%
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% Quantum Dynamics and Decompositions of Singular Continuous Spectra %
% By: Y. Last %
% This is a (plain) TeX document %
% last revised by YL; July 5, 1995 %
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\def \spec {{\rm Spec\,}}
\def \irrh {H_{\beta, \lambda, \theta}}
\def \dimh {{\rm dim}_{\rm H}}
\def \hils{{\cal H}}
\def \cntl {\centerline}
\def\no{\noindent}
\def\today{\ifcase\month\or
January\or February\or March\or April\or May\or June\or
July\or August\or September\or October\or November\or December\fi
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\def\gap{\bigskip\no}
\def\ref#1#2#3#4#5#6{#1:\ #2.\ #3\ {\bf #4},\ #5 \ (#6) }
%References
\def\amr {1} % Amrein's book.
\def\astrs {2} % Avron + Simon: transient and recurrent spectrum.
\def \ids {3} % Avron and Simon, IDS
\def \avs {4} % Avron, van Mouche, and Simon
\def \cala {5} % Carmona and Lacroix
\def \cey {6} % Choi, Elliot, and Yui
\def\comb {7} % Combes
\def \cyc {8} % Cycon, Froese, Kirsch, and Simon
\def\djls {9} % del Rio, Jitomirskaya, Last, and Simon
\def\djlsp {10} % del Rio, Jitomirskaya, Last, and Simon - PRL
\def\delms {11} % del Rio, Makarov, and Simon
\def\fal {12} % Falconer's Fractal Geometry.
\def\gkpa {13} % Geisel, Ketzmerick, and Petschel, 1991
\def\gord {14} % Gordon
\def\guara {15} % Guarneri, 1989
\def\guarb {16} % Guarneri, 1993
\def\gumaa {17} % Guarneri and Mantica
\def\gumab {18} % Guarneri and Mantica
\def \nmt {19} % Hardy and Wright (h)
\def\hols {20} % Holschneider
\def \jisi {21} % Jitomirskaya and Simon
\def\gkpb {22} % Ketzmerick, Petschel, and Geisel, 1992
\def \yortp {23} % yortp
\def\sri {24} % Reed and Simon I
\def\srii {25} % Reed and Simon II
\def\sriii {26} % Reed and Simon III
\def\rodg {27} % Rogers's book.
\def\rta {28} % Rogers and Taylor, 1959
\def\rtb {29} % Rogers and Taylor, 1963
\def\strib {30} % Simon's trace ideals book
\def\sssg {31} % Simon's semi-group paper
\def \srv {32} % Simon's AP review
\def\sabm {33} % Simon: absence of ballistic motion.
\def\ssci {34} % Simon's SC spec. I
\def\sbtd {35} % Simon's borderline time decay
\def\strich {36} % Strichartz
\def \tha {37} % Thouless, 1983
\def\toda {38} % Toda's book
\def\wiau {39} % Wilkinson and Austin
\hfill July 5, 1995
\bigskip
\bigskip
\cntl{\bigrm Quantum Dynamics}
\cntl{\bigrm and Decompositions of Singular Continuous Spectra}
\gap
\gap
\cntl{Y. Last}
\cntl{Division of Physics, Mathematics, and Astronomy}
\cntl{California Institute of Technology}
\cntl{Pasadena, California 91125}
\bigskip
\vskip 2.3cm
\noindent
{\bf Abstract. } We study relations between quantum dynamics and spectral
properties, concentrating on spectral decompositions which arise from
decomposing measures with respect to dimensional Hausdorff measures.
\vfill\eject
\no
{\bf 1. Introduction}
\medskip\no
Let $\hils$ be a separable Hilbert space, $H:\hils\to\hils$ a self
adjoint operator, and $\psi\in\hils$ (with $\|\psi\|\;=\;1$).
The spectral measure $\mu_\psi$
of $\psi$ (and $H$)
is uniquely defined by [\sri]:
$$\langle\psi\,,\,f(H)\psi\rangle\;=\;\int_{\sigma (H)} f(x)\,d\mu_\psi(x)
\;, \eqno (1.1)$$
for any measurable (Borel) function $f$.
The time evolution of the state $\psi$, in the \break
Schr\"odinger picture
of quantum mechanics, is given by
$$\psi(t)\;=\;e^{-iHt}\psi\;. \eqno (1.2)$$
The relations between various properties of the spectral measure $\mu_\psi$
(with an emphasis on ``fractal'' properties) and the nature of the time
evolution have been the subject of several recent papers
[\comb,\gkpa,\guara--\gumab,\hols,\gkpb,\sabm,\strich,\wiau].
Our purpose in this paper is twofold: First, we use a theory, due to
Rogers and Taylor [\rta,\rtb],
of decomposing singular continuous measures with
respect to Hausdorff measures to introduce a corresponding family
of spectral decompositions of $\hils$. These decompositions, which always
fall within the singular continuous subspace, extend the usual
spectral decomposition into pure point, singular continuous, and absolutely
continuous subspaces, and reflect the possibly rich nature of singular
continuous spectra. Second, we show that the above decompositions provide
a natural framework for the discussion and
formulation of some recent results
which relate quantum dynamics to spectral properties. Moreover, the
decomposition theory allows us to extend some of these results, and,
in particular, we get a strengthened version of the Guarneri-Combes
theorem [\comb,\guara].
