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\def\muef{{\rm \mu \, ess.inf}}
\def\mes{{\rm \mu \, ess.sup}}
\def\nmuef{{\rm \mu_{\psi_0} \, ess.inf}}
\def\nmes{{\rm \mu_{\psi_0} \, ess.sup}}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
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\newtheorem{lem}{Lemma}[section]
\newtheorem{prop}{Proposition}[section]
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}{Corollary}[section]
\newtheorem{defn}{Definition}[section]
\newtheorem{rem}{\it {Remarks.}}
\begin{document}
\begin{titlepage}
\begin{center}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
{\twelve Centre de Physique Th\'eorique\footnote{
Unit\'e Propre de Recherche 7061
}, CNRS Luminy, Case 907}
{\twelve F-13288 Marseille -- Cedex 9}
\vspace{3 cm}
{\fifteen REMARKS ON THE RELATION BETWEEN \\
QUANTUM DYNAMICS AND FRACTAL SPECTRA}
\vspace{0.3 cm}
\setcounter{footnote}{0}
\renewcommand{\thefootnote}{\arabic{footnote}}
{\bf
J.M. BARBAROUX$^1$, J.M. COMBES\footnote{
and PHYMAT, Universit\'e de Toulon et du Var, B.P.132, F-83957 La Garde Cedex,
France}, R. MONTCHO$^1$
}
\vspace{2,3 cm}
{\bf Abstract}
\end{center}
We investigate the large time behaviour of various solutions of the
Schr\"odinger equation in terms of some local and global dimensions of the
spectral measure. We emphasize in particular the r\^ole of the Hausdorff and
correlation dimensions for the growth exponents of position moments. We also
discuss the stability of such exponents under local perturbations.
\vspace{2,3 cm}
\noindent January 1996
\noindent CPT-96/P.3303
\bigskip
\noindent anonymous ftp or gopher: cpt.univ-mrs.fr
\end{titlepage}
\setcounter{footnote}{0}
\renewcommand{\thefootnote}{\arabic{footnote}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%% CHAPTER I %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%% INTRODUCTION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\setcounter{page}{1}
\section{Introduction}
${}^{}$
The connections between quantum dynamics and spectral properties of the
Hamiltonians generating the time evolution has a long history going back to
the celebrated RAGE theorem \cite{RS}. The occurence of unusual types of
spectra in various models from solid state physics, e.g. singular
continuous or dense pure point, has stimulated a renewed interest for
a deeper investigation of the transport properties associated to such spectra.
A crucial step in this direction was accomplished by Guarneri \cite{Guarneri} who
investigated the r\^ole of a global scaling exponent for the spectral measures
of the Floquet operators associated to pulsed systems for the time behaviour of
various dynamical quantities such as correlation functions or mean square
displacement. Those results were extended by one of us \cite{Combes} to
continuous dynamics on $\R^d$ using results of Strichartz \cite{Strichartz}
on the asymptotics of Fourier transforms of uniform $\gamma$-dimensional
measures (see definition 2.2 below).
Soon after, Holschneider
\cite{H} using wavelet transform and Ketzmerick et al. \cite{Geisel}
emphasized the r\^ole of the correlation dimension of the spectral measures
for such asymptotics. More recently, new contributions were provided by
\cite{BMZ}, \cite{Gm}, \cite{DelRio Simon} and \cite{Last}. One of the aim
of this paper is to clarify the links between the whole set of results
obtained so far and in particular to derive from them the fact that the
information dimension of the spectral measures provides a universal lower
bound on the growth exponents of the moments for the position operator.
Then using some kind of stability property for the
growth exponent of moments, we will provide some explicit examples showing
that in general the information dimension does not provide upper bounds. In
particular it is easy to construct discrete Schr\"odinger operators
with pure point spectrum having in some sense almost ballistic growth of
high moments. This has already been obtained by del Rio et al. \cite{DelRio
Simon}
on a specific example; we show here
that this occurs for almost every one site perturbations of Schr\"odinger
operators whose spectral measures have information dimension close to 1.
The plan of this paper is as follows; in chapter 2 we recall the definition of
local and global exponents of spectral measures and give a rigourous proof that
the
correlation dimensions control time-decay of correlation functions. In
chapter 3 we consider growth exponents for moments of position operators; our
result in terms of information dimension is in fact a more elaborate form of a
recent result of Last \cite{Last}. In chapter 4 we present a stability result
under rank
one perturbations which shows that dynamics are in a weak sense invariant
under such perturbations. Finally in Appendix A we relate theorem 2.2 of
chapter 2 to those obtained in \cite{H} using wavelet transform and we prove
that the mean quadratic dimensions are equal to the correlation dimensions;
as this paper was written we learned of a preprint of Guerin and Holschneider
showing similar estimates \cite{GH}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%% CHAPTER II %%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%% FRACTAL DIMENSION OF MEASURE %%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%% AND ASYMPTOTIC BEHAVIOUR %%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%% OF FOURIER TRANSFORM %%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Fractal dimension of measure and asymptotic behaviour of Fourier
transform}
${}^{}$
%%%%%%%%%%%%%%%%%%%% LOCAL EXPONENTS %%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let us first recall the definition of local exponents for measures and their
relation to Hausdorff and correlation dimensions.
\begin{defn}
Local Exponents
Consider a positive finite Borel measure $\mu$ on
$\R $.
The upper and lower scaling exponents are defined as follow:
\begin{eqnarray}
\gamma^-(x) =\liminf_{\varepsilon\rightarrow 0}
\frac{\log\mu(B_\varepsilon(x))}{\log\varepsilon} \\
\gamma^+(x) =\limsup_{\varepsilon\rightarrow 0}
\frac{\log\mu(B_\varepsilon(x))}{\log\varepsilon}
\end{eqnarray}
where $B_\eps(x)$ is the open ball of center x and radius $\eps/2$.
\end{defn}
\begin{rem}
${}^{ }$
\noindent
{\rm
\begin{description}
\item[(a)] It is easy to see that $\gamma^-(x)$ is a measurable function;
indeed, it is a consequence of
standard arguments and of the fact that for
all $x$, the limit
in the definition of $\gamma^-(x)$
remain unchanged if the continuous variable $\eps$ is replaced by any sequence
$\{\eps_n\}$ with $\eps_n\searrow 0$ and $\log\,\eps_{n+1}/\log\,
\eps_n\rightarrow\,1$ (see \cite{Young}). In particular for any $\alpha$, the
sets $\{x\; | \; \gamma^-(x)\geq
\alpha\}$ are $\mu$ measurable.
\item[(b)] As a consequence of Radon-Nykodim decomposition of a measure, one
obtains that:
$$ \mu\; a.e.x\, :\;\;\;\;\;0\leq\gamma^-(x)\leq\gamma^+(x)\leq 1 $$
\item[(c)] It is well known that for $\mu$ a probability measure:
\begin{eqnarray}
\muef \,\gamma^-(x)\leq\dim_H(\mu)\leq\mes\,\gamma^+(x)
\end{eqnarray}
where $\dim_H(\mu)$ is the Hausdorff dimension of $\mu$
(see \cite{Falconer} and \cite{Pesin}), also
called information dimension.
\item[(d)] In the particular case where the scaling exponents are $\mu$-almost
everywhere constant,
a theorem of Young asserts that if
\begin{eqnarray}
\gamma^-(x)=\gamma^+(x)=\overline{\gamma}\;\;is\;constant\;for\;
\mu\, a.e.x,
\end{eqnarray}
then $$\dim_H(\mu)=\overline{\gamma}$$
Relation (4) is
exceptional; it is satisfied however for any absolutely continuous measure with
$\overline{\gamma} =1$. We will extend this result in chapter 3 by showing
that
\begin{eqnarray*}
\dim_H(\mu)=\mes\,\gamma^-(x)
\end{eqnarray*}
\end{description}
}
\end{rem}
%%%%%%%%%%%%%% MEASURE LOC UNIF ALPHA DIMENSIONAL %%%%%%%%%%%%%%%%%%%%%
\begin{defn} Let $0\leq \alpha\leq 1$;
$\mu$ is said to be
locally $\alpha-dimensional\;at\;x_0 $ if:
\smallskip
\noindent
$\exists c<+\infty\;\;such\;that\;\;\forall\,0<\varepsilon<1\;\;
\mu(B_\varepsilon(x_0))\leq\, c\,\varepsilon^\alpha$
\smallskip
\noindent
$\mu$ is said to be locally uniformly $\alpha-dimensional$ (L.U.$\alpha$) if:
\smallskip
\noindent
$\exists c<+\infty\; and\; 0<\eps_0<1\;such\; that\;for\;\mu\; a.e.x\;
,\;\forall
\eps ,\;\,0<\eps<\eps_0\, ,\;\;\;\mu(B_\varepsilon(x)\leq
\, c\,\varepsilon^\alpha$
\end{defn}
In \cite{Strichartz}, Strichartz proves the following (see also \cite{Last}
for a simple proof):
%%%%%%%%%%%%%%%%%% STRICHARTZ THEOREM %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thm}
If $\mu$ is a measure on $\R$ which is L.U.$\alpha$, and $f\in L^2(d\mu )$,
then there exists $C_1$ depending only on $\mu$ such that:
\begin{eqnarray}
\sup_{T\geq 1}\, T^{\alpha -1}\int_{-T}^T\, |(f\mu )\,\hat{}
\, (t)|^2dt\leq C_1\,\| f\|_2{}^2
\end{eqnarray}
where $\hat{}$ denotes the Fourier transform.
