \sum_{k=1}^n |(\psi, U^k \psi)|^2$ for tests for point spectrum.
We illustrate such a application below. \\
2) {\bf Unitary}.
While numerical approximations of $\exp(itL)$ are not unitary, the evolution
(\ref{1}) is conjugated to a unitary evolution.
Other discrete unitary time evolutions have been considered in \cite{Be+94}
for some operators. The problem to preserve unitarity has analogues in
numerical integration problems for ODE's, where properties like symplecticity
should be preserved by the discretisation. \\
3) {\bf Locality}.
If $\psi(0)$ has compact support, then $\psi(n)$ has this property also.
The propagation of speed is finite as in a relativistic set-up.
This fact has computational advantages. For example, we know exactly,
after which time, boundary effects begin to influence the value of a wave at
some point. \\
4) {\bf Algebraic nature}.
Evolution (\ref{1}) preserves the algebraic field in which $L$ is defined.
For example, if $L$ is an operator defined over the rationals $\QQ$
and if the coordinates of $\psi(0)$ are rational, then $\psi(t)$ is rational.
Also the Fourier coefficients of spectral measures could then be computed
{\bf exactly} because they are rational numbers.
Evolution~(\ref{1}) is defined also in an algebra over
a not necessarily closed algebraic field. \\
5) {\bf Extendability}.
The evolution (\ref{1}) can be defined on all bounded sequences
$l^{\infty}(\ZZ^d)$ and not
only on $l^2(\ZZ^d)$. For example, the evolution leaves almost periodic
configurations invariant, elements $x \in l^{\infty}(\ZZ^d)$, for which
the closure of all translated sequences $(T^nx)(k)=x(k+n)$
is compact in the uniform topology $d(x,y)={\rm max}_{k \in \ZZ^d}|x(k)-y(k)|$).
This is useful, because as we will see below,
solutions of (\ref{1}) define generalized
eigenfunctions $K\psi=0$ of an operator $K$ defined on space-time, where
$\psi$ does in general not converge to zero at infinity. \\
6) {\bf Essentially unchanged dynamics}.
It is in many respects irrelevant, whether we compute the time evolution
$\exp(it L)$ of $L$ or of a solution of $\tilde{L}=\arccos(L)$,
because spectral properties and so part of the
long time behavior of the dynamics are the
same: the spectral measures of the solution $L_+=\arccos(L)$ satisfying
$\sigma(\tilde{L}) \subset (0,\pi)$ are
just distorted versions of the spectral measures of $L$ because
in general, for any smooth map $f$, $d\mu_{\psi}(f(L))$ is the push-forward
$f_* d\mu_{\psi}(L)$
(and $\arccos: (0,\pi) \rightarrow [0,1]$ is smooth).
Topological or potential theoretical properties which are dynamically
relevant are the same for $L$ and $\arccos(aL)$. \\
7) {\bf Second order approximation}.
The unitary operator $V=iU$ solves $i (V-V^*) = 2 L$
which is a discretisation of the
Schr\"odinger equation $i \dot{U} = 2 L$. Since
$V^4=U^4$, the evolutions $U$ and $V$ are essentially the same.
The evolution (\ref{1}) is a second order approximation to $\exp(it 2L)$
in the sense that $\exp(i \arcsin( 2 \epsilon L ))
= 1-2i \epsilon L -2 \epsilon^2 L^2 - 2 \epsilon^4 L^4 \dots$
and $u \mapsto \exp(2i \epsilon L)
= 1-2i \epsilon L-2 \epsilon^2 L^2
+\frac{4}{3} i \epsilon^3 L^3 + \frac{2}{3} \epsilon^4 L^4 \dots$
are the same up to second order. This second order approximation is
more efficient than methods used like the Cayley method
$ (\epsilon L-i) (\epsilon L+i)^{-1} = \exp(2i \arctan( \epsilon L ))
= 1-2i \epsilon L -2 \epsilon^2 L^2 + 2i \epsilon^3 L^3
+ 2 \epsilon^4 L^4 \dots \; $
which is more stable but costly. \\
{\bf Illustration}. \\
We illustrate evolution~\ref{1} in the case, when $L$ is the free Laplacian
on a graph $(V,E)$ where $V$ is the set of vertices and $E$ the set
of edges. We assume that every vertex $v$ has maximally $d$ neighbors.
The operator $L \psi(v) = (d+1)^{-1} \sum_{(w,v) \in E} \psi(w)$
has norm $<1$. Let $\psi_v$ be a wave vector which is localized
initially at a vertex $v$. The value of ${\rm Re}(\exp(i t \arccos(L))) \psi$
is at integer times given by $T_n(L) \psi$ and can be obtained by
a finite computation. The quantum evolution of a wave should be compared with the
random walk, where $L^n \psi$ is computed. What changed, is that
the polynomial $x^n$ is replaced by the Chebychev polynomial $T_n(x)$.
Opposit to the irreversible random walk,
the quantum evolution is invertible because the knowledge
of the wave at two consecutive times determines it at $t=0$.
The Fourier coefficients $\hat{\mu}_n = (\psi,\psi_n)$
of the measure $\mu$ are easy so compute and so are also the spectral
measures $\nu= 1/2(\mu + \overline{\mu})$ of $\psi$ with respect to $L$.
Experimental investigations of free Laplacians on infinite graphs are
needed because in many examples of regular infinite graphs, one does not know
the spectral type. An example, where nothing about the
spectral type is known, are aperiodic graphs defined by aperiodic tilings of
$\RR^d$. The spectrum of a graph can have all kind of spectral types:
it can be pure point like in the case of a finite
graph or certain self-similar graphs, it can be
absolutely continuous like for $\ZZ^d$ or Cayley graphs of
infinite abelian discrete groups, it can also be singular continuous as
shown recently in \cite{Sim95z}.
\section{Time dependent Hamiltonians}
For numerical simulations of a time-dependent
Schr\"odinger operator, the discrete time evolution (\ref{1})
is useful too. In the time-dependent case, the Schr\"odinger equation
$i \hbar \dot{\psi} = L(t) \psi$
has a first order approximation by the Euler steps:
$U(t) \sim \prod_{j=1}^n (1 + \frac{i t}{n} L( \frac{j t}{n}))$.
Again, there is the problem that the accuracy depends on the size of $n$.
An adaptation of \ref{1} to the time-dependent situation is
\begin{equation}
\label{2}
\left( \begin{array}{c} \psi \\ \phi \\ \end{array} \right) \mapsto
\left( \begin{array}{c} \psi(n) \\ \phi(n) \\ \end{array} \right) =
\left( \begin{array}{cc} 2L(n) & -1 \\ 1 & 0 \\ \end{array} \right)
\cdots
\left( \begin{array}{cc} 2L(2) & -1 \\ 1 & 0 \\ \end{array} \right)
\left( \begin{array}{cc} 2L(1) & -1 \\ 1 & 0 \\ \end{array} \right)
\left( \begin{array}{c} \psi \\ \phi \\ \end{array} \right) \; .
\end{equation}
This evolution is no more conjugated to a unitary evolution as it was when $L(t)=L$
is time independent. Consider a $(d+1)$-dimensional Schr\"odinger operator
of the form $K=\Delta + V = \Delta_1 + L$.
The first coordinate represents "time" and the other $d$ coordinates
represent "space".
If $V$ is independent of this coordinate, then (\ref{2}) is equivalent to
the equation $K u = 0$ if $||L|| < 1$:
\begin{propo}
Given a discrete Schr\"odinger operator
$K=\sum_{i=0}^d h_i \Delta_i +V = h_0 \Delta_0 - 2L$ on $l^2(\ZZ^{d+1})$, where
$\Delta_i \psi_n=\psi_{n+e_i}+\psi_{n-e_i}$ and $V$ is time independent
(that is independent of the first coordinate). If $||L/h_0||<1$, then
solving the problem
$$ K \psi = 0 $$
given $\psi$ on $\{n_0=0\},\{n_0=1\}$
is equivalent to solve the time-dependent Schr\"odinger evolution (\ref{1}).
\end{propo}
\begin{proof}
The time evolution (\ref{2}) is
$h_0 ( \psi(n_0+1) + \psi(n_0-1)) = 2 L(n_0) \psi(n_0)$
where $n_0 \in \NN$ is the time. With
$n=(n_0,n_1, \dots n_d)$, this is obviously equivalent to
\begin{equation}
\label{3}
K \phi(n)=\psi(n+e_0)+\psi(n-e_0) - 2 \frac{L(n)}{h_0} \psi(n) = 0 \;,
\end{equation}
where $e_0=(1, 0, \dots, 0 )$.
\end{proof}
Remark. \\
Even if $||L||$ has arbitrary small norm, an orbit $\psi(n)$
of (\ref{2}) can get unbounded.
An example is, when $L_i$ are IID matrix-valued random variables
because the Lyapunov exponent can then be positive by a theorem of
F\"urstenberg and Kesten. Since the evolution (\ref{2}) can also be
extended to all bounded sequences,
the question arrizes, under which conditions on a given almost periodic operator
$K$, there exists bounded or even almost periodic solutions of $K u = 0$ and so
bounded solutions of the time-dependent Schr\"odinger evolution. \\
{\bf Illustration}. \\
Let us illustrate the evolution (\ref{2}) in the situation
when the Laplacian $K$ is obtained from a lattice gauge field
on a $(d+1)$-dimensional (space-time) lattice $\ZZ^{d+1}$ with unit vectors
$e_i$.
At every bond $(n,n+e_i)$ of $\ZZ^d$ is attached an element $A_i(n)$
of a compact matrix group $G$ like for example $G=SU(N)$ or $U(1)$.
If $G=U(1)$ and $d=4$, then $A$ is a discrete electromagnetic potential
which generates the electro magnetic field $F_{ij}(n) =
A_j(n)^* A_i(n+e_j)^* A_j(n+e_i) A_i(n)$ which is also $G$-valued in this
discrete setup.
Consider the bounded selfadjoint operator
$(K \psi)(n) = \sum_{i=0}^d h_i(A_i(n) \psi(n+e_i)
+ A_i^*(n-e_i) \psi(n-e_i))$
on $l^2(\ZZ^{d+1},\CC^N)$. At least if $G$ is abelian or $d=2$, all
spectral properties of $K$ are entirely determined by the field $F$
(see \cite{Kni95}). Using a gauge transformation $K \mapsto g K g^{-1}$ with
$g: \ZZ^{d+1} \rightarrow G$, one can
achieve $A_0(n)=1$ for all $n \in \ZZ^{d+1}$.
Let $L= \sum_{i=1}^d h_i(A_i + A_i^*)$ be the time-dependent operator on space.
If $||L/h_0||<1$, then $K \psi=0$ can be interpreted as
a discrete time evolution
$$ A_0(n) \psi(n+e_0)+A_0^*(n-e_0) \psi(n-e_0) = 2 \frac{L(n)}{h_0} \psi(n) \;. $$
This equation is gauge invariant and with the special gauge
$A_0(n)=1$, it becomes equivalent to (\ref{3}).
\section{Spectral properties of $L$ read off from measurements}
The discrete evolution (\ref{1}) gives an improved
tool to determine numerically the spectral type of an operator and its
transport properties. We review in this section shortly
some spectral properties which can be
deduced from the Fourier coefficients $(\psi, \psi(n))$. \\
1) {\bf The discrete spectrum}.
Wiener's theorem in Fourier theory
$$\lim_{n \rightarrow \infty}
n^{-1} \sum_{k=1}^n |(\psi, U^n \psi)|^2
= \lim_{n \rightarrow \infty} n^{-1} \sum_{k=1}^n |\hat{\mu}_k|^2
= \sum_{x \in \TT} \mu(\{x\})^2 $$
allows to determine, whether there is some discrete part in the spectral
measure $\mu_{\psi}$ and so some eigenvalues of $L$.
This tool for detecting point spectrum is used in quantum dynamics
(see \cite{Art+94}). If the potential takes finitely many (rational) values,
then $n^{-1} \sum_{k=1}^n |\hat{\mu}_k|^2$ is a (rational) number which can
be computed exactly. Evolution (\ref{1})
allows so to treat the evolution of any bounded discrete Schr\"odinger
operators in one dimension with the same efficiency as
the popular kicked systems like kicked quantum oscillators or
kicked Harper models. \\
2) {\bf The $L^2$-absolutely continuous spectrum}.
If there exists a constant $C$,
such that $\sum_{k=1}^n |(\psi, U^k \psi)|^2 \leq C$
then $\mu_{\psi} \in L^2$,
because of Plancherel's theorem
$\sum_{k=1}^n |(\psi, U^k \psi)|^2 \rightarrow \int |f_{\psi}|^2 \; d\theta$
with $\mu_{\psi}(\theta)= f d \theta$.