While most of our discussion of quantum
dynamics applies to any self-adjoint
Hamiltonian, and is therefore rather general; the primary example we have
in mind is that of a single electron in an external potential. That is,
$H = -\Delta + V$ on $\ell^2(Z^d)$ or $L^2(R^d)$. An initial state $\psi(0)
=\psi$, which is localized in space, will usually spread under the time
evolution (1.2). It is the {\it rate} of this spreading and its relations
with properties of the spectral measure $\mu_\psi$, which is of major
interest to us in this paper. Unlike propagation of classical point
particles, the time evolution of wave packets, which describe quantum
mechanical particles, is ,in general, a complex phenomena,
involving infinitely many degrees of freedom. Various quantities can be
introduced for characterizing this time evolution, and their behavior
need not be coherent. A very elementary quantity, in this context, is the
{\it survival probability},
which gives the probability of finding the particle at time $t$ in its
initial state $\psi$. It is given by:
$$|\langle\psi (0)\,,\,\psi (t)\rangle|^2\;=\;
|\langle\psi\,,\,e^{-iHt}\psi\rangle|^2\;=\;
\Big|\int_{\sigma (H)} e^{-ixt}\,d\mu_\psi(x)\Big|^2\;=\;
|{\hat\mu_\psi} (t)|^2 \;. \eqno (1.3)$$
As is seen from (1.3), the survival probability coincides with the squared
absolute value of the Fourier transform of the spectral measure $\mu_\psi$.
Another family of interesting quantities comes from looking at
the expectations for various operators. Namely,
$$\langle A\rangle\;\equiv\;\langle A\rangle(t)\;\equiv\;\langle\psi(t)\,,\,
A\psi(t)\rangle\;, \eqno (1.4)$$
where $A$ is a (usually self-adjoint) operator. Some special cases of
interest are compact $A$'s, such as finite dimensional projections,
and certain unbounded $A$'s, such as moments of the position operator in
$\ell^2(Z^d)$:
$$|X|^m\;\equiv\;\sum_{n\in Z^d} |n|^m \langle\delta_n\,,\,\cdot\rangle
\delta_n\;, \eqno (1.5)$$
where $\delta_n(k)=\delta_{nk}$ and $m > 0$.
Our discussion will mainly involve {\it time averaged} quantities,
whose behavior seems
to be naturally related to {\it continuity} properties of spectral
measures. For any function $f$ of time, we denote its Ces\`aro time average
by:
$$\langle f\rangle_T\;\equiv\;\langle f(t)\rangle_T\;\equiv\;
{1\over T}\int_0^T f(t)\,dt\;. \eqno (1.6)$$
We also remark that by ``measure,'' we always mean a positive
measure. We shall not consider any other kind of measures in this paper.
The rest of this paper is organized as follows: In Section 2, we give
a brief introduction to spectral decompositions, and
describe some central results which relate spectral decompositions to
dynamics. This includes some classical results, as well as some of the
recent results which motivated our study. In Section 3, we derive
some central results which characterize the dynamics for vectors with
uniformly H\"older continuous spectral measures. In Section 4,
we describe parts of the rich theory, due to to Rogers and Taylor
[\rta,\rtb], of decomposing Borel measures with respect to Hausdorff
measures. In particular, for each $\alpha\in [0,\,1]$, a finite Borel
measure $\mu$ can be decomposed as
$d\mu = d\mu_{\alpha s} + d\mu_{\alpha c}$, where $\mu_{\alpha s}$ is
singular with respect to the $\alpha$-dimensional Hausdorff measure
(namely, supported on a set with zero $\alpha$-dimensional Hausdorff
measure), and $\mu_{\alpha c}$ is continuous with respect to the
$\alpha$-dimensional Hausdorff measure (namely, it does not give weight
to sets with zero $\alpha$-dimensional Hausdorff measure). Measures
which are continuous with respect to Hausdorff measures must be limits
of uniformly H\"older continuous measures, and this link is of major
importance in this paper. In Section 5, we use the decomposition theory
of Section 4 to introduce corresponding decompositions of $\hils$ into
closed, invariant, mutually orthogonal subspaces. In Section 6, we
combine the decomposition theory with the results of Section 3 to
obtain a strengthened version of the Guarneri-Combes theorem, which
is a lower bound on the growth rate of
$\langle\langle |X|^m\rangle\rangle_T$. The main
point in this strengthening is that the growth rate of
$\langle\langle |X|^m\rangle\rangle_T$ is shown to be governed by the
``most continuous'' component of $\psi$,
and unaffected by the existence of
more singular components. In Section 7, we study an interesting example
of the Almost Mathieu operator at large coupling and certain irrational
frequencies which are extremely well approximated by rationals. The
spectrum in this case is long known [\ids] to be purely singular
continuous, and we
show that it is, in fact, purely zero-dimensional, in
the sense that all spectral measures are supported on sets of zero
Hausdorff dimension. Nevertheless, we show that the growth rate
of $\langle\langle |X|^2\rangle\rangle_T$ can be arbitrarily close
to ballistic (namely, to $T^2$), for some time scales. This indicates
that the Guarneri-Combes theorem is a strictly one-sided inequality,
as far as singularity or continuity with respect
to Hausdorff measures is concerned. In Section 8, we discuss ergodic
Schr\"odinger operators, and show that the spectra, arising from
the various spectral decompositions introduced earlier, are the same
for almost all the realizations of the potential. This generalizes
known results about the decomposition into pure point,
singular continuous, and absolutely continuous spectra. Finally,
in Section 9, we discuss some results beyond this paper and some
open problems.
\gap\gap\gap
{\bf 2. Spectral decompositions and quantum dynamics: A brief introduction}
\medskip\no
By Lebesgue's decomposition theorem, any finite Borel measure on $R$
can be decomposed into absolutely continuous, singular continuous, and
pure point parts. Namely,
$$d\mu\;=\;d\mu_{ac}+d\mu_{sc}+d\mu_{pp}\;. \eqno (2.1)$$
Absolutely continuous here, means with respect to Lebesgue measure,
such that $d\mu_{ac}(x)=f(x)\,dx$ for some measurable function $f$.