\end{thm}
\begin{rem}
{\rm
Each finite positive measure is L.U.$0$.
A measure $\mu$ which has a pure point component on $x_0$ has a local exponent
equal to zero at $x_0$.
An absolutely continuous measure (w.r.t. Lebesgue measure) with a density
$f(x)$
has a
local exponent equal to 1 at $x_0$ if $f$ is locally bounded near $x_0$.
A measure which is locally of dimension $\alpha$ at $x$ is locally of dimension
$\beta$ for any $\beta\leq\,\alpha$, and has a scaling exponent
$\gamma^-(x)\geq\,\alpha$ }
\end{rem}
Theorem 2.1 has been used to study the time behaviour of moments $\la\la
|X^m|\ra\ra_T$ (see (24) and (25) for the definition) and to extend
the result of Guarneri \cite{Guarneri} and Last \cite{Last} to $L^2(\R^d)$
(see \cite{Combes}). An immediate consequence of this theorem is the following:
%%%%%%%%%%%%%%%%%%%% CORRELATION FUNCTION %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{cor}
Let ${\cal H}$ be a Hilbert space and $H$ a self-adjoint operator on ${\cal H}$.
Let $E_\lambda(.)$ be the spectral family of the hamiltonian $H$, and define,
for $\psi_0\in{\cal D}(H)$, $d\mu_{\psi_0}=d\la E_\lambda\psi_0,\psi_0\ra $
where $\la\; ,\;\ra$ denotes the scalar product on ${\cal H}$.
For fixed $\psi_0$ we denote by $\psi(t)$ the solution of the Schr\"odinger
equation :
$$\left\{
\begin{array}{l}
\ds \imath\frac{d}{dt}\,\psi (t)=H\,\psi(t)\\
\ds {} \\
\ds\psi(t=0)=\psi_0
\end{array}
\right.$$
Assume that the spectral measure $d\mu_{\psi_0}$ is L.U.$\alpha$; then if
\begin{eqnarray*}
p(t)=|\la\psi_0,\, \psi_t\ra |^2
\end{eqnarray*}
the correlation function
\begin{eqnarray}
\ds C(T)\equiv\frac{1}{T}\int_0^T\, p(t)\, dt
\end{eqnarray}
satisfies:
\begin{eqnarray}
\exists C_1<+\infty\; s.t.\;\;\forall T>1,\;\;C(T)\leq\, C_1\,
T^{-\alpha}
\end{eqnarray}
and for any $\vp\in {\cal D}(H)$
\begin{eqnarray}
\frac{1}{T}\int_0^T |\la\vp\, ,\psi_t\ra |^2\, dt\leq\, C_1\,
T^{-\alpha}\|\vp \|
\end{eqnarray}
\end{cor}
The proof follows obviously from:
\begin{eqnarray*}
\la\vp,\, \psi(t)\ra=\int_{\R} f(\lambda)e^{-\imath t\lambda}d\mu_{\psi_0}
(\lambda )
\equiv (f\mu_{\psi_0})\, \hat{} \,(t)
\end{eqnarray*}
for some $f$ in $L^2(d\mu_{\psi_0})$.
Unfortunately, it appears that the greatest $\alpha$ such that $\mu$ is
L.U.$\alpha$ can be rather small in comparison for example with the
$\mu$-essential infimum of the lower
local exponents $\gamma^-(x)$ (see example of appendix B).
\smallskip
One can try to improve the situation by considering other characteristic
exponents
of the measure $\mu$. This has been done in particular by Geisel et al.
\cite{Geisel} and
Holschneider \cite{H}. The first one emphasized the role of the so-called
correlation
dimension defined as follow:
%%%%%%%%%%%%%%%%%%%% CORRELATION DIMENSIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{defn}
\smallskip
Let $\mu$ be any finite positive Borel Measure on $\R$ and
\begin{eqnarray}
\gamma_2(\varepsilon)=\int_R\;\mu ((\omega-\frac{\varepsilon}{2},\omega+
\frac{\varepsilon}{2}])
\, d\mu(\omega)
\end{eqnarray}
then:
\begin{eqnarray}
D_2^-=\liminf_{\varepsilon\rightarrow 0}\frac{\log\gamma_2(\varepsilon)}
{\log\varepsilon}\;is
\;the\;lower\;correlation\;dimension \\
D_2^+=\limsup_{\varepsilon\rightarrow 0}\frac{\log\gamma_2(\varepsilon)}
{\log\varepsilon}\;is\; the\;upper\;correlation\;dimension
\end{eqnarray}
If $D_2^+=D_2^-=D_2$, then $D_2$ is called the correlation dimension of $\mu$.
\end{defn}
\begin{rem}
{\rm
${}^{}$
\begin{description}
\item[(i)] $D_2^+$ and $D_2^-$ remain unchanged if one replaces
$(\omega-\frac{\varepsilon}{2},\omega+\frac{\varepsilon}{2}]$ by
$B_\eps(\omega)$ in the definition of $\gamma_2(\eps)$. It is obvious if
$\mu$ is a purely continuous measure; if the pure point part $\mu_p$ of $\mu$
is
such that $\mu_p\neq 0$, then
$D_2^+=D_2^-=0$ for $\gamma_2$ defined with
$(\omega-\frac{\varepsilon}{2},\omega+\frac{\varepsilon}{2}]$ or
$B_\eps(\omega)$.
\item[(ii)] In \cite{Geisel}, Ketzmerick et al. infer from a formal calculation
that:
\begin{eqnarray}
\gamma_2(\varepsilon)\sim\eps^{D_2}\Leftrightarrow C(T)\sim T^{-D_2}
\end{eqnarray}
where $C(T)$ is defined as in (6) and $D_2$ is associated to $\mu_{\psi_0}$.
This is confirmed by some
numerical computations; an analytical proof will be given and generalized
below in theorem 2.2. Our results are strongly related to those obtained by
Holschneider in \cite{H} with the wavelet transform. Comparison with
Holschneider 's results is presented in Appendix A.
\end{description}
}
\end{rem}
\begin{prop} If $\mu$ is a finite Borel measure on $\R$, then:
\begin{eqnarray}
D_2^-\leq\, \muef\,\gamma^-(x)
\end{eqnarray}
\end{prop}
\noindent{\bf Proof: }
By the definition of $D_2^-$, for fixed $\eta$ such that $D_2^->\eta>0$,
$$\exists\eps(\eta)\, ,\;\forall\eps<\eps(\eta)\, :\;\;\int \,\frac{\mu(B_
\varepsilon(x))}{\varepsilon^{D_2^--3\eta/4}}\,d\mu(x)\leq\varepsilon^{\eta/4}$$
$$\Rightarrow\int\,d\mu(x)\int_0^{\varepsilon(\eta)}\frac{\mu(B_
\varepsilon(x))}{\varepsilon^{D_2^--3\eta/4}}\,\frac{d\eps}{\eps}<+\infty$$
\begin{equation}
{\Rightarrow\;\;\mu-a.e.x\;\;\int_0^{\varepsilon(\eta)}
\frac{\mu(B_\varepsilon (x))}{\varepsilon^{D_2^--3\eta/4}}\,\frac{d\eps}{\eps}
<+\infty}
\end{equation}
\noindent
Assume that for some $x$ one has $D_2^--2\eta>\gamma^-(x)$; then
by the definition of $\gamma^-(x),\;\exists\;(\eps_n)_{n\in\N},\eps_n\searrow
0$, such that
$$\varepsilon_n<\frac{\varepsilon_{n-1}}{2}\;\;and\;\;\mu(B_{\varepsilon_n}(x))>
\varepsilon_n^{\gamma^-(x)+\eta /2}>\varepsilon_n^{D_2^--3\eta /2}$$
\noindent
Then since $\mu(B_\eps (x))$ is a non decreasing function of
$\eps$ one has
$\mu(B_\varepsilon (x))>\varepsilon^{D_2^--\eta}$ at least on the
interval $[\eps_n,
\eps'_n]=[\eps_n,\eps_n^{1-\eta /2}]$.