It is more difficult to detect other absolutely
continuous spectrum. While $\hat{\mu}_n$ goes to zero by the Riemann-Lebesgue
lemma, if $\mu$ is absolutely continuous, the decay can be arbitary slow and
decay is also possible if $\mu$ is singular continuous.
The $L^2$ absolutely continuous spectrum is however important because by a
result of Kato, the closure of all vectors $\psi$ with $L^2$-absolutely
continuous spectral measures $\mu_{\psi}$ is the absolutely
continuous subspace. \\
3) {\bf Singular continuous spectrum}.
If $(\psi, U^n \psi)$ does not converge to zero, then by the
Riemann-Lebesgue theorem, $\mu_{\psi}$ must have some singular continuous
spectrum. If also $n^{-1} \sum_{k=1}^n |(\psi, U^k \psi)|^2$
converges to zero, then $L$ has purely singular continuous spectrum, a property,
which is generic in many cases (see \cite{Sim95,Sim95p}).
However, if $L$ has purely singular continuous
spectrum, $(\psi, U^n \psi) \rightarrow 0$ still can happen.
The question, whether $(\psi, U^n \psi)$ converges to zero or not can be
subtle and there are both singular continuous measures for which
$(\psi, U^n \psi)$ does or does not converge to zero.
Singular continuous spectrum occurs often in solid state
physics. The dynamics of $U$ on the singular continuous subspace is the
least understood, evenso the ergodic theory of a unitary operator is
clear and the topological entropy an unitary operator acting on the
weakly compact unit ball is zero \cite{Kni96p}. \\
4) {\bf Spectrum with weak continuity properties}.
The quantum dynamics of operators with singular continuous spectrum
has been developed quite well (see \cite{Gua89,Com93,Hol94,Las95}).
A discrete version of a result of Stricharz \cite{Str90} tells that if
there exists a constant $C$ and a function $h \in C(\RR)$ with $h(0)=0$
such that $\mu([a,b]) \leq C h(|b-a|)$ for
all intervals $[a,b]$ on the circle $\RR/(2\pi \ZZ)$ (here identified
with $[0,2\pi)$), then
\begin{equation}
\label{strich}
n^{-1} \sum_{k=1}^n |(\psi, U^n \psi)|^2 \leq C h(n^{-1}) \;
\end{equation}
for all $n$. By a converse of Last \cite{Las95}, if Equation~(\ref{strich})
is satisfied, then $\mu([a,b]) \leq C \sqrt{h(|b-a|)}$ for all intervals
$[a,b]$. In this sense, H\"older continuity properties of the distribution
function $x \mapsto \int_a^x d\mu_{\psi}(t) \; dt$ can be detected by computing
$(\psi, U^n \psi)$. \\
5) {\bf Hausdorff dimension of the spectral measures}.
The $\alpha$-energy of a spectral measure $\mu$ on $\TT$ is for $\alpha>0$
defined by
$I_{\alpha}(\mu) =
\int_{\TT^2} \phi_{\alpha}(\sin(\frac{x-y}{2})) \; d\mu(x) \; d \mu(y)$
with $\phi_{\alpha}(x)=x^{-\alpha}, \phi_0(x)=-\log|x|$.
A measure $\mu=\mu_f$ has finite $\alpha$-energy, if and only if
$\sum_{k=1}^{\infty} k^{\alpha-1}
| \hat{\mu}_n |^2 < \infty$. (\cite{Kahane}).
The Hausdorff dimension of $\mu$,
(the minimum of all Hausdorff dimensions of Borel sets $S$ satisfying
$\mu(S)=1$) is bigger or equal to $\alpha$ if $\mu$ has finite $\alpha$-energy
(see Theorem 4.13 in \cite{Falconer}).
By finding out, where the energy blows up, a lower bound on the
Hausdorff dimension of $\mu$ can be established
and so a lower bound on the Hausdorff dimension
of the support of $\mu$.
\section{Experiments with an electron in a magnetic field}
We illustrate the idea, how to use evolution~(\ref{1}) in two experiments. \\
(i) The first experiment deals with a model for an electron in the plane under
a constant magnetic field $B$ reduced by a Landau gauge
to a one dimensional situation, where
one considers the one dimensional Schr\"odinger operator
$L$ on $l^2(\ZZ)$ of the form
$$ L u(n) = \Delta u(n) + V(n) u(n) \; $$
and take an initial condition which is a localized wave
$\psi(0)= ( \cdots, 0,0,0,1,0,0,0, \cdots )$ at the origin $k=0$.
We use Wiener's theorem
to get numerically information about the discrete part of
the spectral measure. For this illustration, we take the
almost Mathieu (or Harper) operator $V_n=\lambda \cdot \cos(\theta + \alpha n)$,
where much about the spectrum is known (see \cite{Shu94,Las95p,Jit95p}
for reviews. Note that most of the known results hold only for almost all
or generic $\theta$ and under some assumptions on the magnetic flux $\alpha$.
\begin{center} \fbox{ \parbox{15cm}{
{\bf THIS PS FIGURE IN NOT INCLUDED IN THIS VERSION !} \\
Fig 1. A numerical determination
of $S_n(\lambda)=n^{-1} \sum_{k=1}^n |\hat{\mu}_k|^2$ using~(\ref{1}),
for $n=10'000$ as a function of $\lambda \in [0,4]$ in the almost Mathieu
operator with $\theta=\sqrt{3}, \alpha=\sqrt{7}$.
$\mu$ is the spectral measure on the circle belonging
to the vector $\psi=\delta_0$ localized at
the origin in $\ZZ$. The value of $\hat{\mu}_n=(\psi,U^n \psi)$
was computed using evolution~(\ref{1})
with initial condition $(\psi,0)$ on the grid $[-n/2,n/2]$ so that the
boundary effects the value $\hat{\mu}_n$ only after $n$ steps:
$\psi(n)$ has support in $[-n,n]$ and the boundary begins to
affect $\psi$ after $n/2$ time steps and so to influence
$\hat{\mu}_n$ after $n$ steps.
The experiment is in agreement with the now established fact
that there is no point spectrum
for $\lambda \leq 2$ for almost all $\alpha$ (for $\lambda=2$
see \cite{Go+96}) and some point spectrum for $\lambda > 2$ \cite{Jit95}.
Longer runs, ($S_{10000}(2)=0.004858, S_{20000}(2)=0.001900,
S_{40000}(2)=0.001068$) indicated that indeed $S_n(2) \rightarrow 0$
for $n \rightarrow \infty$.
}} \end{center}
(ii) In a second experiment, we take a two dimensional operator $L$ which is
the Hamiltonian for an electron in the discrete plane, where the magnetic field
$B$ is randomly taking values in $U(1)$ (see \cite{Kni96p} for some theoretical
results and \cite{Bar94,Bar94b} for other numerical experiments on this model).
If the distribution of $B(n), n \in \ZZ^2$
is a Haar measure of $U(1)$, then this field can be generated with a vector
potential $A_i(n) = e^{i \theta(n)}$ with independent random variables
$\theta(n)$ having the uniform distribution in $[0,2\pi]$. There is no
free parameter. The ergodic operator, which we consider, is
$$ L \psi(n) = A_1(n) \psi(n+e_1) + \overline{A_1(n-e_1)} \psi(n-e_1)
+ A_2(n) \psi(n+e_2) + \overline{A_2(n-e_2)} \psi(n-e_2) \; $$
and the open question is what is the spectral type of $L$.
\begin{center}
\setlength{\unitlength}{0.004500in}
\begin{picture}(405,302)(140,370)
\thicklines
\put(460,400){\vector( 0, 1){240}}
\put(220,640){\vector( 0,-1){240}}
\put(220,400){\vector( 1, 0){240}}
\put(460,640){\vector(-1, 0){240}}
\put(215,380){\makebox(0,0)[lb]{{\small $n$}}}
\put(445,380){\makebox(0,0)[lb]{{\small $n+e_1$}}}
\put(210,650){\makebox(0,0)[lb]{{\small $n+e_2$}}}
\put(440,650){\makebox(0,0)[lb]{{\small $n+e_1+e_2$}}}
\put(320,515){\makebox(0,0)[lb]{{\large $B(n)$}}}
\put(320,410){\makebox(0,0)[lb]{$A_1(n)$}}
\put(475,515){\makebox(0,0)[lb]{$A_2(n+e_1)$}}
\put(120,515){\makebox(0,0)[lb]{$A_2(n)^*$}}
\put(290,600){\makebox(0,0)[lb]{$A_1(n+e_2)^*$}}
\end{picture}
\end{center}
Fig. 2. The magnetic field $B(n)$ at the plaquette $n \in \ZZ^2$
is obtained from the vector potential $A=(A_1,A_2)$ by
$B(n)=A_2(n)^* A_1(n+e_2)^* A_2(n+e_1) A_1(n)$. \\
While one knows the moments of the density of states of $L$ (the $n$-th
moment of the density of states is the number of closed paths in $\ZZ^2$ of
length $n$ starting at $0$ which give zero winding number to all plaquettes
\cite{Kni96p}), nothing about the spectral type of $L$ seems to be known.
For our now two dimensional experiment, memory limitations allowed us to
experiment only on a $400 \times 400$ lattice. The experimental
values of $S_n$ do grow like $C n^{-1/4}$ in this range,
suggesting that the operator $L$ has no eigenvalues but also no absolutely
continuous spectrum.
Longer runs on larger lattices are needed to get a
clearer picture. \\
Remark. We made also experiments with a Aharonov-Bohm problem on the
lattice. This is the situation, the magnetic field $B$ is different
from $1$ only at one plaquette $n=(0,0)$.
The vector potential $A$ in this situation
can not be chosen differently from $1$ in a compact set. However, in a suitable
gauge, the operator $L$ is a compact perturbation of the free operator by
a result of Mandelstam-Jitomirskaja \cite{MaZh91} see \cite{Kni96p} for an
other proof of this fact).
As expected, there was no indication of some discrete spectrum. The numerical
experiments suggest that $\sum_k |\hat{\mu}_k|^2$ is bounded which would mean
that the spectral measures are in $L^2$.
\section{Discussion}
We pointed out that by changing the
quantum mechanical Hamiltonian, the computation of the unitary quantum
dynamics can be simplified drastically. While the
change of the Hamiltonian alters the dynamics, the long time features
of the dynamics which have relations with the spectral measures,
stay the same. This method allows very well to check
numerically for bound states. It has theoretical interest, because we
can compute the Fourier coefficients of some spectral measures exactly. \\
Because a discrete time evolution is convenient for experiments,
kicked quantum oscillators are well studied in quantum dynamics.
With (\ref{1}), there is a discrete evolution for every bounded operator and
for discrete Schr\"odinger
operators $L$ and waves with compact support $\psi$, the computation of
$U^n \psi$ is a finite process.
In order to compute Wiener averages, we do not
have to determine the integral
$T^{-1} \int_0^T |(\psi, U^t \psi)|^2 \; dt$
but more conveniently a sum $n^{-1} \sum_{k=1}^n |(\psi, U^k \psi)|^2$.
Like this, error sources either due to a lack of unitariness or integration
processes are avoided. \\
Properties of spectral measures $\nu_{\psi}$ of $L^n$ could also be obtained
by determining the moments
$(\psi, L^n \psi) = \int x^n \; d\nu_{\psi}(x)$.
However, moments give only indirectly information about the
nature of the spectrum. The
Stieljes transform $(\psi, (L-E)^{-1} \psi)
= \int (E-x)^{-1} \; d\nu_{\psi}(x)$ of the measure $\nu_{\psi}$ is analytic
at $\infty$ but only by analytic continuation (as Stieljes did with a
continued fraction expansion), one obtaines the behavior
of this function on the real axes, which is relevant for the spectrum.
There is a sufficient condition due to Hausdorff for absolutely continuity of
$\nu$ with respect to any other measure. However, it is for example not
clear, how one reads from the moments, whether the measure $\nu$ has
some discrete part. Szeg\"o's idea to push forward the measure $\nu$
onto the circle is helpful, because on the circle, one has Wiener's theorem
to decide the question about discrete spectrum. This is the idea behind the discrete
time evolution discussed in this letter. \\
We believe that the evolution (\ref{1})
has not only a practical numerical value but illustrates in a different way,
how localisation can disappear by rank one perturbations
(see \cite{Sim95p} and references).
For such perturbations, the Fourier coefficients
$\hat{\mu}_{n}$ of spectral measures are explicitly known polynomials in the
perturbation parameter. If the operator has pure point spectrum, then
$\mu_{\psi}$ is discrete and therefore, the sequence $\hat{\mu}_{n}$
is almost periodic.
\begin{thebibliography}{10}
\bibitem{Bar94b}
R.Fleckinger A.~Barelli and T.Ziman.