The pure point part, $d\mu_{pp}$, is a countable sum of atomic (Dirac)
measures. The singular continuous part, $d\mu_{sc}$, is supported on
some set of zero Lebesgue measure, and does not give weight to any
individual points ($\mu(\{x\})=0\,,\;\forall\; x\in R$).
In classical spectral theory, one uses this measure decomposition
to establish a corresponding decomposition of the Hilbert space [\sri]:
$$\hils\;=\;\hils_{ac}\oplus\hils_{sc}\oplus\hils_{pp}\;, \eqno (2.2)$$
where:
$$\eqalign{\hils_{ac}&\;=\;
\{\psi\,|\, d\mu_\psi\; \hbox{\rm is purely absolutely continuous}\}\cr
\hils_{sc}&\;=\;
\{\psi\,|\, d\mu_\psi\; \hbox{\rm is purely singular continuous}\}\cr
\hils_{pp}&\;=\;
\{\psi\,|\, d\mu_\psi\; \hbox{\rm is purely point}\}\;.\cr} \eqno (2.3)$$
$\hils_{ac}$, $\hils_{sc}$, and $\hils_{pp}$ are closed (in norm),
mutually orthogonal
subspaces, which are {\it invariant} under $H$. The corresponding spectra
$\sigma_{ac}$, $\sigma_{sc}$, and $\sigma_{pp}$ are defined as the spectra
of $H$ restricted to the corresponding subspaces.
$\sigma$, the spectrum of $H$, obeys
$$\sigma\;=\;\sigma_{ac}\cup\sigma_{sc}\cup\sigma_{pp}\;. \eqno (2.4)$$
The continuous subspace is $\hils_c\equiv\hils_{ac}\oplus\hils_{sc}$, and
the singular subspace is $\hils_s\equiv\hils_{sc}\oplus\hils_{pp}$.
The corresponding spectra obey $\sigma_c=\sigma_{ac}\cup\sigma_{sc}$,
$\sigma_s=\sigma_{sc}\cup\sigma_{pp}$.
Note that, while the various subspaces are mutually
orthogonal, the corresponding spectra need not be disjoint.
There is also another standard decomposition of the spectrum into
a discrete spectrum $\sigma_{disc}$, and an essential spectrum
$\sigma_{ess}$. $\sigma_{disc}$ is the union of all isolated eigenvalues
of finite multiplicity. $\sigma_{ess}$ contains the rest of the spectrum,
namely, the continuous spectrum, eigenvalues of infinite multiplicity,
and limit points of eigenvalues. One has
$\sigma_{disc}\cup\sigma_{ess}=\sigma$ and
$\sigma_{disc}\cap\sigma_{ess}=\emptyset$.
({\it Remark.} Some authors,
e.g. [\sri], use slightly different conventions, such as calling
our $\hils_{sc}$, $\hils_{sing}$, and defining $\sigma_{pp}$
as the set of eigenvalues. Our definition of $\sigma_{pp}$
coincides with the closure of the set of eigenvalues.)
In the words of Reed and Simon [\sriii]: ``Spectral analysis of an
operator $A$ concentrates on identifying the five sets
$\sigma_{ess}(A)$, $\sigma_{disc}(A)$, $\sigma_{ac}(A)$, $\sigma_{sc}(A)$,
$\sigma_{pp}(A)$.'' Nevertheless,
a further decomposition of the spectrum was suggested by Avron and Simon
[\astrs], who decomposed $\hils$ into a transient subspace $\hils_{tac}$
and a recurrent subspace $\hils_{rec}\equiv\hils_{tac}^\bot$. $\hils_{tac}$
is a subspace of $\hils_{ac}$, which, in some sense, extracts its smoothest
component. It is given by $\hils_{tac}=P_{ei}\hils_{ac}$,
where $P_{ei}$ is the spectral projection on the essential interior of
the essential support of the absolutely continuous part of the spectral
measure class of $H$. The spectra $\sigma_{tac}\cup\sigma_{rec}=\sigma$
are defined accordingly, and $\sigma_{tac}\subseteq\sigma_{ac}$. It should
be pointed out, however, that this decomposition behaves badly under taking
direct sums, and it is therefore somewhat less fundamental than the previous
decompositions we have discussed.
There are some classical results relating the dynamics to the above spectral
decompositions (where by ``classical'' we mean, roughly, that they are over
ten years old and have appeared in books). The
Riemann-Lebesgue lemma, dating back to 1903 (see [\srii]), states:
\gap
{\bf Theorem 2.1. }(Riemann-Lebesgue lemma)
{\it If $\mu$ is a finite absolutely continuous
measure, then its Fourier transform ${\hat\mu}(t)$ obeys
${\hat\mu}(t)\;\to\;0$ as $t\to\infty$.}
\gap
So, for any $\psi\in\hils_{ac}$ the survival probability vanishes as
$t\to\infty$. Wiener's theorem, dating back to 1935 (see [\sriii]),
states:
\gap
{\bf Theorem 2.2. }(Wiener's theorem)
$\displaystyle{\quad\lim_{T\to\infty}\langle|{\hat\mu}(t)|^2\rangle_T
\;=\;\sum_{x\in R}|\mu(\{x\})|^2}$ .