Now let $N$ be the smallest integer such that $\eps_N<\eps(\eta)$ and
$ \eps_N^{1-\eta /2}>2\eps_N$; then for all
$n>N$, $A_n\equiv [\eps_n,2\eps_n] \subset [\eps_n,\eps_n^{1-\eta /2}]$, and
all the $A_n$ are
disjoint, thus:
\begin{equation}
{
\int_0^{\varepsilon(\eta)}\,\,\frac{\mu(B_\varepsilon(x))}
{\varepsilon^{D_2^--3\eta/4}}\,\frac{d\eps}{\eps}\geq\sum_{n>N}\,
(2\varepsilon_n)^{-1}\,
\frac{\varepsilon_n
^{D_2^--\eta}}{(2\varepsilon_n)^{D_2^--3\eta /4}}\,\varepsilon_n\,= +\infty
}
\end{equation}
>From (14) and (15) one obtains:
$$\;\;\mu-a.e.x:\;\;\gamma(x)\geq D_2^--2\eta$$
This proves the proposition since the above result is valid for all $\eta>0$.
\begin{rem}
{\rm With the same arguments as before, one can prove that $\forall S\in{\cal
B}(\R)$, $\mu(S)>0$:
$$ \liminf_{\eps\rightarrow 0}\frac{\log \int_S\,\mu(B_\eps (x))\, d\mu
(x)}{\log \eps}
\leq\muef\{\gamma^-(x)\, ,\;x\in S\}$$
}
\end{rem}
We are now ready to prove the following:
%%%%%%%%%%%%%%%%%%%% THEOREME 2.2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thm}
Let $\psi_0\in {\cal D}(H)$. With the notations of corollary 2.1,
if $D_2^+$ and $D_2^-$ are the upper and lower correlation dimensions of
the measure $\mu_{\psi_0}$, one has:
\begin{eqnarray}
\liminf_{T\rightarrow\infty}\frac{log\, C(T)}{log\,T}=-D_2^+ \\
\limsup_{T\rightarrow\infty}\frac{log\, C(T)}{log\,T}=-D_2^-
\end{eqnarray}
\end{thm}
\noindent
For the proof of this theorem, we need the following preliminary result:
\begin{lem}
For $\mu\equiv\mu_{\psi_0}$ and $\eps>0$, let
\begin{eqnarray}
A(\eps)=\int_{\R}\, |F_{\mu}(x+\imath\eps)|^2\, dx
\end{eqnarray}
where $F_{\mu}
(z)$ is the
Borel transform of $\mu$: $\ds F_{\mu}(z)=\int_{\R}\frac{d\mu
(y)}{y-z}$.
Then, with the same notations as in corollary 2.1:
\begin{description}
\item[(i)]
\be
A(\frac{1}{T})=4\pi\int_0^\infty\, e^{-\frac{t}{T}}p(t)\, dt
\ee
\item[(ii)]
\be
A(\eps)=4\pi\eps\int_{\R\times\R}\frac{d\mu(y)\, d\mu(y')}{(y-y')^2+4\eps^2}
\ee
\end{description}
\end{lem}
\begin{rem}
{\rm
A similar quantity is used in \cite{DelRio Simon} for a rank one
perturbation theory, and in
\cite{BMZ} to relate the behaviour of the Green function to singularities of
the spectral measure. }
\end{rem}
\noindent
{\bf Proof of lemma:}
(i) is well known from Kato's theory of smooth operator \cite{RS}
\noindent
(ii) follows from a straightforward calculation using Cauchy formula.
\medskip
\noindent
{\bf Proof of theorem 2.2: }
\smallskip
Let $\displaystyle{c^-=\liminf_{\varepsilon\rightarrow 0} \frac{\log
C(1/\varepsilon)} {\log\varepsilon}}$ and
$\displaystyle{c^+=\limsup_{\varepsilon\rightarrow 0}
\frac{\log C(1/\varepsilon)}{\log\varepsilon}}$; then for all $\nu>0$:
\begin{eqnarray*}
\frac{1}{T}\int_0^T|\hat{\mu}(t)|^2\, dt
& \leq & \frac{e}{T}\int_{\R}e^{-t^2/T^2}\int e^{-\imath (x-y)t}d\mu(x)
d\mu(y)\, dt \\ & = & e\sqrt{\pi}\int d\mu(x) d\mu(y)e^{-(x-y)^2T^2/4} \\
& = & e\sqrt{\pi}\int_{|x-y|0\, ,\;\exists T(\nu)<\infty$
such that $\forall T>T(\nu)$:
\begin{eqnarray*}
\gamma_2(\frac{1}{T})\leq a\, T^{\nu} C(T^{1+\nu})
\end{eqnarray*}
and then
\begin{eqnarray}
c^+\leq -D_2^- \; {\rm and} \;c^-\leq -D_2^+
\end{eqnarray}
This inequality together with (22) gives the expected result.
\begin{rem}
{\rm
One can also obtain the result (17) with the help of Wiener Tauberian
theorem \cite{Wiener} and the mean quadratic dimensions (see appendix A).
}
\end{rem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%% CHAPTER III %%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%% LOWER BOUNDS ON MOMENTS %%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Lower bounds on moments}
In this chapter, we consider self-adjoint operator $H$ on ${\cal H}=l^2(\Z^d)$
or ${\cal H}=L^2(\R^d)$ and the moments of $\psi_0$:
\be
\la |X|^m(t) \ra = \la\psi(t),\, |X|^m\psi(t)\ra
\ee
Here $\psi_0$ and $\psi(t)$ are defined as in corollary 2.1, as well as the
measure
$\mu_{\psi_0}$, and $X$ is the position operator. The time averaged
moments are:
\be
\la\la |X|^m \ra\ra_T \equiv \frac{1}{T}\int_0^T\, \la |X|^m(t) \ra\,dt
\ee
The main result of this chapter is
\begin{thm}
With the same notations as above, if $H$ is self-adjoint on $l^2(\Z ^d)$ and
$\psi_0\in{\cal D}(H)$,
$$\forall m>0\;and\;\forall\nu>0\, ,\;\exists \tilde{C}(m,d,\nu,\psi_0)\;
\;such\;that:$$
\begin{eqnarray}
\la\la |X|^m\ra\ra_T\geq \tilde{C}\, T^{\frac{m}{d}(\gamma_0-\nu)}
\end{eqnarray}
where $\ds\gamma_0\equiv \nmes\,\gamma^-(x)$
\end{thm}
{\bf Proof: }
The proof of the theorem consists of three parts
\begin{description}
\item[(1)]
Let $(e_n)$ be an orthonormal basis of $l^2(\Z^d)$. If for
some $\psi_1$, $\mu_{\psi_1}$ is $L.U.\alpha$, then corollary 2.1 tells
us that for $p_n(T,\psi_1)\equiv\frac{1}{T}\int_0^T|\la e_n,\, \psi_1(t)\ra
|^2\, dt$, $\exists C_1<\infty$ such that $\forall N\in \Z$, $\forall T>1$:
\begin{eqnarray}
\sum_{|n|\leq N} p_n(T,\psi_1) \leq N^d C_1\, T^{-\alpha}
\end{eqnarray}
\item[(2)]
Let $\gamma_0=\nmes \gamma^-(x)$, for fixed $\nu>0$, $\exists S\in\R$ such
that $\mu_{\psi_0}(S)>0$ and $\forall x\in S$, $\gamma_0\geq\gamma^-(x)
>\gamma_0-\nu$. Moreover, Egoroff theorem implies that
$\exists S'\subset S$, $\mu(S')>0$, such that the measurable functions
$\ds{f_\eps(x)\equiv\inf_{\eps '<\eps}\frac{\log \mu(B_{\eps '}(x))}{\log
\eps '}}$ converge uniformly on $S '$ when $\eps\rightarrow 0$. Let
$\psi_1=E(S')\psi_0$, where $E(\;)$ is the spectral projection for $H$; then
$\mu_{\psi_1}$ is
$L.U.\, \gamma_0-\nu$ and $\|\psi_1\|>0$.
\item[(3)] Now define $\ds\psi_{1,t}=E(S')\psi_t$, $\ds\psi_{2,t}
=\psi_t-\psi_{1,t}$ and
\begin{eqnarray*}
n_0(\psi_1,T)=\max
\Big\{ k \; | \; \frac{1}{T}\int_0^T\big ( \sum_{|n|< k}
|\la \psi_{1,t},\, e_n\ra |^2\big ) dt\leq\frac{\|\psi_1\|^4}{9} \Big\}
\end{eqnarray*}
then for
$\psi_{1,t}(n)=|\la \psi_{1,t} ,\, e_n\ra |$ and
$\psi_{2,t}(n)=|\la \psi_{2,t} ,\, e_n\ra |$, one has:
\begin{eqnarray}
\sum_{|n|< n_0}\!\!\! p_n(T,\psi_0)
& \leq & \frac{1}{T}\int_0^T\Big( \sum_{|n|< n_0}
(\psi_{1,t}(n)+\psi_{2,t}(n))^2 \Big) dt \nonumber \\
& \leq & \!\frac{1}{T}\int_0^T\Big(\!\!\!\! \sum_{|n|< n_0}\!\!\!