\newblock Europhys. lett.
\newblock 27:531--536, 1994.
\bibitem{Bar94}
R.Fleckinger A.~Barelli and T.Ziman.
\newblock Two-dimensional electron in a random magnetic field.
\newblock {\em Phys. Rev. B}, 49:3340--3349, 1994.
\bibitem{Art+94}
R.~Artuso, G.~Casati, F.~Borgonovi, L.Rebuzzini, and I.~Guarneri.
\newblock Fractal and dynamical properties of the kicked {Harper} model.
\newblock {\em Intern. J. Mod. Phys. B}, 8:207--235, 1994.
\bibitem{Be+94}
C.M. Bender, L.R. Mead, and K.A. Milton.
\newblock Discrete time quantum mechanics.
\newblock {\em Computers Math. Applic}, 28:279--317, 1994.
\bibitem{Com93}
J-M. Combes.
\newblock Connections between quantum dynamics and spectral properties of time
evolution operators.
\newblock In {\em Differential equations with applications to Mathematical
physics}, volume 192 of {\em Mathematics in science and engineering}, 1993.
\bibitem{Falconer}
K.~Falconer.
\newblock {\em Fractal Geometry, Mathematical Foundations and Applications}.
\newblock John Wiley and Sons, Chichester, 1990.
\bibitem{GaLa84}
J.P. Gaspard and P.~Lambin.
\newblock Generalized moments: application to solid-state physics.
\newblock In {\em Polyn\^omes Orthogonaux et Applications}, volume 1171 of {\em
Lecture Notes in Mathematics}. Springer Verlag, 1984.
\newblock Proceedings, Bar-le-Duc.
\bibitem{Go+96}
A.~Gordon, S.~Jitomirskaya, Y.~Last, and B.~Simon.
\newblock Duality and singular continuous spectrum in the almost mathieu
equation.
\newblock preprint 1996, available in {mp\_arc\@math.utexas.edu}, 1996.
\bibitem{Gua89}
I.~Guarneri.
\newblock Spectral properties of quantum diffusion on discrete lattices.
\newblock {\em Europhysics letters}, 10:95--100, 1989.
\bibitem{Hol94}
M.~Holschneider.
\newblock Fractal wavelet dimension and localization.
\newblock {\em Commun. Math. Phys.}, 160:457--473, 1994.
\bibitem{Jit95p}
S.Ya. Jitomirskaya.
\newblock Almost everything about the almost {Mathieu operator II}.
\newblock 1995.
\newblock Proceedings of the {XI'th international} congress of {Mathematical
Physics}, {P}aris {J}uly 18-23, 1994.
\bibitem{Jit95}
S.Ya. Jitomirskaya.
\newblock Almost everything about the almost {Mathieu operator II}.
\newblock To be published in the {Proceedings of the XI'th international}
congress of {Mathematical Physics}, {P}aris {J}uly 18-23, 1994, 1995.
\bibitem{Kahane}
J-P. Kahane and R.~Salem.
\newblock {\em Ensembles parfaits et s\'eries trigonom\'etriques}.
\newblock Hermann, 1963.
\bibitem{Ke+92}
R.~Ketzmerick, G.~Petschel, and T.~Geisel.
\newblock Slow decay of temporal correlations in quantum systems with cantor
spectra.
\newblock {\em Phys. Rev. Lett.}, 69:695--698, 1992.
\bibitem{Kni95}
O.~Knill.
\newblock Discrete random electromagnetic {Laplacians}.
\newblock submitted to CMP, (preprint available in the Mathematical Physics
Preprint Archive, {\tt mp\_arc\@math.utexas.edu}, document number: 95-195),
1995.
\bibitem{Kni96p}
O.~Knill.
\newblock On the dynamics of a general unitary operator.
\newblock Caltech 1996, submitted, 1996.
\bibitem{Las95}
Y.~Last.
\newblock Quantum dynamics and decompositions of singular continuous spectra.
\newblock To appear in J. Funct. Anal. Caltech Preprint, April, 1995.
\bibitem{Las95p}
Y.~Last.
\newblock Almost everything about the almost {Mathieu operator I}.
\newblock 1995.
\newblock {Proceedings of the XI'th international} congress of {Mathematical
Physics}, {P}aris {J}uly 18-23, 1994.
\bibitem{Man96}
V.~Mandelshtam.
\newblock Global recursion polynomial expansions of the {Green}'s function and
time evolution operator.
\newblock In {\em To be published in Multiparticle Quantum Scattering with
Applications to Nuclear, Atomic and Molecular Physics}. Springer-Verlag, New
York, 1996.
\bibitem{MaZh91}
V.A. Mandelshtam and S.Ya.Zhitomirskaya.
\newblock 1d-quasiperiodic operators. {L}atent symmetries.
\newblock {\em Commun. Math. Phys.}, 139:589--604, 1991.
\bibitem{Ric91}
J.L. Richardson.
\newblock Visualizing quantum scattering on the cm-2 supercomputer.
\newblock {\em Comp. Phys. Comm.}, 63:84--94, 1991.
\bibitem{Shu94}
M.A. Shubin.
\newblock Discrete magnetic {L}aplacians.
\newblock {\em Commun. Math. Phys.}, 164:259--275, 1994.
\bibitem{Sim95}
B.~Simon.
\newblock Operators with singular continuous spectrum: {I}. {General}
operators.
\newblock {\em Annals of Mathematics}, 141:131--145, 1995.
\bibitem{Sim95z}
B.~Simon.
\newblock Operators with singular continuous spectrum {VI}, {Graph Laplacians
and Laplace-Beltrami operators}.
\newblock To appear in Proc. Amer. Math. Soc, 1995.
\bibitem{Sim95p}
B.~Simon.
\newblock Spectral analysis of rank one perturbations and applications.
\newblock volume~8 of {\em CRM Proceedings and Lecture Notes}, pages 109--149,
1995.
\bibitem{Str90}
R.S. Strichartz.
\newblock Fourier asymptotics of fractal measures.
\newblock {\em J. Func. Anal.}, 89:154--187, 1990.
\bibitem{Szego}
G.~{Szeg{\"o}}.
\newblock {\em Orthogonal Polynomials}, volume~23 of {\em American Mathematical
Society, Colloquium Publications}.
\newblock American Mathematical Society, 1939.
\end{thebibliography}
\end{document}
End of discrete.tex
echo nonlinear.tex 1>&2
cat >nonlinear.tex <<'End of nonlinear.tex'
\documentstyle[]{amsart}
\title{Nonlinear dynamics from the Wilson Lagrangian}
\author{Oliver Knill
\thanks{Division of Physics, Mathematics and Astronomy,
Caltech, Pasadena, CA 91125, USA,
\newline
Current address:
Department of Mathematics,
University of Arizona, Tucson, AZ 85721, USA
}}
\date{July 22, 1996}
\begin{document}
\bibliographystyle{plain}
\maketitle
\begin{abstract}
A nonlinear Hamiltonian dynamics is derived from
the Wilson action in lattice gauge theory. Let $\mbox{$\cal D$}$ be a
linear space of lattice Dirac
operators $D(a)$ defined by some lattice gauge field $a$.
We consider the Lagrangian
$D \mapsto {\rm tr}((D(a)+i m)^4)$ on $\mbox{$\cal D$}$,
where $m \in {\bf C}$ is a mass
parameter.
Critical points of this functional are given by solutions of a nonlinear
discrete wave equation which describe the time evolution of the gauge
fields $a$. In the simplest case, the dynamical system is a cubic
Henon map. In general, it is a symplectic coupled map lattice.
We prove the existence of nontrivial critical points in two examples.
\end{abstract}
PACS: 02.30.-f, 03.20+i, 03.40.Kf, 03.65.Pm, 05.45.+b,
05.50.+q, 11.15.Ha, 12.90.+b
\section{The problem}
Let $\mbox{$\cal X$}$ be an operator algebra with finite trace ${\rm tr}$.
We introduce the problem to find critical points of the functional
\begin{equation}
\label{Action}
\mbox{$\cal L$}_m: D \mapsto {\rm tr}((D+i m)^4) \;
\end{equation}
on some linear subspace $\mbox{$\cal D$}$ of $\mbox{$\cal X$}$,
where $m$ is a complex mass
parameter. If $\mbox{$\cal D$}$ is formed by discrete
Dirac operators $D= \sum_j a_j \tau_j + (a_j \tau_j)^*$,
where $a_i$ are in a subalgebra $\mbox{$\cal A$} \subset \mbox{$\cal X$}$
and $\tau_j$ are
automorphisms in $\mbox{$\cal A$}$, this functional is an averaged Wilson
action of the lattice gauge field $a_j$.
We demonstrate here that critical points
of $\mbox{$\cal L$}_m$ define a nonlinear dynamical system and look at examples.
In the simplest case, if all the $a_j$ are invariant under space translations,
the time evolution is given by a cubic
Henon twist map in the plane. In general, the dynamics is
an infinite dimensional nonlinear discrete
reversible wave equation. These
discrete partial difference equations are
generalisations of classical coupled map lattices \cite{Kaneko,Gun93a,BuSi88}.
Hamiltonian reversible coupled map lattices appeared in \cite{JaKn95,Kni94}.
Noninvertible coupled map lattices in connection with field theory were
treated in \cite{Bec95}. Here,
we have a both space and time-discrete wave equation, time is physical
time and we work in an ergodic setup.
\section{The motivation}
Nonrelativistic quantum dynamics deals with the Schr\"odinger dynamical
system $i \hbar \dot{\psi} = L \psi$ in a
Hilbert space $\mbox{$\cal H$}$. If $L$ is a bounded operator,
the discrete time version
\begin{equation}
\label{DiscreteEvol}
i \frac{\hbar}{2\epsilon} [\psi(t+\epsilon)-\psi(t-\epsilon)] = L \psi(t)
\end{equation}
defines a unitary evolution and
is useful for studiing spectral measures of $L$ (\cite{Kni96q}).
If the left hand side of Equation~(\ref{DiscreteEvol})
is $i a^{-1} (U-U^*)$ and $V=iU$,
then Equation~(\ref{DiscreteEvol}) is $V+V^* = a L$
a discretisation of a relativistic wave equation. Using the discrete operator
$K=V+V^*-aL$ on space-time, the evolution~(\ref{DiscreteEvol})
is equivalent to $K \psi = 0$, so that a wave $\psi$ is a critical
point of the formal functional $\psi \mapsto (\psi, K \psi)$.
In quantum field theory, waves $\psi$ become operators and
contribute to the Hamiltonian. We assume that both the wave
$\psi=D$ and the Hamiltonian $K=D^2-m^2$
are in an $C^*$ algebra $\mbox{$\cal X$}$ with finite trace ${\rm tr}$.
$\mbox{$\cal X$}$ is a
not yet completed normed space with scalar product $(A,B) = {\rm tr}(A,B)$.
We replace the functional
$\psi \mapsto (\psi, K \psi)$ leading to a linear unitary
quantum evolution~(\ref{DiscreteEvol}) by
the smooth bounded functional $D \mapsto (D, (D^2- m^2) D) = \mbox{$\cal L$}_m-m^4$.
It gives a nonlinear evolution on $\mbox{$\cal D$}$ and
on its completion, the Hilbert space $(\mbox{$\cal H$},(\cdot,\cdot))$.
While the functional~(\ref{Action})
has on $\mbox{$\cal X$}$ only the trivial critical points $D = - i m $,
it becomes interesting when restricted to a linear space
$\mbox{$\cal D$}
= \{ D = \sum_{j=1}^d a_j \tau_j + (a_j \tau_j)^*, a_j \in \mbox{$\cal A$} \}
\subset \mbox{$\cal X$}$, where $\tau_j$ are fixed unitary
elements in $\mbox{$\cal X$}$ defining
automorphisms in $\mbox{$\cal A$}$ and $\mbox{$\cal D$}$
by $a \mapsto \tau_j a \tau_j^*$ rsp.
$D \mapsto D(T_j)= \tau_j D \tau_j^*$.
Every $D \in \mbox{$\cal D$}$ defines a real-valued lattice
gauge field on the Cayley graph of the group $\mbox{$\cal Z$}$
generated by the unitaries
$\tau_j$. At the bond connecting $D$ with $D(T_j)$ is attached the field
$a_j$. The value of $\mbox{$\cal L$}_m$ is a sum over all parallel
transports of closed paths of
length $4$. If the gauge fields $a_j$ are unitary and $\Omega$ is finite
then $\mbox{$\cal L$}_m$ is up to a linear transformation exactly the Wilson action.