\gap
In particular, it implies:
\gap
{\bf Corollary 2.2.1. } {\it
$\displaystyle{\quad
\lim_{T\to\infty}\langle|{\hat\mu_\psi}(t)|^2\rangle_T
\;=\;0}$ if and only if $P_{pp}\psi=0$, where $P_{pp}$ is the orthogonal
projection on $\hils_{pp}$.}
\gap
It should be pointed out that Theorem 2.1 does not have an inverse, namely,
the Fourier transform of singular continuous measures (which by Theorem
2.2 vanishes on the average) may or may not tend to zero in the limit
$t\to\infty$. As shown by Amrein [\amr], the set $\hils_w$ defined
by
$$\hils_w\;=\;\{\psi\,|\, {\hat\mu_\psi}(t)\to 0\;\hbox{\rm as}
\;t\to\infty\} \eqno (2.4)$$
is a closed invariant subspace, obeying
$\hils_{ac}\subseteq\hils_w\subseteq\hils_c$. It is an open question
%(as far as we know)
whether or not $\hils_w$ can be obtained from a spectral
decomposition, namely, from the spectral projection on some definite
subset of $\sigma$.
A more modern result, from the late 70's, is the celebrated RAGE theorem,
named after Ruelle, Amrein, Georgescu, and Enss (see [\sriii]):
\gap
{\bf Theorem 2.3. } (RAGE theorem) {\it
$\displaystyle{\lim_{T\to\infty}\langle\langle A\rangle\rangle_T\;=\;0}$
for any compact operator $A$ if and only if $\mu_\psi$ is purely
continuous.}
\gap
Theorem 2.3 has the corollary:
\gap
{\bf Corollary 2.3.1. } {\it Let $\hils\;=\;\ell^2(Z^d)$.
If $P_c\psi\not= 0$, where $P_c$ is the
projection on $\hils_c$, then, for any positive $m$,
$\displaystyle{\lim_{T\to\infty}\langle\langle
|X|^m\rangle\rangle_T\;=\;\infty}$.
\gap
Proof. } Let $\psi_1=P_c\psi$, $\psi_2=(1-P_c)\psi$, and denote
$\psi_j(t)\equiv e^{-iHt}\psi_j$, for $j=1,\,2$. Since
$\psi_1$ and $\psi_2$ belong to mutually orthogonal (invariant)
subspaces, and since $e^{-iHt}$ is unitary,
$\psi_1(t)$ and $\psi_2(t)$ are orthogonal for any $t$,
$\|\psi_1(t)\|^2=\|\psi_1\|^2$, $\|\psi_2(t)\|^2=\|\psi_2\|^2$,
and $\|\psi_1\|^2+\|\psi_2\|^2=\|\psi\|^2=1$. Let $P_N$
denote the projection on a sphere of radius $N$, namely,
$$P_N\;\equiv\;\sum_{|n|\leq N}
\langle\delta_n\,,\,\cdot\rangle\delta_n\;. \eqno (2.5)$$
We have $\|P_N \psi_1(t)\|^2 =
\langle\psi_1(t),\,P_N\psi_1(t)\rangle$, and thus, by Theorem 2.3,
$\langle\|P_N \psi_1(t)\|^2\rangle_T\to 0$ as $T\to\infty$. Since
$\langle\|P_N\psi(t)\|^2\rangle_T\;\leq\;
\langle(\|P_N\psi_1(t)\|+\|P_N\psi_2(t)\|)^2\rangle_T$, this implies
$$\limsup_{T\to\infty}\langle\|P_N\psi(t)\|^2\rangle_T\;\leq\;
\limsup_{T\to\infty}\langle\|P_N\psi_2(t)\|^2\rangle_T\;\leq\;
\|\psi_2\|^2\;. \eqno (2.6)$$
Since also
$1=\langle\|\psi(t)\|^2\rangle_T = \langle\|P_N \psi(t)\|^2\rangle_T
+ \langle\|(1-P_N)\psi(t)\|^2\rangle_T$, we obtain
$$\liminf_{T\to\infty}\langle\|(1-P_N)\psi(t)\|^2\rangle_T
\;\geq\;1-\|\psi_2\|^2=\|\psi_1\|^2\;. \eqno (2.7)$$
This implies
$$\liminf_{T\to\infty}\langle\langle
|X|^m\rangle\rangle_T\;\geq\;\|\psi_1\|^2 N^m\;, \eqno (2.8)$$
and since N is arbitrary we obtain
$$\lim_{T\to\infty}\langle\langle
|X|^m\rangle\rangle_T\;=\;\infty \;.\quad\bigcirc \eqno (2.9)$$
A slightly different type of results characterize certain subspaces
as {\it closures} of sets of vectors which have certain dynamical
characteristics. A result going back at least to Kato (see [\sriii])
states:
\gap
{\bf Theorem 2.4. } $\quad\displaystyle{\hils_{ac}\;=\;
\overline{\{\psi\,|\,\hat\mu_\psi(t)\;\in\;L^2 \,\}}}$.
\gap
Similarly, Avron and Simon [\astrs] have shown that $\hils_{tac}\;=\;
\overline{\{\psi\,|\,\hat\mu_\psi(t)\;\in\;L^1 \,\}}$.
More recently, several additional results in this spirit have been
published.
Simon, in a 1990 paper [\sabm],
has shown that, for Schr\"odinger operators,
pure point spectrum implies the absence of ballistic motion, namely,
for $H = -\Delta + V$ on $\ell^2(Z^d)$ which has pure
point spectrum, and for any initially localized state $\psi$ ($\psi$
in the domain of $X$ suffices), one has
$\displaystyle{\lim_{t\to\infty}t^{-2}\langle|X|^2\rangle(t) = 0}$.