\psi_{1,t}^2(n) \Big)\, dt
+ 2\|\psi_2\| \Big( \frac{1}{T}\int_0^T\Big(\!\!\!\! \sum_{|n|< n_0} \!\!\!
\psi_{1,t}^2(n)\Big)\,dt
\Big)^{\frac{1}{2}} \nonumber \\
& &\; +\|\psi_2\|^2
\end{eqnarray}
By the definition of $n_0$, one obtains from (28) that
\begin{eqnarray}
\sum_{|n|\geq n_0}p_n(T,\psi_0) \geq \frac{1}{6} \|\psi_1\|^2
\end{eqnarray}
then
\begin{eqnarray}
\la\la |X|^m \ra\ra_T & \equiv & \sum_{n\in\Z^d} |n|^m p_n(T,\psi_0) \nonumber
\\ & \geq & n_0(\psi_1,T)^m \sum_{|n|\geq n_0} p_n(T,\psi_0)
\end{eqnarray}
Since $\mu_{\psi_1}$ is
$L.U.\, \gamma_0-\nu$, one obtains from (27) and the definition of $n_0$:
\begin{eqnarray}
n_0(T,\psi_1)\geq \big ( T^{\gamma_0-\nu}\frac
{\|\psi_1\|^4}{9 C_1} \big )^{1/d}
\end{eqnarray}
which implies together with (29) and (30) that
\begin{eqnarray}
\la\la |X|^m\ra\ra_T \geq \tilde{C}\, T^{m(\gamma_0-\nu)/d}
\end{eqnarray}
where $\tilde{C}$ is a finite constant depending on $\psi_1$, $m$, $d$, and
$\nu$.
\end{description}
\begin{rem}
{\rm
This result can also be deduced from Last's theorem, which is presented in
term of most continuous component of the measure; indeed,
the $\alpha$-continuous part $\mu^{c,\alpha}_{\psi_0}$
of $\mu_{\psi_0}$ is supported on a set
$T_{\alpha ,c}$ (in the sense $\mu^{c,\alpha}_{\psi_0}(\R -
T_{\alpha ,c})=0$), such that
$ \{x\, |\, \gamma^-(x)>\alpha\}\subset T_{\alpha ,c} \subset
\{x\, |\, \gamma^-(x)\geq\alpha\}$ (see \cite{Rogers}). Then,
$\forall\alpha<\gamma_0$, $\mu^{c,\alpha}_{\psi_0}\neq 0$.
If $\exists S$, $\mu(S)>0$ such that $\forall x\in S$, $\gamma^-(x)=\gamma_0$,
then one can take $\nu=0$ in theorem 3.2 and corollary 3.1. This is the case for
example if the absolutely continuous part of $\mu$ is not zero, and for which
$\gamma_0=1$.
}
\end{rem}
\medskip
One can relate the dimension $\gamma_0$ to the information dimension as
follows:
\begin{thm}
For $\ds\mu_{\psi_0}$ and $\gamma_0$ as above:
\begin{eqnarray}
\gamma_0=\dim_H(\mu_{\psi_0})
\end{eqnarray}
\end{thm}
{\bf Proof} Let $\mu\equiv\mu_{\psi_0}$ and
$S\equiv S(\nu)=\{x\in\R\; |\;\gamma^-(x)\geq\gamma_0-\nu\}$.
Let $\ds \mu |_{S}$ be the probability measure defined as: for all
subset $A$ of $\R$,
$\mu |_{S}(A)=\mu(S\cap A)/\mu(S)$. For every $A$ such that $\mu(A)=1$,
one has $\mu|_S(A)=1$; then
\be
\dim_H(\mu |_{S})\leq\dim_H(\mu)
\ee
(3) and the definition of $\mu |_S$ gives:
\begin{eqnarray}
\gamma_0-\nu\leq{\rm \mu |_{S} \, ess.inf}\gamma^-(x)\leq\dim_H(\mu |_{S})
\end{eqnarray}
Now, for fixed $\nu>0$, let $S'(\nu)\equiv \{x\; |\;
\gamma^-(x)<\gamma_0+\nu\}$.
One has $\ds S'(\nu)\subset T_{s,\gamma_0 +\nu}\equiv \{x \; |
\;\limsup_{\eps\rightarrow 0}
\frac{\mu(B_\eps(x))}{\eps^{\gamma_0 +\nu}}=\infty\}$, where $T_{s,\gamma_0
+\nu}$ is a set that supports the singular part of $\mu$ with respect to
$H^{\gamma_0+\nu}$ (see \cite {Rogers} and \cite{Last}),
$H^{\gamma_0+\nu}$
beeing the $({\gamma_0+\nu})$-Hausdorff measure.
Then
\begin{eqnarray}
H^{\gamma_0+\nu}(S'(\nu))=0
\end{eqnarray}
Now since
$\mu(S'(\nu))=1$, (36) implies:
\begin{eqnarray}
\gamma_0+\nu\geq\dim_H(S'(\nu))\geq\dim_H(\mu)
\end{eqnarray}
(33) follows from (35) and (37).
\begin{rem}
{\rm
I.Guarneri informed us that he has obtained independently results similar to
theorem 3.1 and 3.2 \cite{Guarneri2}.
\noindent
One can also relate $\gamma_0$ to the correlation dimension introduced in
chapter 2 (Notice that one already has from proposition 2.1 that
$D_2^-\leq\gamma_0$). Define:
\begin{eqnarray}
D_2^-(\psi_0)\equiv\sup\{\,\liminf_{\eps\rightarrow 0}\frac{\log
\int_A\, \mu_{\psi_0}(B_\eps(x))\, d\mu_{\psi_0}(x)}{\log \eps}
\;\; ;\;A\in{\cal B}(\R)\;and\;\mu_{\psi_0}(A)>0\;\}
\end{eqnarray}
$(Here\; {\cal B}(\R) \; is\; the\; Borel\; algebra\; of\; \R)$
\noindent
Then:
\begin{eqnarray}
D_2^-\leq\nmuef\gamma^-(x)\leq D_2^-(\psi_0)=\gamma_0
\end{eqnarray}
$\gamma_0$ defined as above.
}
\end{rem}
\begin{cor}
If $H$ is self-adjoint on $l^2(\Z^d)$, then $\forall\psi_0\in{\cal D}(H),
\;\forall
m>0$ and $\forall \nu>0,\; \exists\, \tilde{C}(m,d,\nu,\psi_0)
<\infty\; s.t.$
\begin{eqnarray}
\forall T>1,\;\;\;\la\la |X|^m \ra\ra_T >\tilde{C}\, T^{\frac{m}
{d}(\dim_H(\mu_{\psi_0})-\nu)}
\end{eqnarray}
\end{cor}
\begin{rem}{\rm The result (40) also holds in the continuous case when
$H=-\Delta+V$ on $L^2(\R^d)$
if $V^+\in K_d^{loc}$ and $V_-\in K_d$ (\cite{Last}, \cite{Combes}).}
\end{rem}
\begin{cor}
Consider a self-adjoint operator $H=-\Delta +V$ on $l^2(\Z^d)$, with $V$
bounded. Let $\psi\in {\cal D}(H)$. If the information dimension
$\gamma_0=\dim_H(\mu_\psi)$ verifies
$\gamma_0>0$ then
for $\eps>0$, $\eps<\gamma_0$ and $\nu>0$, there exists $C_1(\nu)>0$,
$C_2(\eps\, ,\nu)>0$,
$C_3(\nu)>0$ and
$T(\eps \, ,\nu)$ such that with the notations of corollary 2.1, $\forall
T>T(\eps\, , \nu)$:
\begin{eqnarray}
\frac{1}{T}\int_0^T\,\sum_{C_2\, t^{\frac{\gamma_0-\eps}{d}}<|n|0$, $\exists
C_1(m)$, $a(m,\nu)>1$ and $t_0(m)<\infty$ such that $\forall t>t_0$:
\begin{eqnarray}
\sum_{|n|>C_1\,t}\,|n|^m\,|\la\psi_t\, ,\delta_n
\ra |^2 0$:
\begin{eqnarray}
\frac{1}{T}\,\int_0^T\,\sum_{|n|0$, one obtains the result (41) for $m$ small enough.
\begin{rem}
{\rm
For $H$ self-adjoint on ${\cal H}$, and some $\psi\in{\cal D}(H)$, define the
minimal growth rate of
$|X|^m$ for fixed $m>0$ as:
\begin{eqnarray}
\beta_m =\liminf_{T\rightarrow\infty} \frac{\log \la\la |X|^m\ra\ra_T}{\log T}
\end{eqnarray}
then relation (42) tells
us that for ${\cal H}=l^2(\Z^d)$, $\beta_m\leq m$. Moreover, it is easy to see
(proposition C.2) that
$\frac{\beta_m}{m}$ is increasing with $m$. Thus $\frac{\beta_m}{m}$ admit a
limit value
when $m$ tends to infinity. We will see later (theorem 4.1) that this value is
in some sense stable under
rank one perturbation.