We considered in \cite{Kni95e} the
variational problem
\begin{equation}
\label{DetAction}
D \mapsto {\rm Det}(D+m) = \exp({\rm tr}(\log|(D+m)|)) \; ,
\end{equation}
which is not smooth in infinite dimensions.
Minimizers of (\ref{DetAction})
exist for all $m \in {\bf C}$ in a statistical mechanical set-up, where $D$
is defined by a translational invariant measure on the set of unitary gauge field
configurations. The functional~(\ref{DetAction}) defines no dynamics.
${\rm tr}(D^2)$ gives no interesting
evolution. The expansion
$m^{-1} {\rm Det}(D+m) = 1-{\rm tr}(D^2) m^{-2}/2
+ ({\rm tr}(D^2)^2-2{\rm tr}(D^4)) m^{-4}/8+O(m^{-6})$
makes (\ref{Action}) the simplest substitue for the
determinant~(\ref{DetAction}). Higher
order polynomial actions do not lead to wave equations:
critical points would lead to unphysical higher order PDE's
and violate so the primitive causality axiom \cite{Haag}.
Another motivation is of course that the Wilson action is the
Hamiltonian in lattice gauge theory. It is fundamental
because it becomes in the continuum limit the Maxwell-Yang-Mills functional.
A more general problem which we do not address in this paper is to allow
$\mbox{$\cal D$}$
to be a manifold in $\mbox{$\cal X$}$ like for example
$ \mbox{$\cal D$}
= \{ D = \sum_j a_j \tau_j + (a_j \tau_j)^*, \; a_j^{-1} = a_j^* \in
\mbox{$\cal A$} \; \}$ leading to a variational problem with constraints.
\section{The functional}
Let $\mbox{$\cal R$}$ be any group acting as automorphisms on a probability space
$(\Omega,\nu)$.
Let $\mbox{$\cal Z$}$ be a discrete subgroup of $\mbox{$\cal R$}$ with generators
$T_k:\Omega \to \Omega$ and automorphisms
$\tau_k: a \mapsto a(T_k)$ on $\mbox{$\cal A$}=L^{\infty}(\Omega,\nu)$.
Let $\mbox{$\cal X$}$ be the crossed product of
$\mbox{$\cal A$}$ with the $\mbox{$\cal Z$}$ action. We call elements
$D=\sum_k a_k \tau_k + (a_k \tau_k)^* \in \mbox{$\cal X$}$
discrete Dirac operators. In the discrete case, matrix-valued
coefficients $\gamma_k a_k$ with Dirac matrices $\gamma_k$ are not necessary because
Clifford relations can be achieved on a
doubled lattice, which leads for $dim=4$ to operators on spinors with $16$
components. Let $\mbox{$\cal D$}$
be a linear space of such operators. We look at the problem
to find critical points of the action~(\ref{Action}) on
$\mbox{$\cal D$}$, where $m \in {\bf C}$
is a parameter. Physically motivated is the choice
$\mbox{$\cal R$} = \mbox{$\cal P$} \times \mbox{$\cal G$}$, where
$\mbox{$\cal P$}$ is the Poincar\'e group and where $\mbox{$\cal G$}$ is a
product of compact Lie groups. $\mbox{$\cal Z$} \subset \mbox{$\cal R$}$
is the product of a discrete lattice ${\bf Z}^4$ in the translation group generated
by $\tau_0, \dots, \tau_3$ of
$\mbox{$\cal P$}$ and a countable group in $\mbox{$\cal G$}$ generated
by $\sigma_j$. Every $D
=\sum_l a_l \tau_l + (a_l \tau_l)^*=\sum_{j=0}^3 a_j \tau_j + (a_j \tau_j)^*
+ \sum_k b_k \sigma_k + (b_k \sigma_k)^*$,
is the sum of a kinematic part
and a part responsible for internal degrees of freedom.
If $a_j$ and $b_k$ commute with involutions $\sigma_k$,
we write $D=\sum_j a_j \tau_j + (a_j \tau_j)^*
+ \sum_k 2b_k \sigma_k$. \\
1) {\bf Critical points}.
Let $\pi$ be the projection from $\mbox{$\cal X$}$ to $\mbox{$\cal D$}$
defined by $(\cdot,\cdot)$.
A functional $D \mapsto \mbox{$\cal L$}_f(D)={\rm tr}(f(D))$ on $\mbox{$\cal D$}$
has the Fr\'echet derivative $d \mbox{$\cal L$}(D)(U) = {\rm tr}(U \pi f'(D))$
and so the functional derivative
$\delta \mbox{$\cal L$}_f = \pi f'(D)$. Especially,
the Euler equations of the functional $\mbox{$\cal L$}_m={\rm tr}((D+i m)^4)$ are
$\pi(D+im)^3 = 0$. Explicitely,
\begin{eqnarray*}
0 &=& -3m^2 a_k
+ a_k ( \sum_{j \neq k} a_j^2(T_k) + a_j^2(T_kT_j^{-1})
+ a_j^2 + a_j^2(T_j^{-1}) ) \\
&+& a_k ( a_k^2(T_k) + a_k(T_k^{-1})^2 + a_k^2) \\
&+& \sum_{j \neq k} a_k(T_j) a_j a_j(T_k)
+ a_k(T_j^{-1}) a_j(T_j^{-1}) a_j(T_j^{-1} T_k) \; .
\end{eqnarray*}
Let $\mbox{$\cal D$}_0 = \{ D \in \mbox{$\cal D$} \; | \; D(T_0)=D \; \}$.
Because $a_k(T_0)$ and $a_l(T_0)$ do not
occur simultaneously in one equation for $k \neq l, k \neq 0,l \neq 0$,
one can solve for $a_k(T_0)$ and then for $a_0(T_0)$. This gives a symplectic
map $S$ on $\mbox{$\cal D$}_0 \times \mbox{$\cal D$}_0$.
We will write it down only
in examples.
Given an $S$-invariant measure $\mu$ on $\mbox{$\cal D$}_0 \times \mbox{$\cal D$}_0$,
which is
$T_i$-invariant for $i \geq 1$. Define a new probability space
$\Omega = \mbox{$\cal D$}_0 \times \mbox{$\cal D$}_0$ and $D((\omega_1,\omega_2))
= \omega_1$. $\Omega$ carries an additional action
$S$ replacing $T_0$ and commuting with $T_j$. Denoting again by $\mbox{$\cal X$}$
the crossed product of $\mbox{$\cal A$}=L^{\infty}(\Omega,\mu)$
with this new $\mbox{$\cal Z$}$ action
and by $\mbox{$\cal D$}$ the corresponding subspace, then $D$ is a critical point
of $\mbox{$\cal L$}_m$.
Having in mind the Wightman axioms \cite{GlimmJaffe}, we choose
$\mu$ ergodic. Space and time invariant field $D$ is then essentially unique.
The validity of adapted versions of Wightman or Haag-Kastler axioms will be
discussed elsewhere. \\
2) {\bf The Hessian}.
The map $D \mapsto \pi D^n$ on $\mbox{$\cal D$}$ has the linearisation
$U \mapsto \pi \sum_{k=0}^{n-1} D^k U D^{n-1-k}$.
The Hessian $\delta^2 \mbox{$\cal L$}(D)$
at a critical point $D$ is therefore the linear map on $\mbox{$\cal D$}$
$$ U \mapsto L U = \pi [(D+im) U (D+im) + U (D+im)^2 + (D+im)^2 U] \; . $$
For $U = u_k \tau_k + u_k \tau_k^* \in \mbox{$\cal D$}$, we obtain
$LU = v_k \tau_k + (v_k \tau_k)^*$ with
\begin{eqnarray*}
v_k &=& -3m^2 u_k
+ \sum_{j \neq k} (a_j^2 + a_j^2(T_k T_j^{-1})
+ a_j^2(T_k) + a_j^2(T_j^{-1})) u_k \\
&+& (a_k^2(T_k) + a_k^2(T_k^{-1}) + 3 a_k^2) u_k \\
&+& \sum_{j} a_j a_j(T_k) u_k(T_j)
+ a_j(T_j^{-1}) a_j(T_k T_j^{-1}) u_k(T_j^{-1})\;.
\end{eqnarray*}
If $L$ is invertible, the critical point is structurably stable with respect
to changes in $m$. This fact
can be useful for constructing critical points perturbatively as we will see in
an example. \\
3) {\bf Gauge invariance}.
For $g \in \mbox{$\cal A$}$, define the gauge transformation
$D \mapsto g D g^{-1}$
on $\mbox{$\cal D$}$. The lattice gauge field $a$ defined by $D$ transforms like
$a_k \mapsto g a_k g(T_k^{-1})^{-1}$.
The trace property
${\rm tr}(g f(D) g^{-1}) = {\rm tr}(f(D))$ implies that
$\mbox{$\cal L$}_m$ is gauge invariant. Especially, if $D$ is a critical point,
then $gDg^{-1}$ is a critical point too. Gauge transformations are
unitary with respect to the scalar product ${\rm tr}( \cdot, \cdot )$ on
$\mbox{$\cal D$}$. \\
4) {\bf Fields}.
Each operator $D$ defines a lattice gauge field or discrete $1$-form
$a = \sum_j a_j \tau_j$.
If $\mbox{$\cal Z$}$ is abelian, define the field
$F = da = \sum_{i1$, then
the topological entropy $h_{top}(S)$ is positive \cite{BlFr80}.
Moreover, $h_{top}(S)$ takes the maximal value $\log(3)$
if $|4-3m^2|>3 g^{2/3}$ due to an embedded horseshoe.
\cite{FrMi89}.
$h_{top}(S)>0$ implies the existence of
a compact invariant hyperbolic invariant set \cite{Kat80}.
(ii) If $|4-3m^2|<2$, then $0$ is a linearly stable fixed point. It is
stable for the generic set of $m$'s with nontrivial Birkhoff normal form
\cite{Moser}. This assures the existence of invariant measures even
absolutely continuous with respect to Lebesgue measure.
The Hessian is the bounded random operator on $l({\bf Z})$
$ \delta^2 \mbox{$\cal L$}_m(D):
u \mapsto u_{n+1} + u_{n-1} - 2 u_n + V''(x_n) u_n$,
where $x_n = b(T^n)$ is obtained from the function $b$ defining the critical point
$D$.\\
2) {\bf A higher dimensional example}.
Given
$\mbox{$\cal D$}= \{ D = \tau+ \tau^*
+ \alpha \sum_{j=1}^d (\tau_j + \tau_j^*) + b \sigma \;\}$.
A critical point of $\mbox{$\cal L$}_m$ satisfies the Euler equations
\begin{equation}
\label{Eulerb}
b(T_0) + b(T_0^{-1}) + (4 - 3 m^2) b + \alpha^2 \Delta b
+ 4 d \alpha^2 b + b^3 = 0 \; ,
\end{equation}
where $\Delta b= \sum_{j=1}^d b(T_j) + b(T_j^{-1})$.
Writing $x=b,y=b(T_0^{-1})$, we
obtain the discrete Hamiltonian system
\begin{equation}
\label{S}
S: \left( \begin{array}{c} x \\ y \end{array} \right) \mapsto
\left( \begin{array}{c} (- \alpha^2 (\Delta + 4d) x
- (4 -3 m^2) x - x^3 - y \\
x \end{array} \right) \; .
\end{equation}
$S$ defines an invertible {\it coupled map lattice}. In \cite{JaKn95},
invertible coupled map lattices of similar type have been considered.
For $\alpha=0$, $S$ is an array decoupled cubic Cremona maps.
They are for $\alpha>0$ linked
through a linear nearest neighbor coupling.
(\ref{S}) is a discrete version of the
nonlinear $\phi^4$ wave equation $(\Box +m^2) \phi = p(\phi)$, where
$\Box=\partial_t^2-\Delta$ and $p$ is a cubic polynomial.
If $\Omega$ is a finite set, $S$ is a symplectic map
$(x,y) \mapsto (f(x)-y,x)$
on ${\bf R}^{2 |\Omega|}$. The linearization of $S$ is conjugated
to a decoupled system of two unitary Schr\"odinger evolutions if $m$ is choosen
so that the Hessian is not invertible. While the nonlinear Schr\"odinger
equation is integrable by an infinite dimensional Siegel theorem \cite{Zeh78},
the nonlinear $\phi^4$ wave analogue is not. An analogue fact holds
here in the special case of functions which are constant in space:
while a polynomial map $x \mapsto f(x)$ can be integrable near $0$ by Siegel's
theorem, the symplectic map $(x,y) \mapsto (f(x)-y,x)$ is nonintegrable and only
a deeper KAM argument can establish stability of an elliptic fixed
point. In higher dimensions, a linearly stable fixed point $0$ is in general
unstable. We prove now that for large $m$,
there are aperiodic nontrivial solutions of the dynamical system~(\ref{S})
using an argument of Aubry \cite{AuAb90,MaAu94,JaKn95,Kni94a,Kni94}.