His result also extends to continuous Schr\"odinger operators on
$L^2(R^d)$ with some mild restrictions on $V$ and $\psi$.
Another set of results, which gives the main motivation to our
current paper, involves relations between power-law rates associated
to some dynamical quantities and what appears to be ``fractal'' (or
dimension related) spectral properties. In order to describe some
of these results, let us start with a definition:
\gap
{\bf Definition 2.1. } {\it Let $\mu$ be a Borel measure on R,
$\alpha\in [0,\,1]$, and let $|\cdot|$ denote Lebesgue measure.
\item{\rm (i)}
We say that $\mu$ is uniformly $\alpha$-H\"older continuous (denoted
{\rm U$\alpha$H}) if there exists a constant $C$ such that for every
interval $I$ with $|I|<1$, $\mu(I)0$ there exists
$\delta>0$ such that for every interval $I$ with $|I|<\delta$,
$\mu(I)<\epsilon |I|^\alpha$.
\gap}
Strichartz, in a 1990 paper [\strich], has proven the following:
\gap
{\bf Theorem 2.5. } {\it Let $\mu$ be a finite {\rm U$\alpha$H} measure,
and for each $f\in L^2(R,\,d\mu)$ denote
$$\widehat{f\mu}\;\equiv\; \widehat{f\mu} (t)\;\equiv\;
\int e^{-ixt}f(x)\,d\mu(x)\;,$$
then there exists a constant $C$, depending only on $\mu$,
such that for any $f\in L^2(R,\,d\mu)$ and $T>0$
$$\langle|\widehat{f\mu}|^2\rangle_T\;<\;C\|f\|^2 T^{-\alpha}\;,$$
where $\|f\|$ is the $L^2$ norm of $f$.
\gap
Remark. } Strichartz's result is actually more general,
involving $\sigma$-finite measures on $R^d$.
\gap
The special case $f=1$ in Theorem 2.5 yields:
\gap
{\bf Corollary 2.5.1. } {\it If $\mu$ is a finite {\rm U$\alpha$H} measure,
then there exists a constant $C$ such that, for any $T>0$,
$\langle|\hat\mu|^2\rangle_T\;<\;C T^{-\alpha}$.}
\gap
Guarneri, in a 1989 paper [\guara], used a weak version of Theorem 2.5
to show that, for any self-adjoint $H$ on $\ell^2(Z^d)$,
if $\mu_\psi$ is U$\alpha$H, then
$\langle\langle|X|^2\rangle\rangle_T > CT^{2\alpha/d}/\ln^2 T$,
where $C$ is a constant depending only on $\psi$. In 1993, Combes [\comb]
used Theorem 2.5, as proven by Strichartz, to slightly improve this
estimate and show:
\gap
{\bf Theorem 2.6. } (Guarneri-Combes theorem) {\it If $H$
is self-adjoint on $\ell^2(Z^d)$ and $\mu_\psi$ is {\rm U$\alpha$H},
then, for each $m>0$, there exists a constant $C_{\psi,m}$, depending
on $\psi$ and $m$, such that for every $T>0$
$$\langle\langle|X|^m\rangle\rangle_T > C_{\psi,m} T^{m\alpha/d}\;.$$
\gap
Remarks.}
\item{(i) } The formulations of Theorem 2.6 in the various papers by
Guarneri [\guara,\guarb] and by Combes [\comb] vary between considering
only the second moment $m=2$ and considering more general monotonely
increasing functions of $|X|$. Our formulation
in terms of any positive moment is a somewhat arbitrary compromise
between generality and transparency. The full nature of the underlying
estimates should become clear from our discussion in Sections 3 and 5.
\item{(ii) } Combes has also proven a version of Theorem 2.6 for
continuous Schr\"odinger operators on $L^2(R^d)$ with potentials
bounded from below. The requirement of $\psi$, in this case, is that
it lies in the domain of $e^H$ and that $\mu_{e^H\psi}$ is U$\alpha$H.
\gap
A somewhat different variant of Theorem 2.6 has been developed by Guarneri
in a 1993 paper [\guarb]. It removes the requirement for uniform
$\alpha$-H\"older continuity, and replaces it by a requirement
for a certain $\alpha$ scaling behavior, that only needs to hold for
a.e.\ $x$ with respect to $\mu_\psi$. Another related result was
introduced in a 1992 paper [\gkpb] by Ketzmerick, Petschel, and Geisel,
who gave a formal calculation showing that if
$\langle|{\hat\mu}(t)|^2\rangle_T \sim T^{-\alpha}$,
then $\alpha$ is the correlation dimension of $\mu$. A rigorous
version of their result has been obtained by Holschneider [\hols], who
has shown that the upper and lower power decay rates which can
be associated with $\langle|{\hat\mu}(t)|^2\rangle_T$ are the
same as the upper and lower correlation dimensions of $\mu$.
\gap\gap
\vfil\eject
\gap
{\bf 3. Quantum dynamics for U$\alpha$H measures}
\medskip\no
In this Section we summerize some central results which characterize
the dynamics for vectors with U$\alpha$H spectral measures. For the
reader's convenience we include complete proofs, and, in particular,
we give a proof of Theorem 2.5.
\gap
{\bf Theorem 3.1. } {\it Let $\mu$ be a finite Borel measure on $R$
and $\alpha\in [0,\,1]$.
\item{\rm (i) } If $\mu$ is {\rm U$\alpha$H},
then there exists a constant $C$ such that, for any $T>0$, \hfil\break
$\langle|\hat\mu|^2\rangle_T < C T^{-\alpha}$.