}
\end{rem}
\noindent
{\bf Applications of corollary 3.2:}
The Fibonacci chain is a model for a quasicrystal in dimension 1. This model is
defined by the discrete Schr\"odinger operator on $l^2(\Z)$:
\begin{eqnarray}
(H\psi)(n)\equiv\psi(n+1)+\psi(n-1)+(V\psi)(n)
\end{eqnarray}
where $(V\psi)(n)=V_n\psi(n)$ and $(V_n)_{n\in\Z}$ takes values $+V$ or $-V$ at
site n,
arranged in a Fibonacci sequence. This model has extensively been studied, and
it has been proven that it's spectrum is singular continuous and is a Cantor set
of Lebesgue measure zero
\cite{Bellissard}, \cite{Suto}. Numerical computations \cite{Zhong} show that
for $\psi_0(n)=\delta_{n,0}$,
$C(T)$ behaves
like $c_0T^\beta$ for large T, where $c_0$ is a constant and $0<\beta<1$,
with $\beta$ approaching 1 when the modulation strengh $V$ tends to zero.
In particular, $\forall\nu>0,\;\exists V$ small enough
such that the spectral measure $\mu_{\psi_0}$ has a lower correlation dimension
$D_2^-\geq 1-\nu$ (see theorem 2.2); this gives a quasi-ballistic motion
using corollary 3.2,
since $\dim_H(\mu_{\psi_0})\geq D_2^-$.
\smallskip
Another example is the Harper model described by:
\begin{eqnarray}
(H\psi)(n)\equiv\psi(n+1)+\psi(n-1)+\lambda\cos (2\pi n\sigma)\psi(n)
\end{eqnarray}
on $l^2(\Z^d)$, $\ds\sigma=\frac{\sqrt{5}-1}{2}$ is the golden mean, and
$\lambda$
is a parameter. This model describes two dimensional electrons in a uniform
magnetic field with periodic potential. The spectrum of this operator is a
singular continuous Cantor set which has Lebesgue measure zero only if
$\lambda=2$. If $\lambda <2$, the spectrum is absolutely continuous, and
numerical computations obtained in \cite{Zhong}
gives $C(T)\sim (c_1\log T)T^{-1}$ so that $D_2^-=\dim_H(\mu_{\psi_0})=1$,
($\psi_0$ as above). By corollary 3.2
the motion is ballistic (which is an expected result for an absolutely
continuous spectrum with extended states).
\medskip
In these two examples, the behaviour of $C(T)$ is closely linked with the one
of
$\la\la |X|^2\ra\ra_T$. This is due to the fact that in these examples $D_2^-$
is close to $\dim_H(\mu_{\psi_0})$; obviously for arbitrary $H$ and $\psi_0$
this needs
not be so. As a purely theoretic example consider $\mu={\cal L}_{[1/2,1]}
+\frac{1}{2}\delta_0$ where
${\cal L}_{[1/2,1]}$ is the Lebesgue measure on $[1/2,1]$ and $\delta_0$ is
the Dirac
measure. In this example, the best upper bound for the correlation
function is given by $\ds \limsup\frac{\log C(T)}{\log T}=0$,
whereas $\la\la |X|^2\ra\ra_T\geq T^{2-\eps}$, $\forall \eps>0$.
In general, $D_2^-$ picks the "less continuous" part of the measure
$\mu_{\psi_0}$ whereas $\dim_H(\mu_{\psi_0})$ does the opposite (In the last
example, one has
$D_2^-=0$ whereas $\dim_H(\mu_{\psi_0})=1)$.
\smallskip
Unfortunately, $\dim_H(\mu_{\psi_0})$ is not yet the best exponent to describe
moments. On the
one hand, in
\cite{DelRio Simon}, there is an example of a Schr\"odinger
operator with pure point spectrum for which the maximal growth rate of
$\la\la|X|^2\ra\ra_T$ is close to ballistic, that is $\forall \nu>0$, $\exists
(T_n)_{n\in\N}$ , $T_n\rightarrow
\infty$, such that $\la\la |X|^\ra\ra_{T_n}\geq T_n^{2-\nu}$.
Such a spectrum implies $D_2^-(\psi_0)=0$, and the corollary 3.2 and (1)
would give only
$\la\la |X|^2\ra\ra_T\geq\; constant$. This shows that the lower bounds
obtained are strictly one sided.
Furthermore, if one recalls the example
of \cite{Simon Wolff}: Let $\mu$ be the conventional
Cantor measure. For the associate operator $A$ and vector $\psi_0$:
$\dim_H(\mu_{\psi_0})
\geq D_2^-
=\frac{\log 2}{\log 3}$,
and the perturbed measure $\mu_\lambda$ associated
to the perturbed operator $A_\lambda=A+\lambda
\la\psi_0,\, .\ra\psi_0$ is pure point and then
local exponents for $A_\lambda$ are $\mu$-almost everywhere zero.
This example proves the instability of $\dim_H(\mu_{\psi_0})$.
On the other hand, one expects that the lower
bound of $\la\la |X|^2\ra\ra_T$ does not move too much
under small perturbation. It then
appears trough this simple example that the lower bound
obtained may be not sharp. Our
purpose in the next chapter is to illustrate this point of view by a simple
perturbative result.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%% CHAPTER IV %%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%% STABILITY THEOREM %%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ Stability Theorem}
We wish here to exhibit a stability theorem under rank one perturbation $P$
for the moments $~\ml |X|^m\mr_T~$.
One then obtains a
relation between the dynamical quantities$~\ml |X|^m\mr_T~$ respectively
associated to the operators $~H_0~$ and $~H_\lambda~$ where
$\ds H_\lambda=H_0+\lambda P \, ,\;\lambda\in \R $.
Then, using
a rank one perturbation's theory \cite{Simon Wolff}, one can
show that in a one dimensional case, for the perturbed Fibonacci hamiltonian,
the dynamical quantity
$~\ml |X|^2\mr_T~$ behaves like $~\ml |X|^2\mr_T\ge C\, T^{1-\eps}~$,
$\eps> 0~$ small, although the spectral measure is pure point.
Let $~H_0~$ be a self-adjoint operator on $L^2(\R^d)$ or $l^2(\Z^d)$. For
$~\psi\in L^2( R^d)~$ (resp. in $l^2(\Z^d))$ let $~P=\la\; ,\, \psi\ra\psi~$ be
a rank one self-adjoint projection onto a vector $~\psi~$ and
$~H_{\lambda}=H_0+\lambda P~~,\lambda\in R~$ a family
of self-adjoint operator with the same domain $~{\cal D}(H_0)~$.
We denote by$~\psi^0_t~$ and $~\psi^\lambda_t~$ the respective
solutions of the Schr\"odinger equation associated to $~H_0~$ and $~H_\lambda~$
with the same initial conditions $~\psi^0_0=\psi~$,$~\psi^\lambda_0=\psi~$,
and by
$~\ml |X|^m_0\mr_T~,~\ml |X|^m_\lambda
\mr_T~$, $m>0$, the corresponding moments:
$$
\ml |X|^m_0(t)\mr_T=\frac{1}{T}\int^T_0{
\langle\psi^0_t,|X|^m\psi^0_t\rangle}dt
$$
$$
\ml |X|^m_\lambda\mr _T=\frac{1}{T}\int^T_0{
\langle\psi^\lambda_t,|X|^m\psi^\lambda_t\rangle}dt
$$
One can prove:
\begin{thm}
$\forall\lambda$, $\forall\nu>0$, there exists $T_0(\nu)$ and $~C(\lambda,\nu)~$
such that:
$~\forall m>0\, ,\;\forall~ T>T_0$:
\be
\ml |X|^m_\lambda
\mr_T\ge C(\lambda,\nu)\, T^{D_2^--2-\nu}\ml |X|^m_0\mr_T
\ee
where $D_2^-$ is the lower correlation dimension of $\mu_{\psi_0}$ associated
to $\psi$ and $H_0$.