Equation~(\ref{Eulerb}) is equivalent
to $F(\epsilon,q)=0$, where $q =b/m, \epsilon = m^{-3}$ and
$$ F(\epsilon,q)=
\epsilon [q(T_0) + q(T_0^{-1}) + \alpha^2 \Delta q + (4d \alpha^2+4) q]
- 3q + q^3 \;.$$
For $\epsilon=0$, solve $-3q+q^3=0$ by any function
$q: \Omega \rightarrow {\bf R}$ taking values in $\{\pm \sqrt{3}\}$.
The linear map
$(\partial/\partial q) F(0,q)$ on $L^{\infty}(\Omega)$
is invertible. The implicit function theorem
gives solutions $q$ for small $\epsilon=m^{-2}$.
This argument generalizes to find critical points for large $m$ for
$\mbox{$\cal D$}= \{ D = \tau+ \tau^* + \sum_{j=1}^d \alpha (\tau_j + \tau_j^*)
+ \sum_{j=1}^n b_j \sigma_j \}$, where
critical points satisfy
$b_i(T_0) + b_i(T_0^{-1}) + (4-3 m^2) b_i + \alpha^2 \Delta b + b_i^3
+ 3 b_i \sum_{k \neq i} b_k^2 = 0$. The
corresponding symplectic map $S$ has also the generating function
$\sum_j (\frac{(b_j(T_0)-b_j)^2}{2}
+ \alpha^2 \sum_k \frac{(b_j(T_k)-b_j)^2}{2}
+ (2 \alpha^2 + 6-3m^2) \frac{b_j^2}{2} + \frac{b_j^4}{4}
+ \frac{3}{2} \sum_k b_j^2 b_k^2 )$.
\begin{thebibliography}{10}
\bibitem{AuAb90}
S.~Aubry and G.Abramovici, {\em Physica D}, 43:199--219, 1990.
\bibitem{MaAu94}
R.S. MacKay and S.Aubry.
{\em Nonlinearity}, 6:1623--1643, 1994.
\bibitem{Bec95}
C.~Beck. {\em Nonlinearity}, 8:423--441, 1995.
\bibitem{BlFr80}
P.~Blanchard and J.~Franks.
{\em Inv. Math.}, 62:333--339, 1980.
\bibitem{BuSi88}
L.A. Bunimovich and Ya.~G. Sinai.
{\em Nonlinearity}, 1:491--516, 1988.
\bibitem{Fel+78}
J.~Feldman, P.~Hahn, and C.~Moore.
{\em Adv. in Math.}, 28:186--230, 1978.
\bibitem{FeMo77}
J.~Feldmann and C.Moore.
{\em Trans. Am. Math. Soc.}, 234:289--359, 1977.
\bibitem{FrMi89}
S.~Friedland and J.~Milnor.
{\em Ergod. Th. Dyn. Sys.}, 9:67--99, 1989.
\bibitem{GlimmJaffe}
J.~Glimm and A.~Jaffe. {\em Quantum physics, a functional point of view}.
Springer Verlag, New York, second edition, 1987.
\bibitem{Gun93a}
V.M. Gundlach and D.A. Rand.
{\em Nonlinearity}, 6:165--230, 1993.
\bibitem{Haag}
R.~Haag. {\em Local Quantum Physics}.
Texts and Monographs in Physics. Springer, New York, 1992.
\bibitem{JaKn95}
A.~Jakobsen and O.~Knill.
{\em Phys. Lett. A}, 205:179--183, 1995.
\bibitem{Kaneko}
K.~Kaneko.
{\em Theory and applications of coupled map lattices}.
Chichester : New York : John Wiley and Sons, 1993.
\bibitem{Kat80}
A.~Katok.
{\em Pub. I.H.E.S.}, 51:137--172, 1980.
\bibitem{Kni94}
O.~Knill.
In M.Fannes et~al., editor, {\em On Three Levels}, pages 321--330.
Plenum Press, New York, 1994.
\bibitem{Kni94a}
O.~Knill.
Topological entropy of some standard type monotone twist maps.
To appear in Trans. Amer. Math. Soc, 1996
\bibitem{Kni95e}
O.~Knill.
Determinants of random {Schr\"odinger} operators arizing from lattice
gauge fields. Preprint, Caltech 1995.
\bibitem{Kni95}
O.~Knill. Discrete random electromagnetic {Laplacians}.
Preprint, Caltech 1995.
\bibitem{Kni96q}
O.~Knill. A remark on quantum dynamics.
Preprint, Caltech 1996,
\bibitem{Mos60}
J.~Moser. {\em Bol. Soc. Mat. Mex.}, 5:176--180, 1960.
\bibitem{Moser}
J.~Moser.
{\em Stable and random Motion in dynamical systems}.
Princeton University Press, Princeton, 1973.
\bibitem{Hoo96}
G~t'Hooft. {\em Class. Quantum Grav.}, 13:1023--1039, 1996.
\bibitem{Zeh78}
E.~Zehnder. {\em Manuscripta Math.}, 23:363--371, 1978.
\end{thebibliography}
\end{document}
End of nonlinear.tex
echo koop.tex 1>&2
cat > koop.tex <<'End of koop.tex'
\documentstyle[]{amsart}
\newcommand{\Acal}{\mbox{$\cal A$}} \newcommand{\Fcal}{\mbox{$\cal F$}}
\newcommand{\Gcal}{\mbox{$\cal G$}} \newcommand{\Hcal}{\mbox{$\cal H$}}
\newcommand{\Kcal}{\mbox{$\cal K$}} \newcommand{\Xcal}{\mbox{$\cal X$}}
\newcommand{\RR}{{\bf R}} \newcommand{\ZZ}{{\bf Z}}
\newcommand{\NN}{{\bf N}} \newcommand{\CC}{{\bf C}}
\newcommand{\QQ}{{\bf Q}} \newcommand{\TT}{{\bf T}}
\title{Singular continuous spectrum and quantitative rates of weakly mixing}
\author{Oliver Knill \thanks{Division of Physics, Mathematics and Astronomy,
California Institute of Technology, 253-37,
Pasadena, CA, 91125 USA. } }
\date{Feb 21, 1996}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{propo}[thm]{Proposition}
\newtheorem{coro}[thm]{Corollary}
\newenvironment{proof}{\begin{trivlist}\item[]{\em Proof.\/\ }}%
{\hfill$\Box$ \end{trivlist}}
\newenvironment{remark}{\begin{trivlist}\item[]{\em Remark.\/\ }}
{\hfill \\ \end{trivlist}}
\newtheorem{definition}[thm]{Definition}
\newtheorem{example}[thm]{Example}
\def\marnote#1{\marginpar{\scriptsize\raggedright #1}}
\begin{document}
\bibliographystyle{plain}
\maketitle
\begin{abstract}
Using a result of Simon, we give a new proof that in the weak topology of
measure preserving transformations, a dense $G_{\delta}$
has purely singular continuous spectrum in the orthogonal complement $H_0$ of the
constant functions in $L^2(X)$. \\
We prove that for a dense $G_{\delta}$ of shift-invariant
measures on $A^{\ZZ^d}$, all $d$ shifts have purely singular continuous spectrum
on $H_0$. We also give new examples of smooth unitary cocycles over an
irrational rotation which have purely singular continuous spectrum. \\
Quantitative weak mixing properties are related by results of Strichartz and Last
to spectral properties of the spectral measures of $U_T$.
\end{abstract}
\begin{center} {\bf AMS Classification}. 28D05, 28D20, 58F11 \end{center}
{\bf Keywords}. Ergodic theory, spectral theory, mixing, invariant measures,
singular continuous spectrum.
\section{Introduction}
The spectral theory of a unitary operator $U_T: f \mapsto f(T)$
belonging to a measure-preserving transformation $T$ which started with
Koopman and von Neumann is now an important part in
ergodic theory (see \cite{CFS,Queffelec,Sinai}).
One of the issues in this subject is to relate dynamical properties
of $T$ with spectral properties of $U_T$. An other question is to find
out what happens Bair typically in the set $\Gcal$ of all transformations.
In this paper we first discuss spectral questions in ergodic theory
in the context of recent developments in operator theory
\cite{Sim95,Sim95a,De+94,De+94a,JiSi94,Hof+95,Las95}.
This alternative approach leads also to new results like for example
that a generic shift-invariant measure has purely singular continuous
spectrum. \\
Having established
that most transformations are weakly mixing but do not have eigenvalues
on $H_0=\{ g \; | \; \int g \; dm = 0 \} \subset L^2(X)$, we consider then
quantitative average decay rates of the Fourier coefficients
$$ c_n=\int \overline{g(x)} g(T^nx) \; dm(x) = (g, U^n_T g) $$
of a spectral measure $\mu_g$ for $g \in H_0$. Such growth rates were
investigated in quantum dynamics, where $U=e^{iL}$
(\cite{Gua89,Com93,Hol94,Las95}).
While the Riemann-Lebesgue lemma assures that strong mixing holds
if $\mu_{g}$ has absolutely continuous spectrum, the sequence
$c_n$ might or might not
converge to zero, if $T$ has purely singular continuous spectrum and as
Fourier theory indicates, this question can depend in a subtle way on parameters
(\cite{Kahane}). Averages of $|c_n|^2$ behave better. By Wiener's theorem,
$\lim_{n \rightarrow \infty} n^{-1} \sum_{k=1}^n |c_k|^2 = 0$
if $\mu_g$ is continuous. On the other hand, if
the decay of $c_k$ is sufficiently fast (for example
$|c_k| \leq k^{-1}$), then the spectral measure is absolutely continuous.
Singular continuous spectrum is therefore
related with a more moderate decay rate. Power law decays of
the C\'esaro averages and the spectral properties of $\mu$ have been
investigated by Strichartz \cite{Str90} in the case of Fourier transforms.
These results influenced the research in quantum dynamics, where the decay
rate of the survival probability $|(g, U^n g)|^2$ is a measure for transport
the system (see \cite{Las95} for more details). \\
Also in ergodic theory, the decay rates of the Fourier coefficients
$(g, U^n g)=\hat{\mu}_g$ are interesting.
Since a generic set of dynamical systems is weakly mixing
but not strongly mixing, a finer quantitative subdivision of weakly mixing
is desirable. We call $T$ partially weakly $h$-mixing, if
$n^{-1} \sum_{k=1}^n |(g,U^n g)|^2 \leq C h(n^{-1})$ for at least one
vector $g$ orthogonal to the constants and uniformly weakly $h$-mixing,
if this decay holds for all such vectors. These properties are invariants of the
dynamical system. \\
This paper is organized as follows. In section 2, we give new proofs for
genericity of singular continuous spectrum in ergodic theory.
The proofs are new and are easy corollaries
from corresponding results in the spectral theory of Schr\"odinger operators.
In section 3, we introduce classes of weakly $h$-mixing
dynamical systems and give an interpretation
of Strichartz result in ergodic theory and a consequence
of a recent RAGE theorem of Last. In section 4, we show that
a generic shift invariant measure gives a dynamical system which has
purely singular continuous spectrum. This result as well as a generalization
to higher dimensions in section 5 are new and should also be seen in the
context of statistical mechanics. In section 6, we give new cocycles
over an irrational rotation with purely singular continuous spectrum.
In an appendix, a result of Strichartz as well as the converse due to Last
are proven in the case of Fourier series.
Since the discrete reformulations do not follow as
results from the corresponding Fourier transform results,
a translation of the proofs is included.
\section{Generic purely singular continuous spectrum for
measure preserving transformations}
A measure preserving automorphism $T$ of a probability space $(X,\Acal,m)$
defines the unitary Koopman operator $U_T: f \mapsto f(T)$ on $L^2(X)$.
Because $1$ is always an eigenvalue, it is convenient to restrict $U_T$ to
the orthogonal complement $H_0=\{ f \; | \; \int f \; dm =0 \; \}$
of the constant functions and to consider the spectrum of this operator.
Let $\Gcal$ be the complete topological
group of automorphisms of
$(X,\Acal,m)$ with the weak topology: $T_j$ converges to $T$ weakly, if
$m(T_j(A) \Delta T(A)) \rightarrow 0$ for all $A \in \Acal$
(\cite{Halmos}).
\begin{thm}
\label{generic}
In the weak topology on $\Gcal$,
a dense $G_{\delta}$ of transformations $T:X \rightarrow X$ is
in $\Gcal$ and has purely singular continuous spectrum.