\item{\rm (ii) } If there exists a constant $C$ such that
$\langle|\hat\mu|^2\rangle_T < C T^{-\alpha}$ for any $T>0$, \hfil\break
then $\mu$ is {\rm U${\alpha\over 2}$H}.
\gap
Remark. } (i) coincides with Corollary 2.5.1 and is included here
for completeness.
\gap\gap
{\bf Lemma 3.1. } {\it If $\mu$ is a finite Borel measure on $R$, then
$$\langle|\hat\mu|^2\rangle_T\;\geq\;{1\over 2}\int d\mu(x)\,d\mu(y)\,
{{\sin^2 ((x-y)T/2)}\over{((x-y)T/2)^2}}\;.$$}
\gap
{\it Proof. } We have
$$\eqalign{\langle|\hat\mu|^2\rangle_T\;=&\;
{1\over T}\int_0^T dt\,\left|\int d\mu(x)\,e^{-ixt}\right|^2\cr
=&\;{1\over T}\int_0^T dt\, \int d\mu(x)\,d\mu(y)\,e^{-i(x-y)t}\cr
=&\;{1\over T}\int d\mu(x)\,d\mu(y)\, \int_0^T dt\, \cos ((x-y)t)\cr
=&\;\int d\mu(x)\,d\mu(y)\,{{\sin ((x-y)T)}\over{(x-y)T}}\;,\cr}
\eqno (3.1)$$
and similarly,
$$\eqalign{{1\over {T^2}}\int_0^T dt\,t\, |\hat\mu(t)|^2\;
=&\;{1\over {T^2}}\int d\mu(x)\,d\mu(y)\, \int_0^T dt\,t\,
\cos ((x-y)t)\cr
=&\;{1\over {T^2}}\int d\mu(x)\,d\mu(y)\,\left(
{{T\sin ((x-y)T)}\over{(x-y)}}+
{{\cos ((x-y)T)-1}\over{(x-y)^2}}\right)\cr
=&\;\int d\mu(x)\,d\mu(y)\,\left({{\sin ((x-y)T)}\over{(x-y)T}}-
{{2\sin^2 ((x-y)T/2)}\over{((x-y)T)^2}}\right)\cr
=&\;\int d\mu(x)\,d\mu(y)\,{{\sin ((x-y)T)}\over{(x-y)T}}\cr
&\qquad\qquad\quad\;-\;
{1\over 2}\int d\mu(x)\,d\mu(y)\,
{{\sin^2 ((x-y)T/2)}\over{((x-y)T/2)^2}}\;.\cr} \eqno (3.2)$$
Since the left-hand side of (3.2) is clearly positive, (3.1) and
(3.2) imply the lemma. $\quad\bigcirc$
\gap
{\it Proof of Theorem 3.1. } (i) will follow from proving Theorem 2.5
below. It remains to prove (ii). Suppose that $\mu$ is not
U${\alpha\over 2}$H, then there exists a sequence of intervals
$\{I_n\}_{n=1}^\infty$, such that $|I_n|\to 0$ as $n\to\infty$
and $\mu(I_n) > n|I_n|^{\alpha/2}$. Let $T_n = \pi/(2|I_n|)$, then
for every $x,y\in I_n$, we have $|(x-y)T_n|\leq |I_n|T_n\leq\pi/2$,
and thus
$${{\sin^2 ((x-y)T_n/2)}\over{((x-y)T_n/2)^2}}\;\geq\;
{{\sin^2 (\pi/4)}\over{(\pi/4)^2}}\;=\;{8\over {\pi^2}}\;.
\eqno (3.3)$$
By Lemma 3.1 we have
$$\langle|\hat\mu|^2\rangle_{T_n}\;\geq\;
{1\over 2}\int_{I_n\times I_n} d\mu(x)\,d\mu(y)\,
{{\sin^2 ((x-y)T/2)}\over{((x-y)T/2)^2}}\;, \eqno (3.4)$$
and thus (3.3) implies
$$\langle|\hat\mu|^2\rangle_{T_n}\;\geq\;
{{4(\mu(I_n))^2}\over {\pi^2}}\;\geq\;
{{4n^2}\over {\pi^2}}|I_n|^\alpha \;=\;
{{4n^2 \pi^\alpha}\over {\pi^2 2^\alpha}}T_n^{-\alpha}\;, \eqno (3.5)$$
which proves (ii). $\quad\bigcirc$
\gap
Theorem 3.1 is optimal, in the sense that U$\alpha$H does not
imply more than $\langle|\hat\mu|^2\rangle_T < C T^{-\alpha}$, which
in turn does not imply more than U${\alpha\over 2}$H. Examples of
U$\alpha$H measures for which
$\langle|\hat\mu|^2\rangle_T \sim T^{-\alpha}$ are provided by uniformly
distributed measures on ``nice,'' self similar, Cantor sets. In fact,
Strichartz [\strich] has shown that this must be the case for a large
class of such measures. An example of a measure
for which $\langle|\hat\mu|^2\rangle_T < C T^{-\alpha}$, and yet $\mu$
is no more than U${\alpha\over 2}$H, is given by the following:
\gap
{\bf Example 3.1. } {\it Consider $\beta\in (1/2,\,1)$, $d\mu(x)=
x^{-\beta}dx$ on $(0,\,1]$, and let $\alpha=2(1-\beta)$; then
$|\hat\mu(t)|^2 < Ct^{-\alpha}$ and $\mu$ is no more than
{\rm U${\alpha\over 2}$H}.