\end{thm}
{\bf Proof} By Duhamel's formula , one can write
the solution $~\psi^0_t~$ as :
\be
\psi^0_t=\psi^\lambda_t+i\lambda\int^t_0{\psi^\lambda_{t-s}
\langle\psi^0_s,\psi\rangle}ds
\ee
Thus , multiplying equation (51) by $~\mid X\mid^{m/2}~$ and taking the
norm one has:
$$
\|\mid X\mid^{m/2}\psi^0_t\|^2\leq 2\Big\{\|\mid X\mid^{m/2}\psi^
\lambda_t\|^2+\mid\lambda\mid^2\Big\{\int^t_0{\mid\langle
\psi^0_s,\psi\rangle\mid\|\mid X\mid^{m/2}\psi^\lambda_{t-s}\|}ds\Big\}^2
\Big\}
$$
The Cauchy-Schwarz inequality gives
\be
\|\mid X\mid^{m/2}\psi^0_t\|^2\leq 2\Big\{\|\mid X\mid^{m/2}\psi^
\lambda_t\|^2+\mid\lambda\mid^2\int^t_0{\mid\langle
\psi^0_s,\psi\rangle\mid^2}ds\int^t_0{\|\mid X
\mid^{m/2}\psi^\lambda_{t-s}\|^2}ds
\Big\}
\ee
which implies that
\be
\ml |X|^m_0\mr_T\leq 2\Big\{\ml |X|^m_\lambda\mr_T
+\frac{\mid\lambda\mid^2}{T}\int^T_0{\Big(\int^T_0{\mid\langle
\psi^0_s,\psi\rangle\mid^2}ds\int^T_0{\|\mid X
\mid^{m/2}\psi^\lambda_{t-s}\|^2}ds}\Big)dt
\ee
Recalling that
$$
\int^T_0{\mid\langle
\psi^0_s,\psi\rangle\mid^2}ds=TC(T)
$$
and $\forall\, T>T_0$, $~C(T)\leq c\, T^{-D^-_2+\nu}~$ for some
constants $c$ and $T_0$ depending
only on $\nu$, as proved in
theorem 2.2, one can write the inequality (53)
as:
$$
\ml |X|^m_0\mr_T\leq 2\Big\{\ml |X|^m_\lambda\mr_T
+c\mid\lambda\mid^2 T^{2-D^-_2+\nu}\ml |X|^m_\lambda\mr_T\Big\}
$$
which leads to write, for some constant $C(\lambda,\nu)>0$ and $\forall\,
T>T_0$:
\be
\ml |X|^m_\lambda\mr_T\ge C(\lambda,\nu)\, T^{D^-_2-2-\nu}
\ml |X|^m_0\mr_T
\ee
and proves the statement$~\hspace{9cm}\Box~$
\newline
A direct consequence of this theorem is the following corollary:
\begin{cor}
If $d=1$, for all $\nu>0$, there exists $~C(\lambda,\nu)>0~$ and $T_0>0
\;\;such\;that\;\forall T>T_0~$,
\be
\ml |X|^2_\lambda
\mr_T\ge C(\lambda ,\nu)T^{3{D_2^-}-2-\nu}
\ee
\end{cor}
{\bf Proof} \hspace{1mm}It suffices to apply the corollary 3.1
to $~H_0~$ and remark that the information dimension of $\mu_{\psi}$
is such that $~\dim_H(\mu_{\psi})\ge D_2^-.~$
\hspace{2cm}$~\Box~$
\begin{defn}
Let $H_0$ be a discrete hamiltonian on $l^2(\Z^d)$; given $C_1$, $C_2$ two
positive constants and $0<\rho<1$, define the operator $A_{t,m,\rho }$ as:
\begin{eqnarray}
A_{t,m,\rho}\equiv\sum_{C_2t^\rho\leq |n|\leq C_1t}\!\!
|n|^{m/2}|\delta_n\ra\la\delta_n|
\end{eqnarray}
\end{defn}
\begin{cor}
For any $\nu>0$ and $\lambda\in\R$, $\exists T(\nu)<\infty$ and
$C(\lambda,\nu ,\rho )$ such that $\forall T>T(\nu)$, one has:
\begin{eqnarray}
\frac{1}{T}\int_0^T\| A_{t,m,\rho }\psi^\lambda_t\|^2 dt\geq
C(\lambda,\nu,\rho )\, T^{D_2^--2-\nu}\frac{1}{T}\int_0^T
\|A_{t,m,\rho}\psi^0_t\|^2 dt
\end{eqnarray}
\end{cor}
{\bf Proof:} Same as for theorem 4.1.
\medskip
As a consequence of this corollary and of corollary 3.2 one can easily prove the
following stability result:
\begin{cor}
Let $H_0$ be a self-adjoint operator on $l^2(\Z^d)$ and $H_\lambda$ defined
as for theorem 4.1 , then
$\forall\eps>0$, $\eps<\gamma_0$ and $\nu>0$, $\exists C_1(\nu)>0$,
$C_2(\eps\, ,\nu)>0$, $C_3(\nu)>0$ and
$T(\eps \, ,\nu)$ such that $\forall T>T(\eps\, , \nu)$:
\begin{eqnarray}
\frac{1}{T}\int_0^T\,\sum_{C_2\, t^{\frac{\gamma_0-\eps}{d}}
<|n|0$ there
exists a small enough value of V such that
$~D_2^-\ge 1-\nu~$. Thus for the
perturbed hamiltonian $H_\lambda =H_0+\lambda\, P$ (P defined as above),
corollary 4.1 implies that
for fixed $\lambda>0$ and $\nu>0$, $\exists\, T_0(\nu)<\infty$ and
$C(\lambda\, ,\nu)$ such that:
\be
\forall T>T_0(\nu),\hspace{3mm}
\ml |X|^2_\lambda
\mr_T\ge C(\lambda,\nu)\, T^{1-\nu}
\ee
or equivalently
\begin{eqnarray*}
\liminf_{T\rightarrow\infty}\frac{\la\la |X|^2_\lambda\ra\ra_T}
{T^{1-2\nu}}=\infty
\end{eqnarray*}
One also knows that the spectrum of $H_0$ is a Cantor set ${\cal C}$ of
Lebesgue measure $0$ so that in the terminology of \cite{Simon Wolff}:
$$B(x)\equiv\int\,\frac{d\mu_\psi(y)}{(y-x)^2}<\infty\;\;\;\;\forall x\nin
{\cal C}$$ which implies that $d\mu_{\psi}^\lambda$ associated to $\psi$ and
$H_\lambda$ is pure point for Lebesgue almost every $\lambda$ (see \cite{Simon
Wolff}).
So $d\mu_\psi^\lambda$ has zero Hausdorff dimension but for $\nu<1$ the lower
bound
given by (26) for $m=2$ is strictly less than the one of (59). We refer to
\cite{Last} and
\cite{DelRio Simon} for even more striking examples of hamiltonians with
zero-Hausdorff
dimensional spectral measures having
\begin{eqnarray*}
\limsup_{T\rightarrow\infty}\frac{\la\la |X|^2\ra\ra_T}{T^2/\log T}=\infty
\end{eqnarray*}
\item[(ii)] Let $\beta_{m,0}$ and $\beta_{m,\lambda}$ be the lower growing
exponent defined by (47) for $H_0$ and $H_\lambda$ respectively. Then it
follows from (50) that
\begin{eqnarray*}
\beta_{m,\lambda}\geq\beta_{m,0}+D_2^--2
\end{eqnarray*}
and in particular
\begin{eqnarray*}
\lim_{m\rightarrow\infty}\frac{\beta_{m,\lambda}}{m}\geq
\lim_{m\rightarrow\infty}\frac{\beta_{m,0}}{m}
\end{eqnarray*}
Reversing the r\^oles of $H_0$ and $H_\lambda$ one obtains
\begin{eqnarray}
\lim_{m\rightarrow\infty}\frac{\beta_{m,\lambda}}{m}=
\lim_{m\rightarrow\infty}\frac{\beta_{m,0}}{m}
\end{eqnarray}
This is an interesting stability property although we don't know how
to caracterize these limits in term of suitable dimensions of spectral
measures. Notice
that in the previous example (59) gives $\ds\lim_{m\rightarrow\infty}
\frac{\beta_{m,\lambda}}{m}\geq1-\nu$ instead of the mere
$\beta_{m,\lambda}\geq 0$ given
by (26).