\end{thm}
\begin{proof}
By Halmos second category theorem \cite{Halmos},
the set of weakly mixing transformations in $\Gcal$ (and so with
continuous spectrum by Halmos mixing theorem \cite{Halmos})
is a dense $G_{\delta}$.
Rohlin's lemma (\cite{Halmos}) assures that
the set of periodic transformations (and so with pure point spectrum)
is dense in the uniform topology and so also dense in the weak topology.
If $T_n \rightarrow T$ in $\Gcal$, then $U_{T_n} \rightarrow U$ in the
strong operator topology and so in the strong resolvent topology.
Apply now Simon's "Wonderland theorem" \cite{Sim95,Sim95a}) in the following a
form: let $\Xcal$ be a complete metric space of unitary operators,
where the metric is stronger than the strong resolvent topology. If
there exists a dense set with continuous spectrum and a dense set with
pure point spectrum, then there exists a dense $G_{\delta}$ of oprators with
purely singular continuous spectrum.
(Simon's theorem is formulated for selfadjoint operators. However, the proofs
that the two sets $\{ A \in \Xcal \; | \; A$ has no eigenvalues $\}$,
$\{ A \in \Xcal \; | \; A$ has no absolutely continuous spectrum $\}$
are both $G_{\delta}$'s, hold without any modification also of unitary
operators.)
\end{proof}
\begin{remark}
Theorem~\ref{generic} was proven by Katok-Stepin in
(\cite{KaSt67}, see also \cite{Ka+75} remark 4.1.1)
and by Choksy-Nadkarni in \cite{ChNa90}.
Related to Theorem~\ref{generic} is the result that a generic
volume preserving homeomorphism of a manifold has purely
singular continuous spectrum (\cite{KaSt70} p. 201) which
strengthens Oxtoby-Ulams's theorem about generic ergodicity.
\end{remark}
\begin{remark}
There are examples of interval exchange transformations
with purely singular continuous spectrum (\cite{Rob83} Theorem 7.1).
For other generic spectral properties of such systems \cite{Vee84}.
\end{remark}
\begin{remark}
The Riemann-Lebesgue lemma implies that transformations
with pure absolutely continuous spectrum are strongly mixing. Since
strongly mixing is a set of first category
(\cite{Halmos}), one obtains again that pure ac spectrum is meager in $\Gcal$.
\end{remark}
Theorem~\ref{generic} is not true in the uniform topology
$d(T,S)=m( S(x) \neq T(x))$ on $\Gcal$ because ergodic transformations
are then nowhere dense \cite{Halmos}. One can however restrict to
$\Kcal$, the closure $\Gcal_{aperiodic}$ of the aperiodic transformations
in the uniform topology.
\begin{thm}
\label{uniformgeneric}
In the uniform topology, $\Kcal$ contains
a dense $G_{\delta}$ of transformations
$T$ with purely singular continuous spectrum.
\end{thm}
\begin{proof}
$\Kcal$ is a complete metric
space because it is a closed subset of a complete metric space.
By Rohlin's lemma, it contains a dense set of
periodic transformations.
The conjugacy class $[T]$ of
an aperiodic transformation $T$ is dense in $\Gcal_a$ in the uniform
topology (\cite{Halmos}) so that
the existence of a single weakly mixing transformations,
gives a dense set of weakly mixing transformations.
Simon's theorem implies the claim.
\end{proof}
\begin{remark}
Theorem~(\ref{generic}) (rsp. Theorem~\ref{uniformgeneric})
implies Rohlin's result \cite{Roh59} that
a generic transformation has zero metric entropy because
a transformation with positive metric entropy contains
countable Lebesgue spectrum.
\end{remark}
\begin{remark}
With the induced product topology, the set $\Gcal^d$ of $\ZZ^d$-
actions is a complete metric space. Since all $d$ projections from $\Gcal^d$
to $\Gcal$ are continuous and surjective,
the inverse image of a generic set is generic and
for a dense $G_{\delta}$ of $\ZZ^d$-actions, all the transformations
$T^n, n \neq 0$ have purely singular continuous spectrum on $H_0$. This
generalizes a result of Natarajan (\cite{Ka+75} 4.1.1).
Because systems with positive metric entropy have also for $d>1$ some Lebesgue
spectrum (\cite{KaLi94}), a generic $\ZZ^d$-action has zero entropy.
\end{remark}
\begin{remark}
Theorem~\ref{generic} implies
that most selfadjoint operators of the form $L=U_T+U_T^*$ on $L^2(X)$ have purely
singular continuous spectrum because
$U = L/2+ i \sqrt{1-L^2/4}$ has the same spectral type then $L$.
One can show more generally that for a generic
$\ZZ^d$ action generated by $T_1, \dots, T_d: X \rightarrow X$,
the operator $L=\sum_{i=1}^d U_{T_i} + U_{T_i}^*$ has purely singular
continuous spectrum.
\end{remark}
\section{A scale of weakly mixing dynamical systems}
In this section, a quantitative definition of weak mixing is introduced. It is
motivated by recent developments in research on
transport properties of quantum mechanical systems
({\cite{Gua89,Com93,Hol94,Las95}). \\
\begin{definition} Given $h: \RR \rightarrow [0,\infty)$ which is continuous
and $h(0)=0$. Call $T$ uniformly weakly $h$-mixing, if there exists a
constant $C$ such that for all unit vectors
$f,g \in L^2(X)$ and all $n$
\begin{equation}
\label{relation}
n^{-1} \sum_{k=1}^n |\int \overline{f} g(T^k) \; dm
- \int f \; dm \cdot
\int g \; dm|^2 \leq C \cdot h(n^{-1}) \; .
\end{equation}
If relation (\ref{relation}) holds for at least
one vector $f \neq 0$ and all unit vectors $g$, we say
$T$ has partial weakly $h$-mixing.
If weakly $h$-mixing holds for $h(t)=t^{\alpha}$, call
$T$ weakly $\alpha$-mixing.
\end{definition}
\begin{remark}
Given two isomorphic systems $(X_i,\Acal_i,T_i)$.
If one of them is uniformly weakly $h$-mixing, the other is weakly
$h$-mixing too. The same is true for systems having partial weak $h$-mixing.
Therefore, "uniform weak $h$-mixing" or "partial weak $h$-mixing" are invariants
of dynamical systems in the same way as
ergodicity, weakly mixing, mixing, entropy etc. We will just see that
any weakly mixing system has partial weakly $h$-mixing for some $h$. (This does
not directly follow from the definition because the decay rate
$h$ might be $g$ dependent).
The supremum over all $h$, for
which $T$ has partial weak $h$-mixing is an invariant of the dynamical system.
\end{remark}
\begin{remark}
Obviously, if $h_10$.
\end{remark}
\begin{remark}
For $g=1_A-m(A)$, Proposition~\ref{energy} says that a spectral measure
$\mu_{1_A}$ has finite $\alpha$-energy if and only if
$$ \sum_{k=1}^{\infty} k^{\alpha-1}
(m(A \cap T^k(A)) - m(A)^2)^2 < \infty \; \; . $$
A numerical determination of potential
theoretical properties (and so an estimate of the Hausdorff dimension)
of the spectral measures is in principle possible
because $(m(A \cap T^k(A)) - m(A)^2)^2$ is experimentally accessible.
\end{remark}
\begin{remark}
The kernel $\phi_{\alpha}(t)$ can be replaced by
other kernels $\phi_h(t)=h(\sin(t/2))$ like for example
the logarithmic kernel $\phi_0(t)=-\log(|\sin(t/2)|)$ which is
in an appropriate sense the limit of $\phi_{\alpha}$ for $\alpha \rightarrow 0$
and Proposition~\ref{energy} stays true for $\alpha=0$.
The corresponding energy is the logarithmic potential theoretical
energy of $\mu$.
\end{remark}
An almost immediate consequence of the Wiener theorem is
the classical RAGE theorem in quantum mechanics saying that
$n^{-1} \sum_{k=1}^n$ $(U^k f,A U^k f)$ $\rightarrow 0$ for all compact $A$
if and only if $f$ has a continuous spectral measure $\mu_f$ for the unitary $U$.
The unitary $U$ defines an evolution of probability measures with density
$|f(k)|^2$ on $\ZZ^d$ where $f \in l^2(\ZZ^d)$ is a unit vector.
If $Q_i u(n)=n u(n)$ are the position operators on $l^2(\ZZ^d)$, it is
a consequence of RAGE that $n^{-1} \sum_{k=1}^n (U^nf, |Q|^2 U^nf)
\rightarrow \infty$ if $\mu_f$ has some continuous component.
New quantitative RAGE theorems have appeared in the last years
\cite{Gua89,Com93,Hol94,Las95}
(an example is Theorem~\ref{RAGE} below which leads to the divergence
$n^{-1} \sum_{k=1}^n (U^nf, |Q|^2 U^nf) \geq C n^{2 \alpha/d}$
if the projection of $f$ to the $\alpha$-dimensional subspace $H_{\alpha}
= \overline{\{ f \; | \; I_{\alpha}(\mu_f) <\infty \; \}}$ has positive length.)
\begin{thm}[Last's RAGE theorem, discrete version]
\label{RAGE}
If $f$ is uniformly $h$-continuous for the unitary operator $U$ of a Hilbert
space $H$, then there exists a constant $C$ such
that for all $p-$ Schatten class operators $A$
$$ n^{-1} \sum_{k=1}^n |(U^k f, A U^k f)_H|
\leq C^{1/p} ||A||_p h(n^{-1})^{1/p} \; . $$
\end{thm}
\begin{proof}
With the discrete version of Theorem~(\ref{strichartz}), the proof of
\cite{Las95} translates to the discrete case when
replacing corresponding averaging integrals by sums.
\end{proof}
\begin{coro}
\label{RAGEAPP}
If $f \in H_0$ has a uniformly $h$-continuous
spectral measure for $U_T$, there exists a constant $C$ such that
for all $k \in L^2(X \times X)$
$$ n^{-1} \sum_{k=1}^n |\int_X \int_X f(T^kx) k(x,y) f(T^ky)
\; dm(x) dm(y)| \leq C \cdot ||k||_{L^2(X \times X)}
\sqrt{h(n^{-1})} \; . $$
\end{coro}
\begin{proof}
Follows from Theorem~\ref{RAGE} using that
the integral operator $Kf(x)=\int k(x,y) f(y) \; dm(y)$ has
Hilbert-Schmidt norm $||K||_2 \leq ||k||_{L^2(X \times X)}$.
\end{proof}
\section{Shift invariant measures}
Given a topological shift $(X=A^{\ZZ},T)$ over a finite alphabet
$A$. Consider the space $M(X)$ of shift-invariant measures
with weak-* topology.
\begin{thm}
\label{genericmeasure}
There exists a dense $G_{\delta}$ of $T$-invariant measures $m$ for which
the abstract dynamical system $(X,T,m)$ has purely singular continuous
spectrum.
\end{thm}
\begin{proof}
The claim again follows from Simon's theorem, but not directly because
changing the measure changes also the Hilbert space
on which the Koopman operator acts.
The idea is to construct
a continuous map $F$ from the space of shift invariant measures
to the unitary group of operators on $l^2(\ZZ)$ such that $F(\mu)$ restricted
to the orthogonal complement of the eigenspace of $1$ of $F(\mu)$ is
unitarily equivalent to $U_T:H_0 \rightarrow H_0$ of the system
$(X,T,\mu)$. \\
There exists a homeomorphism $\phi$ which maps $X$ homeomorphically
onto a subset $\phi(X)$ of $[0,1] \subset \RR$. Let
$K$ be the set of shift-invariant measures on $X$.
Define the set of Borel probability measures
$M=\{ 1/2(\nu+\phi^*(\mu)) \; | \; \mu \in K \}$ on $\RR$,
where $\phi^*$ is
the push-forward homeomorphism from $K$ onto $M$ and where $\nu$ is
the Lebesgue measure on $[-1,0]$.
(The combination of $\nu$ and $\phi^*(\mu)$ is taken so that
the functions $x^n$ are always linearly independent in $L^2([-1,1],\mu)$
also if $\mu$ is a pure point measure with finitely many atoms.)