\gap
Proof. } We have
$$\hat\mu(t)\;=\;\int_0^1 e^{-ixt}x^{-\beta}\,dx\;
=\;t^{\beta -1}\int_0^1 e^{-ixt}(xt)^{-\beta}t\,dx\;
=\;t^{-\alpha/2}\int_0^t e^{-iu}u^{-\beta}\,du\;. \eqno (3.6)$$
Since $\int_0^\infty e^{-iu}u^{-\beta}\,du$ converges, we conclude
that $t^\alpha |\hat\mu(t)|^2 \to Const\;$ as $t\to\infty$.
Consider now $I_\epsilon = (0,\,\epsilon)$. We have
$$\mu(I_\epsilon)\;=\;\int_0^\epsilon x^{-\beta}\,dx\;=\;
{{\epsilon^{1-\beta}}\over {1-\beta}}\;=\;
{1\over {1-\beta}}|I_\epsilon|^{\alpha/2}\;, \eqno (3.7)$$
so $\mu$ is U${\alpha\over 2}$H, but does not have any stronger
uniform H\"older continuity. $\quad\bigcirc$
\gap
We now turn to proving Strichartz's theorem.
\gap
{\it Proof of Theorem 2.5. } Suppose that $\mu$ is U$\alpha$H, and
let $f\in L^2(R,\,d\mu)$, then
$$\eqalign{\langle|\widehat{f\mu}|^2\rangle_T\;
=&\;{1\over T}\int_0^T dt\,|\widehat{f\mu}(t)|^2\cr
\leq&\;{e\over T}\int_{-\infty}^\infty dt\,e^{-t^2/T^2}\,
|\widehat{f\mu}(t)|^2\cr
=&\;{e\over T}\int_{-\infty}^\infty dt\,e^{-t^2/T^2}\,
\int d\mu(x)\,d\mu(y)\,f(x)\,\overline{f(y)}\,e^{-i(x-y)t}\cr
=&\;{e\over T}\int d\mu(x)\,d\mu(y)\,f(x)\,\overline{f(y)}\,
\int_{-\infty}^\infty dt\,e^{-t^2/T^2-i(x-y)t}\cr
=&\;e\sqrt{\pi}\int d\mu(x)\,d\mu(y)\,f(x)\,\overline{f(y)}\,
e^{-(x-y)^2 T^2/4}\cr
\leq&\;e\sqrt{\pi}\int d\mu(x)\,d\mu(y)\,(|f(x)|e^{-(x-y)^2 T^2/2})
(|f(y)|e^{-(x-y)^2 T^2/2})\;.\cr} \eqno (3.8)$$
By the Cauchy-Schwartz inequality, (3.8) implies
$$\langle|\widehat{f\mu}|^2\rangle_T\;\leq\;
e\sqrt{\pi}\int d\mu(x)\,|f(x)|^2
\int d\mu(y)\,e^{-(x-y)^2 T^2/4}\;. \eqno (3.9)$$
Since $\mu$ is U$\alpha$H, there exists $C$ such that
$\mu(I) < C|I|^\alpha$ for $|I| < 1$. Without loss, we can
assume $T > 1$, and thus, for every $x$,
$$\int d\mu(y)\,e^{-(x-y)^2 T^2/4}\;=\;
\sum_{n=0}^\infty \int_{{n\over T}\leq |x-y|< {{n+1}\over T}}
d\mu(y)\,e^{-(x-y)^2 T^2/4}\;\leq\;
\sum_{n=0}^\infty 2CT^{-\alpha}e^{-n^2/4}\;. \eqno (3.10)$$
Let $C_1=2C\sum_{n=0}^\infty e^{-n^2/4}$, then we obtain from (3.9)
and (3.10)
$$\langle|\widehat{f\mu}|^2\rangle_T\;\leq\;
e\sqrt{\pi}\int d\mu(x)\,|f(x)|^2\,C_1 T^{-\alpha}\;=\;
e\sqrt{\pi} C_1 \|f\|^2 T^{-\alpha}\;. \quad\bigcirc$$
\gap
Our next theorem can be described as a RAGE-like Hilbert space
adaptation of Strichartz's theorem.
\gap
{\bf Theorem 3.2. } {\it If $\mu_\psi$ is {\rm U$\alpha$H}, then
there exists a constant $C_\psi$ such that for any
compact operator $A$, $p\in N$, and $T>0$:
$$\langle|\langle A\rangle|\rangle_T\;<\;
C_\psi^{1/p}\,\| A\|_p\, T^{-\alpha/p}\;,$$
where $\| A\|_p\equiv ({\rm Tr}(|A|^p))^{1/p}$ denotes the
$p$-th Schatten norm of $A$.
\gap
Remarks. }
\item{(i) } $\|\cdot\|_1$ is the trace norm; $\|\cdot\|_2$
is the Hilbert-Schmidt norm.
\item{(ii) } A compact operator $A$ may have $\| A\|_p=\infty$
for all $p$'s, in which case Theorem 3.2 is empty.
\gap
{\bf Lemma 3.2. } {\it If $\mu_\psi$ is {\rm U$\alpha$H}, then
there exists a constant $C_\psi$ such that for any $\varphi\in\hils$
with $\|\varphi\|\leq 1$:
$$\langle|\langle\varphi,\,\psi(t)\rangle|^2\rangle_T\;<\;
C_\psi T^{-\alpha}\;.$$
\gap
Proof. }
By the spectral theorem [\sri], $H$ restricted to the cyclic subspace
spanned by $\psi$ (and $H$) is unitarily equivalent to multiplication
by $x$ on $L^2(R,\,d\mu_\psi)$.