\end{description}
}
\end{rem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%% APPENDIX A %%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix
\section{Appendix}
In this chapter, we relate correlation dimensions $D_2^+$ and $D_2^-$ to mean
quadratic dimension and wavelet dimensions. First of all, let us define
mean quadratic dimensions already introduced in \cite{Lau} to generalize the
results of Strichartz (5).
\begin{defn}
\begin{eqnarray*}
W(\varepsilon)=\frac{1}{\varepsilon}\int_R\mu^2((\omega-\frac{\varepsilon}{2},
\omega+\frac{\varepsilon}{2}])\, d\omega
\end{eqnarray*}
Then
\begin{eqnarray}
w^+=\,\limsup_{\varepsilon\rightarrow 0}\frac{\log W(\varepsilon)}
{\log\varepsilon}\\ w^-=\,\liminf_{\varepsilon\rightarrow 0}\frac{\log
W(\varepsilon)}{\log\varepsilon}
\end{eqnarray}
are respectively the upper and lower mean quadratic dimensions.
\end{defn}
\begin{prop}
\begin{eqnarray}
D_2^+=w^+ \\
D_2^-=w^-
\end{eqnarray}
\end{prop}
{\bf Proof}
\begin{eqnarray}
{W(\varepsilon)}&=&{\frac{1}{\varepsilon}\int_R\mu^2((\omega-\frac{\varepsilon}
{2},\omega +
\frac{\varepsilon}{2} ])\, d\omega}\nonumber \\
&=&{\frac{1}{\varepsilon}\sum_{k\in\Z}\;
\int_{k\varepsilon}^{(k+1)\varepsilon}\,
d\omega\int_{(\omega-\frac{\varepsilon}{2},\omega+\frac{\varepsilon}{2}]}
d\mu(\omega')
\mu((\omega-\frac{\varepsilon}{2},\omega+\frac{\varepsilon}{2}]) }\nonumber\\
&=&{\frac{1}{\varepsilon}\sum_{k\in\Z}\int_0^
\varepsilon d\xi\int_{(\xi+k
\varepsilon-\varepsilon/2,\xi+k\varepsilon+\varepsilon/2]}\!\!\!\!\!\!\!\!\!\!\!\!\!\!
d\mu(\omega')\mu((\xi+k\varepsilon-\varepsilon/2,
\xi+k\varepsilon+\varepsilon/2])} \\
&{\leq}&{\frac{1}{\varepsilon}\int_0^\varepsilon d\xi
\sum_{k\in\Z}\int_{(\xi+k\varepsilon
-\varepsilon/2,\xi+k\varepsilon+\varepsilon/2]}d\mu(\omega')
\mu((\omega'-\varepsilon,\omega'+
\varepsilon])}\nonumber\\
&{\leq}&{\frac{1}{\varepsilon}\int_0^\varepsilon d\xi\,
\gamma_2(2\varepsilon)}\nonumber\\
&{\leq}&{\gamma_2(2\varepsilon)}
\end{eqnarray}
On the other hand, one has from (65):
\begin{eqnarray}
W(\eps) & {\geq} &{\frac{1}{\varepsilon}\int_0^\varepsilon d\xi\sum_{k\in\Z}
\int_{(\xi+k
\varepsilon-\varepsilon/4,\xi+k\varepsilon+\varepsilon/4]}d\mu(\omega')
\mu((\omega'-\varepsilon/4,\omega' +\varepsilon/4])}
\end{eqnarray}
and
\begin{eqnarray}
{W(\varepsilon)}&=&{\frac{1}{\varepsilon}\sum_{k\in\Z}
\int_{k\varepsilon-\varepsilon/2}^{(k+1)
\varepsilon-\varepsilon/2}\,d\omega\int_{(\omega-\varepsilon/2,
\omega+\varepsilon/2]}\, d\mu(\omega')
\mu((\omega-\varepsilon/2,\omega+\varepsilon/2])}\nonumber\\
&=&{\frac{1}{\varepsilon}\int_0^\varepsilon\sum_{k\in\Z}
\int_{(\xi+k\varepsilon -\varepsilon,\xi+k\varepsilon]}\,
d\mu(\omega')\,\mu((\xi+k\varepsilon-\varepsilon,\xi+k\varepsilon])}
\nonumber\\
&{\geq}&{\frac{1}{\varepsilon}\int_0^\varepsilon
\sum_{k\in\Z}\int_{(\xi+k
\varepsilon-3\varepsilon/4,\xi+k\varepsilon-\varepsilon/4]}\,d\mu(\omega')\,
\mu((\omega'-\varepsilon/4,
\omega'+\varepsilon/4])}
\end{eqnarray}
\smallskip
(67) and (68) imply that :
\begin{eqnarray}
W(\varepsilon)\geq\frac{1}{2}\gamma_2(\frac{\varepsilon}{2})
\end{eqnarray}
Finally, one obtains from (66) and (69):$$\frac{1}{2}
\gamma_2(\frac{\varepsilon}{2})\leq W(\varepsilon)\leq
\gamma_2(2\varepsilon)$$
which proves (63) and (64).
\medskip
Let us now recall some recent results of Holschneider \cite{H},
and give their connections with our results.
The main tool used here is the wavelet transform which is powerfull
in analyzing micro-local
properties of functions or measures. Let us first recall some definitions:
Let ${\cal S}_+(\R)$ be the subspace of Schwartz functions whose Fourier
transform
is supported
by the positive frequencies. For any positive Radon measure $\mu$ in
${\cal S}'_+(\R )$
and any g in
${\cal S}_+(\R )$
, we define the wavelet transform w.r.t. g in the Fourier space as:
\be
{{\cal W}_g\mu (b,a)=\frac{1}{2\pi}\int_R\,\overline{\hat g}
(a\omega)e^{ib\omega} {\hat \mu}(\omega)d\omega}
\ee
Where $\hat {} $ denotes the Fourier transform.
\begin{defn}
${ }^{}$\newline
Let
$\displaystyle{\Gamma(\eps,2)=min\{\int_\eps^1\,\frac{d\alpha}{\alpha}\int_R\,
db\mid {\cal W}_g\mu(b,\alpha)\mid ^2\:
,\:\int_0^\eps\frac{d\alpha}{\alpha}\int_R\, db\mid {\cal W}_g\mu(b,\alpha)\mid
^2\}}$
\noindent
and:
\begin{eqnarray}
\kappa^-=\,\liminf_{\eps\rightarrow 0}\frac{\log\Gamma(\eps,2)}{\log a} \\
\kappa^+=\,\limsup_{\eps\rightarrow 0}\frac{\log\Gamma(\eps,2)}{\log a}
\end{eqnarray}
The last two quantities are called lower and upper 2-wavelet dimensions.
\end{defn}
\begin{rem}
{\rm
$\kappa^+$ and $\kappa^-$ do not depend on the choice of the wavelet $g$,
provided
$g\neq 0$ (see\cite{H}).
The second integral in the definition of $\Gamma(\eps,2)$ is used when the
first converges when
$\eps\rightarrow 0$. This is the case only if $\mu$, as an element of ${\cal
S}'_+(\R )$ is in $L^2(\R )$.}
\end{rem}
%
%
\begin{thm}
With the notations of corollary 2.1,
let $\eta\in S'(\R),\;\eta\neq 0$ satisfying $s\star\eta\in S(\R)$ for all
$s\in S(\R)$.
Assume that $\eta \nin L^2(\R)$. Then $\kappa^-(2)\leq0$ and it follows that
\begin{eqnarray}
-\kappa^+(2)-1=\liminf_{T\rightarrow\infty}\frac{\log C(T)}{\log T}\leq\limsup
_{T\rightarrow\infty}\frac{\log C(T)}{\log T}=-\kappa^-(2)-1
\end{eqnarray}
\end{thm}
We want here to relate the quantities $\kappa^+$ and $\kappa^-$to the
correlation dimensions
$D_2^+$ and $D_2^-$ introduced
in definition 2.3. An immediate consequence of theorem A1 and our theorem 2.2
is that:
\begin{eqnarray*}
D_2^+ = \kappa^++1\;\;\; {\rm and}\;\;\; D_2^- = \kappa^-+1
\end{eqnarray*}
We give here another proof of
similar relations independently of $C(T)$
in the following proposition.