For every $\mu \in M$, let $f_n^{(\mu)}$ be a basis in $L^2([-1,1],\mu)$ obtained
by the Gramm-Schmidt orthogonalisation of the linearly independent
polynomials $p_n: x \mapsto x^n$ with respect to the scalar product
$(f,g)_{\mu} =\int \overline{f} g \; d\mu$. Define a
map $\tilde{F}$ from $M$ to the space $U$ of unitary matrices on the Hilbert
space $l^2(N)$ by
$$ \mu \mapsto \tilde{F}(\mu)_{nm}
= \int_{[-1,1]} \overline{f_n^{\; (\mu)}(x)}
f_m^{\; (\mu)}(Sx) \; d\mu(x) \; , $$
where $S=\phi T \phi^{-1}$ is the conjugation of the shift $T$ on $\phi(X)$
and $S(x)=x$ for $x \in [-1,0]$. Let $F=\pi \tilde{F} \pi$, where $\pi$ is
the projection on the closed space of all functions in $H$ which are
vanishing on $[-1,0]$ or $\mu$ almost everywhere constant in $[0,1]$.
While $\tilde{F}(\mu)$ contains the
eigenvalue $1$ with infinite multiplicity, $F(\mu)$ is conjugated to
the unitary Koopman operator of the dynamical system
$(X,S,\mu)$ expressed in a $\mu$-dependent basis.
For each $n$, the map $\mu \in M \mapsto f_n^{\; (\mu)} \in C([-1,1])$ is
continuous, because of the continuity of $\mu \mapsto (f,g)_{\mu}$ for fixed $f,g$
and because the Gramm-Schmidt orthogonalisation is a
finite process. Also $\mu \mapsto \int h \; d\mu$ is continuous
for fixed $h \in C(X)$. An $\epsilon/3$ argument assures that $\tilde{F}:
\mu \mapsto F(\mu)_{nm}$ is
continuous for all $m,n$. Therefore, $F$ is continuous as a map from $M$ to
the set of unitary operators on $l^2(\ZZ)$.
The image $F(M)$ is a complete metric space of unitary operators.
Because a dense set of invariant measures give
periodic dynamical systems \cite{Par61}
and a dense $G_{\delta}$ of invariant measures are weakly mixing
\cite{Sig72}, a dense set in $F(M)$ has pure point spectrum
and a dense set of operators with continuous spectrum on $H_0$.
By Simon's theorem, there is a generic set of operators
with purely singular continuous spectrum on $H_0$. \\
Because $F$ is continuous and surjective and because $F(M)$ contains a
dense $G_{\delta}$ of operators with purely singular continuous spectrum,
also a dense $G_{\delta}$ if shift invariant measures $m$ give a system
$(X,T,m)$ with purely singular continuous spectrum.
\end{proof}
\begin{remark}
Theorem~\ref{genericmeasure} is not a corollary of Theorem~\ref{generic}.
There is no continuous map from the space of invariant measures
to $\Gcal$: the image of such a map would be a compact subset in $\Gcal$
different from $\Gcal$ because there is an entropy bound. On the other
hand, it must be $\Gcal$ because it
contains the dense set of periodic transformations.
\end{remark}
\begin{remark}
A corollary of Theorem~\ref{genericmeasure},
is Sigmund's result \cite{Sig71} that
the set of shift-invariant measures with
zero metric entropy is a dense $G_{\delta}$.
Summarizing with other known results in literature (see \cite{DGS}):
a generic shift-invariant measure
on $X$ is ergodic, nonatomic, with support $X$, weakly mixing
and not strongly mixing, has zero metric entropy and singular continuous
spectrum.
\end{remark}
\begin{remark}
An other corollary is that
if $T$ is an invertible Axiom-A diffeomorphism restricted
to a basic set $\Omega_i$, then there is a dense $G_{\delta}$ of
$T$-invariant measures on $\Omega_i$ which
have purely singular continuous spectrum (\cite{Sig70,Sig72}).
\end{remark}
\section{Invariant measures of multi dimensional shifts}
A $\ZZ^d$ dynamical system is a $\ZZ^d$ action of automorphisms of
a probability space $(T,\Acal,m)$.
We consider now $\ZZ^d$ dynamical systems obtained by
taking a shift-invariant measure $\mu$ on $K^{\ZZ^d}$, where $K$ is
a compact metric space.
\begin{coro}
\label{multi}
For a generic $\ZZ^d$-shift invariant measure in $K^{(\ZZ^d)}$,
all shifts different from the identity have
purely singular continuous spectrum.
\end{coro}
\begin{proof}
Theorem~(\ref{genericmeasure}) stays true if the finite alphabet $A$ is
replaced by a compact metric space $Y$ because
there is a dense set of measures
in $Y^{\ZZ}$ which have support on some $A^{\ZZ}$, where $A$ is a finite
subset of $Y$. We can especially take $Y=K^{(\ZZ^{d-1})}$, where $K$ is
a compact space. Let $M(Y)$ be the set of $T_1$ invariant measures on
$X=K^{(\ZZ^d)}$. We know
that a dense $G_{\delta}$ in $M_1(Y)$ has purely singular continuous spectrum. Let
$\Lambda_k = [-k,k]^d \rightarrow \ZZ^d$ be a van Hove sequence. Each of the compact
sets $M_k(Y)=\{ |\Lambda_k|^{-1} \sum_{n \in \Lambda_k} (T^n)^* \mu \;
| \; \mu \in M_1(Y) \}$
consists of $T_1$-invariant measures and each contains a dense
$G_{\delta}$ of measures which have purely singular continuous spectrum with respect
to $T_1$ (any of the sets contains a dense set of periodic measures and
a dense set of weakly mixing measures so the proof in \ref{genericmeasure}
leads to generic singular continuity).
The intersection $M_{\infty}(Y)$ of all these sets $M_k(Y)$
is exactly the set of all $\ZZ^d$ invariant measures. It follows that
a dense $G_{\delta}$
in $M_{\infty}(Y)$ has singular continuous spectrum with respect
to the transformation $T_1$ (because if $K_i$ are some closed subsets
of a Baire space containing each a dense $G_{\delta}$ subset $H_i \subset K_i$,
then $\bigcap_i H_i$ is a dense $G_{\delta}$ in $K=\bigcap K_i$.)
Having a dense $G_{\delta}$ of $\ZZ^d$-invariant measures with
purely singular continuous spectrum with respect to $T_1$, we obtain
also a dense $G_{\delta}$ of $\ZZ^d$ invariant measures
with purely singular continuous spectrum with respect to any of the
shifts $T^i \neq Id$.
\end{proof}
\begin{remark}
It follows with \cite{KaLi94} that a generic shift invariant measure
has zero entropy.
\end{remark}
\begin{remark}
Let $A$ be a finite set.
A cellular automaton is a continuous map $\phi: X=A^{\ZZ^d} \rightarrow A^{\ZZ^d}$
which commutes with all shifts. Given a shift
invariant measure $\mu$ on $A^{\ZZ^d}$ for which all shifts have
purely singular continuous spectrum.
The push-forward $\phi^* \mu$ has the same property
because $(X,\phi^* \mu)$ is
a factor of the system $(X,m)$ (\cite{HoKn95}).
There exists therefore a residual set of shift invariant measures on
$\phi^k(X)$, for which the shifts have purely singular continuous spectrum.
\end{remark}
\begin{remark}
Related to Theorem~\ref{multi} is the open problem
in crystallography whether there exist "turbulent but not chaotic
crystals"? A "crystal" is a point $x \in \{0,1\}^{\ZZ^d}$ for which the closure
$X$ of the orbit of the shift $\ZZ^d$ action is uniquely ergodic
(and so minimal). A crystal is "turbulent", if the subshift
has no nontrivial discrete spectrum and it is called "chaotic",
if all the shifts have some absolutely continuous spectrum.
Theorem~\ref{multi} suggests (but does not imply)
that the set of measures
for which the shifts have singular continuous spectrum is generic
in the set of measures with uniquely ergodic support.
\end{remark}
\section{Circle-valued cocycles}
Let $T_{\alpha}$ be the irrational rotation $x \mapsto x+\alpha \;
{\rm mod} \; 1$ on the circle $X=\TT^1$. Define the maps $a_{\beta,s}:
X \rightarrow \TT=\{|z|=1\}$
by $a_{\beta,s}(x)=e^{2\pi i s 1_{[0,\beta)}}x$.
\begin{thm}
\label{cocycle}
Given $s \neq 0$. For a generic pair $(\alpha,\beta)$, the operator
$(U_{\beta,s,\alpha} f)(x) = a_{\beta,s}(x) f(x+\alpha)$
has purely singular continuous spectrum.
\end{thm}
\begin{proof}
If $\alpha$ has bounded partial quotients and $s \neq 0$ and $\beta \notin
\alpha \RR/\ZZ$, then $U_{\beta,s,\alpha}$ has purely continuous spectrum
(\cite{Mer85} Theorem 2.4) using \cite{Bag88} Proposition 2.1).
If $\alpha$ is rational, then $U_{\beta,s,\alpha}$ has pure point
spectrum. $U_{\beta,s,\alpha}$ depends continuously
on $(\alpha,\beta)$ in the strong operator topology.
The claim follows from Simon's theorem.
\end{proof}
\begin{remark}
Riley \cite{Ril78} has proven that
for all irrational $\alpha$, the unitary $U_{\beta,1/2,\alpha}$ has purely
singular continuous spectrum for almost all $\beta$.
\end{remark}
Let $V$ be a measurable map from $\TT^1=\RR/\ZZ$ to $\RR$. Consider a
differential equation $\dot{u} = H_t u$ for a
function $u \in L^2(\TT^1)$, where $H_t$ is the time-dependent Hamiltonian
$H_t = i \left(\frac{d}{d\theta}
+ V(\theta) \delta(t+2 \pi n \alpha) \right)$.
The time one map is a unitary circle-valued cocycle.
\begin{coro}
There are potentials $V=s \cdot 1_{[0,\beta)}$
for which the unitary Floquet operator of $H_t$ has purely
singular continuous spectrum.
\end{coro}
\begin{proof}
The time one map is a unitary circle-valued cocycle of
Theorem~\ref{cocycle}.
\end{proof}
Given $m \in \ZZ \setminus \{0\}$ and
$a_{\phi,m}(x)=e^{2\pi i \phi(x) + m x}$, where $\phi$ is in the topological
space $C(\TT)$ of continuous periodic functions. Denote by
$U_{\phi,m,\alpha}$ the associated unitary weighted composition
operator on $L^2(X)$: $(U_{\phi,m,\alpha} f)(x) = a_{\phi,m}(x) f(x+\alpha)$.
\begin{thm}
\label{continuouscocycle}
For $m \neq 0$, there exists a dense $G_{\delta}$ of points
$(\phi,\alpha) \in C(\TT) \times \RR$ such that the
operator $U_{\phi,m,\alpha}$ has purely singular continuous spectrum.
\end{thm}
\begin{proof}
The complete metric space $C(\TT)$ contains the dense set of functions $\phi$
which have a Fourier series $\hat{\phi}_n=o(1/n)$. For such $\phi$ and irrational
$\alpha$, the operator $U_{\phi,m\alpha}$ has no eigenvalues if $m \neq 0$
(\cite{LeMa94} Corollary 2).
If $\alpha$ is rational, then the operator $U$ has pure point spectrum.
Apply Simon's theorem.
\end{proof}
\begin{remark}
In Theorem~\ref{continuouscocycle},
the space $C(\TT)$ can be replaced with any subset of $L^2(\TT)$
which is equipped with a topology in which functions with Fourier series
of order $o(1/n)$ are dense. Especially, there exist smooth cocycles with
purely singular continuous spectrum.
\end{remark}
\section*{Appendix: Fourier series version of a theorem of Strichartz and its
converse of Last}
Let $D_n(t) = \sum_{k=-n}^n e^{ikt}=\frac{\sin((n+1/2)t)}{\sin(t/2)}$
be the Dirichlet kernel. The relation
$ \sum_{k=-n}^n |\hat{\mu}_k|^2
= \int_{\TT} D_n(y-x) \; d\mu(x) d \mu(y)$
is used in the proof of Wiener's theorem (\cite{Katznelson}).
Because the F\'ejer kernels $K_n(t)$ satisfy
$$ K_n(t) = \frac{1}{n+1} \left( \frac{\sin(
\frac{n+1}{2} t)}{\sin(t/2)} \right)^2
= \sum_{k=-n}^n (1-\frac{|k|}{n+1}) e^{ikt}
= D_n(t) - \sum_{k=-n}^n \frac{|k|}{n+1} e^{ikt} \; , $$
one gets
\begin{eqnarray}
0 &\leq& \frac{1}{n+1} \sum_{k=-n}^n |k| |\mu_k|^2
= \int_{\TT} \int_{\TT} (D_n(y-x) - K_n(y-x)) \nonumber
d \mu(x) d\mu(y) \\
&=& \sum_{k=-n}^n |\mu_k|^2 -
\int_{\TT} \int_{\TT} K_n(y-x) d\mu(x) d\mu(y) \; .
\label{estimate}
\end{eqnarray}
Both of the following proofs are
adaptations of proofs given in \cite{Las95} in the case of Fourier transforms.
Proof of Theorem~\ref{last}:
\begin{proof}
Because $\hat{\mu}_n=\overline{\hat{\mu}}_{-n}$, we can also sum from $-n$ to $n$,
changing only the constant $C$. If $\mu$ is not uniformly $\sqrt{h}$
continuous, there exists a sequence of intervals $|I_k| \rightarrow 0$
with $\mu(I_l) \geq l \sqrt{h(|I_l|)}$. A property of the F\'ejer kernel
$K_n(t)$ is that for large enough $n$, there exists $\delta>0$
such that $\frac{1}{n} K_n(t) \geq \delta>0$ if $1 \leq n |t| \leq \pi/2$.
Choose $n_l$, so that $1 \leq n_l \cdot |I_l| \leq \pi/2$. Using
Estimate~(\ref{estimate}), one gets
$$ \sum_{k=-n_l}^{n_l} \frac{|\mu_k|^2}{n_l}
\geq \int_{\TT} \int_{\TT} \frac{K_{n_l}(y-x)}{n_l} \; d\mu(x) d\mu(y)
\geq \delta \mu(I_l)^2
\geq \delta l^2 h(|I_l|)
\geq {\rm C} \cdot h(\frac{1}{n_l}) \; , $$
which contradicts the existence of $C$ such that
$\frac{1}{n} \sum_{k=-n}^n |\mu_k|^2 \leq C h(n^{-1})$.
\end{proof}
Proof of Theorem~\ref{strichartz}:
\begin{proof}
\begin{eqnarray*}
\frac{1}{n} \sum_{k=-n}^{n-1} |\hat{F \mu}|_k^2
&\leq& e
\int_0^1 \sum_{k=-n}^{n-1}
\frac{e^{-\frac{(k+\theta)^2}{n^2}}}{n} \; d\theta \;
|\hat{F \mu}|_k^2 \\
&=& e
\int_0^1
\sum_{k=-n}^{n-1} \frac{e^{-\frac{(k+\theta)^2}{n^2}}}{n}
\int_{\TT^2} e^{-i(y-x) k} F(x) \overline{F(y)} \;
d\theta d\mu(x) d\mu(y) \\
&=& e
\int_{\TT^2}
\int_0^1 \sum_{k=-n}^{n-1}
\frac{e^{-\frac{(k+\theta)^2}{n^2}-i(x-y) k}}{n} \;
F(x) \overline{F(y)} d\theta d\mu(x) d\mu(y) \\
&=& e \int_{\TT^2}
\int_0^1 e^{-(x-y)^2 \frac{n^2}{4}+i(x-y) \theta}
\sum_{k=-n}^{n-1}
\frac{e^{-(\frac{k+\theta}{n}+i (x-y) \frac{n}{2})^2}}{n} d\theta
F(x) \overline{F(y)} d\mu(x) d\mu(y) \\
&\leq& e \int_{\TT^2}
e^{-(x-y)^2 \frac{n^2}{4}}
| \int_0^1 \sum_{k=-n}^{n-1}
\frac{e^{-(i \frac{k+\theta}{n}+(x-y)
\frac{n}{2})^2}}{n}\;d\theta | \;
|F(x)| |F(y)| d\mu(x) \; d\mu(y) \\
&\leq& e \int_{-\infty}^{\infty}
\frac{e^{-(\frac{t}{n}+ i(x-y) \frac{n}{2})^2}}{n} \; dt
\int_{\TT^2} e^{-(x-y)^2 \frac{n^2}{4}}
|F(x)| \; |F(y)| \; d\mu(x) d\mu(y) \\
&=& e \sqrt{\pi}
\int_{\TT^2} (|F(x)| |F(y)|) \cdot
(e^{-(x-y)^2 \frac{n^2}{4}}) \; d\mu(x) \; d\mu(y) \\
&\leq& e \sqrt{\pi} ||F||_2^2
(\int_{\TT^2} e^{-(x-y)^2 \frac{n^2}{2}} \; d\mu(x) \; d\mu(y))^{1/2} \; \\
&=& e \sqrt{\pi} ||F||_2^2
(\sum_{k=0}^{\infty} \int_{k/n \leq |x-y|
\leq (k+1)/n} e^{-(x-y)^2 \frac{n^2}{2}} \; d\mu(x) \; d\mu(y))^{1/2} \\
&\leq& e \sqrt{\pi} ||F||_2^2 h(n^{-1}) (\sum_{k=0}^{\infty} e^{-k^2/2})^{1/2}
\leq C \cdot ||F||_2^2 \cdot h(n^{-1}) \; .
\end{eqnarray*}
\end{proof}
{\bf Acknowledgements}: I thank A. Hof, Y. Last, B. Simon for valuable discussions,
A. Hof for the literature information \cite{Rob83,Ril78} and
and J.Choksi for sending us a reprint of \cite{ChNa90}.
\begin{thebibliography}{10}
\bibitem{Bag88}
L.~Baggett.
\newblock On circle-valued cocycles of an ergodic measure-preserving
transformation.
\newblock {\em Israel J. Math.}, 61:29--38, 1988.
\bibitem{ChNa90}
J.R. Choksi and M.G.Nadkarni.
\newblock Baire category in spaces of measures, unitary operators and
transformations.
\newblock In {\em Invariant Subspaces and Allied Topics}, pages 147--163.
Narosa Publ. Co., New Delhi, 1990.
\bibitem{Com93}
J-M. Combes.
\newblock Connections between quantum dynamics and spectral properties of time
evolution operators.
\newblock In {\em Differential equations with applications to Mathematical
physics}, volume 192 of {\em Mathematics in science and engineering}, 1993.
\bibitem{Sinai}
I.P. Cornfeld and Ya. G.Sinai.
\newblock Dynamical systems {II}.
\newblock In Ya.G. Sinai, editor, {\em Encyclopaedia of Mathematical Sciences,
{Volume 2}}. Springer-Verlag, Berlin Heidelberg, 1989.
\bibitem{CFS}
I.P. Cornfeld, S.V.Fomin, and Ya.G.Sinai.
\newblock {\em Ergodic Theory}, volume 115 of {\em {Grundlehren} der
mathematischen {Wissenschaften} in {Einzeldarstellungen}}.
\newblock Springer Verlag, 1982.
\bibitem{DGS}
M.~Denker, C.~Grillenberger, and K.~Sigmund.
\newblock {\em Ergodic Theory on Compact Spaces}.
\newblock Lecture Notes in Mathematics 527. Springer, 1976.
\bibitem{Falconer}
K.~Falconer.
\newblock {\em Fractal Geometry, Mathematical Foundations and Applications}.
\newblock John Wiley and Sons, Chichester, 1990.
\bibitem{Gua89}
I.~Guarneri.
\newblock Spectral properties of quantum diffusion on discrete lattices.
\newblock {\em Europhysics letters}, 10:95--100, 1989.
\bibitem{Halmos}
P.~Halmos.
\newblock {\em Lectures on ergodic theory}.
\newblock The mathematical society of {Japan}, 1956.
\bibitem{HoKn95}
A.~Hof and O.~Knill.
\newblock Cellular automata with almost periodic initial conditions.
\newblock {\em Nonlinearity}, 8:477--491, 1995.
\bibitem{Hof+95}
A.~Hof, O.~Knill, and B.~Simon.
\newblock {Singular continuous spectrum for palindromic Schr\"odinger
operators}.
\newblock {\em Commun. Math. Phys.}, 174:149--159, 1995.
\bibitem{Hol94}
M.~Holschneider.
\newblock Fractal wavelet dimension and localization.
\newblock {\em Commun. Math. Phys.}, 160:457--473, 1994.
\bibitem{JiSi94}
S.~Jitomirskaya and B.Simon.
\newblock Operators with singular continuous spectrum: {III.} {Almost} periodic
{Schr\"odinger} operators.
\newblock {\em Commun. Math. Phys.}, 165:201--205, 1994.
\bibitem{Rob83}
E.A.~Robinson Jr.
\newblock Ergodic measure preserving transformations with arbitrary finite
spectral multiplicities.
\newblock {\em Inv. Math.}, 72:299--314, 1983.
\bibitem{Kahane}
J-P. Kahane and R.~Salem.
\newblock {\em Ensembles parfaits et s\'eries trigonom\'etriques}.
\newblock Hermann, 1963.
\bibitem{KaLi94}
B.~Kaminski and P.~Liardet.
\newblock Spectrum of multidimensional dynamical systems with positive entropy.
\newblock {\em Studia Mathematica}, 108:77--85, 1994.
\bibitem{Ka+75}
A.B. Katok, Ya.~G. Sinai, and A.M. Stepin.
\newblock Theory of dynamical systems and general transformation groups with
invariant measure.
\newblock {\em Math. Ann.}, 13:129--262, 1975.
\bibitem{KaSt67}
A.B. Katok and A.M. Stepin.
\newblock Approximations in ergodic theory.
\newblock {\em Russ. Math. Surveys}, 22:77--102, 1967.
\bibitem{KaSt70}
A.B. Katok and A.M. Stepin.
\newblock Metric properties of measure preserving homemorphisms.
\newblock {\em Russ. Math. Surveys}, 25:191--220, 1970.
\bibitem{Katznelson}
Y.~Katznelson.
\newblock {\em An introduction to harmonic analysis}.
\newblock John Wiley and Sons, Inc, New York, 1968.
\bibitem{Las95}
Y.~Last.
\newblock Quantum dynamics and decompositions of singular continuous spectra.
\newblock Caltech preprint, April, 1995.
\bibitem{LeMa94}
M.~Lemanczyk and C.~Mauduit.
\newblock Ergodicity of a class of cocycles over irrational rotations.
\newblock {\em J. London Math. Soc.}, 49:124--132, 1994.
\bibitem{Mer85}
K.D. Merrill.
\newblock Cohomology of step functions under irrational rotations.
\newblock {\em Israel J. Math.}, 52:320--340, 1985.
\bibitem{Par61}
K.R. Parthasarathy.
\newblock On the categorie of ergodic measures.
\newblock {\em Ill. J. Math.}, 5:648--656, 1961.
\bibitem{Queffelec}
M.~Queff\'elec.
\newblock {\em Substitution Dynamical Systems---Spectral Analysis}, volume 1294
of {\em Lecture Notes in Mathematics}.
\newblock Springer, 1987.
\bibitem{Ril78}
G.W. Riley.
\newblock On spectral properties of skew products over irrational rotations.
\newblock {\em J. London Math. Soc.}, 17:152--160, 1978.
\bibitem{De+94a}
R.~Del Rio, S.~Jitomirskaya, N.~Makarov, and B.~Simon.
\newblock Singular continuous spectrum is generic.
\newblock {\em Bull. Amer. Math. Soc.}, 31:208--212, 1994.
\bibitem{De+94}
R.~Del Rio, N.~Makarov, and B.~Simon.
\newblock Operators with singular continuous spectrum ii, rank one operators.
\newblock {\em Commun. Math. Phys.}, 165:59--67, 1994.
\bibitem{Roh59}
V.A. Rohlin.
\newblock On the entropy of a metric automorphism.
\newblock {\em Dokl. Akad. Nauk.}, 128:980 --983, 1959.
\bibitem{Sig70}
K.~Sigmund.
\newblock Generic properties of invariant measures for axiom a diffeomorphisms.
\newblock {\em Inv. Math.}, 11:99--109, 1970.
\bibitem{Sig71}
K.~Sigmund.
\newblock On the prevalence of zero entropy.
\newblock {\em Israel J. Math.}, 10:497--504, 1991.
\bibitem{Sig72}
K.~Sigmund.
\newblock On mixing measures for axiom a diffeomorphisms.
\newblock {\em Proc. Amer. Math. Soc.}, 36:497--504, 1992.
\bibitem{Sim95a}
B.~Simon.
\newblock {$L^p$} norms of the {Borel} transform and the decomposition of
measures.
\newblock {\em Proc. Amer. Math. Soc.}, 123:3749--3755, 1995.
\bibitem{Sim95}
B.~Simon.
\newblock Operators with singular continuous spectrum: {I}. {General}
operators.
\newblock {\em Annals of Mathematics}, 141:131--145, 1995.
\bibitem{Str90}
R.S. Strichartz.
\newblock Fourier asymptotics of fractal measures.
\newblock {\em J. Func. Anal.}, 89:154--187, 1990.
\bibitem{Vee84}
W.A. Veech.
\newblock The metric theory of interval exchange transformations {I}, generic
spectral properties.
\newblock {\em Amer. J. Math.}, 106:1331--1359, 1984.
\end{thebibliography}
\end{document}
End of koop.tex