\begin{prop}
Let $\mu$ be a Borel probability measure on $\R$, such that it is not in
$\L^2(\R)$, then:
\begin{eqnarray}
D_2^+\leq\kappa^++1 \\
D_2^-=\kappa^-+1
\end{eqnarray}
\end{prop}
{\bf Proof:}
By parseval equation, one has
\begin{eqnarray*}
{\Gamma(\eps,2)}&=&{\int_\eps^\infty\frac{da}{a}\int_{\R} \, d\omega\mid
{\cal W}_g\mu (b,a)\mid ^2}\\
&=&{\int_\eps^\infty\frac{da}{a}\int_{\R} \, d\omega\mid\hat
g(a\omega)
\mid ^2\,\mid\hat\mu(\omega)\mid ^2}
\end{eqnarray*}
\smallskip
\noindent
We choose $g\in{\cal S}_+(\R )$ such that supp$\ds(\hat g)\subset[\delta,1]$ for
$\delta>0$, and
$\ds\int_{\R^+}\!\mid\hat g(a)\mid^2\frac{da}{a}\,=\,c_0>0$
\medskip
\noindent
For $\displaystyle{F(\eps)=\int_\eps^\infty\,\mid\hat g(a)\mid^2\frac{da}
{a}}$, we have
$\displaystyle{\Gamma_g(\eps,2)=\int_{\R^+}
\,d\omega\,F(\eps\omega)\mid\hat\mu(\omega)\mid^2}$
and
\begin{eqnarray}
{c_0 \, \int_0^{\frac{\delta}{\eps}}\, \mid\hat\mu (\omega)\mid ^2\, d\omega}&
{\leq} &{\Gamma_g(\eps,2)} \nonumber \\
{}&{}&{} \\
&{\leq }&{c_0\,\int_0^{\frac{1}{\varepsilon}}\, \mid\hat \mu (\omega)\mid ^2\,
d\omega}\nonumber
\end{eqnarray}
On the other hand, one has :
\begin{eqnarray}
{W(\varepsilon)}&=&{\frac{1}{\varepsilon}\int_{\R}\mu^2(\omega-\frac{\eps}{2},
\omega+
\frac{\varepsilon}{2})\, d\omega}\nonumber\\
&=&{\frac{1}{\eps}\int _{\R}\mid (\chi_{(-\frac{\eps}{2} ,
\frac{\eps} {2})}\star\mu)(\omega)\mid ^2 d\omega }\nonumber\\
&=&{\frac{1}{2\pi}\frac{\eps}{2}\int_0^\infty\frac{sin^2(\frac{\eps
\omega}{2})}{{(\frac{\eps\omega}{2})}^2}\mid\hat\mu(\omega)\mid ^2
d\omega}
\end{eqnarray}
>From (76) and (77), one easily has that for some $c_0 '>0$:
\begin{eqnarray}
{W(\eps)\geq c_0 '\, \eps\Gamma_g(\eps,2)\Rightarrow}
&{D_2^+\leq\kappa^++1}\\
{ }&{D_2^-\leq\kappa^-+1}
\end{eqnarray}
Moreover, one has:
\begin{eqnarray}
{\varepsilon\int_{R^+}\frac{sin^2(\frac{\eps\omega}{2})}
{{(\frac{\varepsilon\omega}{2})}^2}\mid
\hat\mu(\omega)\mid ^2 d\omega}
&{\leq}&{\eps\,\sum_n\int_0^{\frac{2\pi n}{\eps}}\,[(\frac{1}{2\pi})^n-(\frac{1}
{2\pi}) ^{n+1}]\mid\hat\mu(\omega)\mid ^2 d\omega}\nonumber\\
{\Rightarrow\;\; W(\varepsilon)}&{\leq}&{\frac{2\pi -1}{2\sqrt{2\pi}}
\varepsilon\sum_n\,
\frac{1}{2\pi ^{n-1}}\,\Gamma _g(
\frac{\eps}{2 \pi \delta n})}
\end{eqnarray}
By the definition of $\kappa^-$, we have for $\eps<\eps_0$
small enough :
$$\forall\nu\, ,0<\nu <\kappa^-+1\, ,\;\exists\;c(\nu)<\infty\;s.t.
\;\Gamma_g(\eps)
\leq c(\nu)\eps^{\kappa^--\nu}$$
So (80) implies that:
\begin{eqnarray*}
{W(\varepsilon)}&{\leq}&{c_1(\nu)\sum_n\,\Big( (2\pi n)^
{(\nu-\kappa^-}\!\!\frac{1}{(2\pi)^{n-1}}\Big)\,
\eps^{\kappa^--\nu+1}}\\
&{\leq}&{c_2(\nu)\, \eps^{\kappa^--\nu+1}}
\end{eqnarray*}
\be
\Rightarrow\;\;D_2^-\geq\kappa^-+1
\ee
The result follws from (78),(79) and (81).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%% APPENDIX B %%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Appendix}
${}^{ }$
Our goal here is essentially to present some results or examples with some
specific measures that permit to understand the link
between the different dimensions introduced in this paper.
Let us first introduce a somewhat surprizing example of measure for which
the largest $\alpha$
such that it is L.U.$\alpha$ is smaller than any local exponents $\gamma^-(x)$.
Consider the absolutely continuous measure $\mu$ defined on $(0,1]$ as follow:
\be
d\mu=\nu\, x^{\nu-1}dx\;\;\;for \; fixed\; \nu\in(0,\frac{1}{2})
\ee
Then $\mu\; a.e.x.:\;
\gamma^-(x)=\gamma^+(x)=1$, but the supremum
over all the $\alpha$ such that $\mu$ is L.U.$\alpha$ is less than $\nu$. That
is,
$\forall x_0\in (0,1],\, \exists\eps_0$ small enough and a
constant $\tilde{c}=\frac{\nu^2}{x_0^{1-\nu}}$
such that
$\forall\eps<\eps_0,\; \mu
(B_\eps(x_0))\leq \tilde{c}\,\eps$
, but one can not hope to find
$\eps_0$ and $\tilde{c}$ uniform in $x$.
Indeed, let $\alpha_0=\nu +\delta$ for
some $\delta>0$, and assume $\exists\, c>0$ and $\eps_0>0$ such that
\begin{eqnarray}
\forall\eps <\eps_0\, ,\;\mu\, a.e. x\, ,\;\;\mu(B_\eps(x))0$. Thus this example shows
that (13) cannot
in general be turned into an equality.
\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%% APPENDIX C %%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Appendix}
\begin{prop}
Let $H$ be defined on $l^2(\Z^d)$ as:
$$H\psi(n)=(-\Delta+V)\psi(n)=\sum_{|n-k|=1}\psi(k)\, +V(n)\psi(n)$$
Assume that $V$ is bounded; then $\forall\psi\in {\cal D}(H)$ with compact
support,
$\forall m\geq 0$, $\exists c_1>0$, $a>1$ and $t_0<+\infty,$ all depending on
$m$, $\| V\|_\infty$ and $\psi$ such that $\forall t>t_0$:
\begin{eqnarray}
\sum_{|n|>c_1 t}|n|^m|\la\psi_t\, ,\delta_n\ra|^22e(2 d+\| V\|_\infty)$. For $n\in\N$, $H^n\psi$ is supported on
$\{k\in\Z^d\, ,\; |k|\leq n\}$, one has:
\begin{eqnarray*}
\Big (\sum_{|k|>c_1t}|k|^m|\la\psi_t\, ,\;\delta_k\ra|^2\Big )^{\frac{1}{2}} &
\leq &
\|\sum_{|k|>c_1t}\;\sum_{n>|k|}\frac{|k|^{\frac{m}{2}}t^n}{n!} |\la H^n\psi
\, ,\;\delta_k\ra | \,\delta_k\| \\
& \leq & \sum_{n>c_1t}\frac{n^{\frac{m}{2}}t^n}{n!}\, (2 d+\| V\|_\infty)^n \\
& \leq & 2 \sqrt {2\pi}\sum_{n>c_1t}n^{\frac{m-1}{2}-n}(t(2 d+\| V\|_\infty)e)^n
\\
& \leq & 4 \sqrt {2\pi}(c_1t)^{\frac{m-1}{2}}e^{-c_1\ln (2)t}
\end{eqnarray*}
In the third inequality, we have used Stirling formula.
\begin{prop}
Let $H$ be any self-adjoint operator on ${\cal H}=l^2(\Z^d)$ (or $L^2(\R^d)$),
and $\psi\in{\cal D}(H)$,
then for $\beta_m$ defined as in {\rm (47)}, $\ds\big (\frac{\beta_m}{m}\big )
_{m>0}$ is an increasing
sequence. Moreover, if ${\cal H}=l^2(\Z^d)$, $\exists\beta_\infty\leq 1$ such
that:
\begin{eqnarray}
\lim_{m\rightarrow\infty}\frac{\beta_m}{m}=\beta_\infty
\end{eqnarray}
\end{prop}
{\bf Proof: }
Assume that $\forall t$, $\psi_t\in{\cal D}(|X|^m)$, then, for $00}$ is an increasing
sequence.
As a consequence of proposition C.1, if ${\cal H}=l^2(\Z^d)$, one has
$\beta_m\leq m$, which implies
the last statement.
\bigskip
\bigskip
\noindent
{\fifteen Acknowledgments}
\medskip
The authors want to thank I. Guarneri and S. Vaienti for helpfull
and enjoyable discussions about fractal dimensions. Two of us
(JMB and JMC) also want to thank the organizers of the workshop
"Singular Spectra" at the Erwin Schr\"odinger Institute of Vienna
where this work has been completed.
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