In particular, for each $\varphi\in\hils$
there exists $f_\varphi\in L^2(R,\,d\mu_\psi)$ with
$\|f_\varphi\|\leq\|\varphi\|$, such that
$$\langle\varphi,\,\psi(t)\rangle\;=\;
\langle\varphi,\,e^{-iHt}\psi \rangle\;=\;
\int e^{-ixt}f_\varphi(x)\,d\mu_\psi(x)\;\equiv\;
\widehat{f_\varphi\mu_\psi}\;. \eqno (3.11)$$
Thus, Theorem 2.5 implies the Lemma. $\quad\bigcirc$
\gap
{\it Proof of Theorem 3.2. } Since $A$ is compact, there exist
orthonormal bases $\{\psi_n\}_{n=1}^\infty$ and
$\{\varphi_n\}_{n=1}^\infty$ of $\hils$, and a monotonely decreasing
sequence $\{E_n\}_{n=1}^\infty$, $E_n\geq 0$, such that $A$ is
given by the (norm-convergent) sum [\strib]
$$A\;=\;\sum_{n=1}^\infty E_n
\langle\varphi_n,\,\cdot\,\rangle\psi_n\;. \eqno (3.12)$$
Moreover, $\|A\|_p=(\sum_{n=1}^\infty E_n^p)^{1/p}$. Thus, we have
$$\eqalign{\langle|\langle A\rangle|\rangle_T\;=&\;
\Biggl\langle\biggl|\sum_{n=1}^\infty E_n\langle\varphi_n,\,\psi(t)\rangle
\langle\psi(t),\,\psi_n\rangle\biggr|\Biggr\rangle_T\cr\leq&\;
\sum_{n=1}^\infty E_n\left\langle\left|\langle\varphi_n,\,\psi(t)\rangle
\langle\psi(t),\,\psi_n\rangle\right|\right\rangle_T\cr\leq&\;
\sum_{n=1}^\infty E_n
\left(\langle|\langle\varphi_n,\,\psi(t)\rangle|^2\rangle_T\right)^{1/2}
\left(\langle|\langle\psi(t),\,\psi_n\rangle|^2\rangle_T\right)^{1/2}\;.
\cr} \eqno (3.13)$$
Let $p,q\in N$ obey $1/p +1/q = 1$, then, by H\"older's inequality,
we obtain from (3.13)
$$\eqalign{\langle|\langle A\rangle|\rangle_T\;\leq&\;
\left(\sum_{n=1}^\infty E_n^p\right)^{1/p}\,
\left\lbrack\sum_{n=1}^\infty
\left(\langle|\langle\varphi_n,\,\psi(t)\rangle|^2\rangle_T\right)^{q/2}
\left(\langle|\langle\psi(t),\,\psi_n\rangle|^2\rangle_T\right)^{q/2}
\right\rbrack^{1/q}\cr\leq&\;
\|A\|_p\, \left\lbrack\left(\sum_{n=1}^\infty
\left(\langle|\langle\varphi_n,\,\psi(t)\rangle|^2\rangle_T\right)^q\right)
\left(\sum_{n=1}^\infty
\left(\langle|\langle\psi(t),\,\psi_n\rangle|^2\rangle_T\right)^q\right)
\right\rbrack^{1/2q}\;.\cr} \eqno (3.14)$$
By Lemma 3.2, we have for every $n$
$$\langle|\langle\varphi_n,\,\psi(t)\rangle|^2\rangle_T\;<\;
C_\psi T^{-\alpha}\;,\qquad
\langle|\langle\psi_n,\,\psi(t)\rangle|^2\rangle_T\;<\;
C_\psi T^{-\alpha}\;. \eqno (3.15)$$
Moreover, since the $\psi_n$'s and $\varphi_n$'s are orthonormal bases
and since $e^{-iHt}$ is unitary, we have:
$$\sum_{n=1}^\infty
\langle|\langle\varphi_n,\,\psi(t)\rangle|^2\rangle_T\;=\;
\sum_{n=1}^\infty
\langle|\langle\psi_n,\,\psi(t)\rangle|^2\rangle_T\;=\;
\|\psi\|^2\;=\;1\;. \eqno (3.16)$$
Thus,
$$\eqalign{\sum_{n=1}^\infty
\left(\langle|\langle\varphi_n,\,\psi(t)\rangle|^2\rangle_T\right)^q
\;<&\;(C_\psi T^{-\alpha})^{q-1}\;,\cr
\sum_{n=1}^\infty
\left(\langle|\langle\psi_n,\,\psi(t)\rangle|^2\rangle_T\right)^q
\;<&\;(C_\psi T^{-\alpha})^{q-1}\;,\cr} \eqno (3.17)$$
and from (3.14) we obtain
$$\eqalign{\langle|\langle A\rangle|\rangle_T\;<&\;
\|A\|_p \left\lbrack
(C_\psi T^{-\alpha})^{q-1}(C_\psi T^{-\alpha})^{q-1}\right\rbrack^{1/2q}
\;=\;\|A\|_p(C_\psi T^{-\alpha})^{(q-1)/q}\cr
\;=&\;\|A\|_p(C_\psi T^{-\alpha})^{1/p}\;.\quad\bigcirc\cr}
\eqno (3.18)$$
\gap\gap\gap
{\bf 4. Hausdorff measures and Rogers-Taylor decomposition theory}
\medskip\gap
Our purpose in this section is to describe the theory of decomposing
finite Borel measures with respect to Hausdorff measures and dimensions.
This theory has been extensively studied by Rogers and Taylor
[\rta,\rtb] in the late 50's and early 60's. For our relatively
restricted purposes, we shall rely only on results which can be found
in the last chapter of Rogers' book [\rodg], and derive whatever we
need beyond that. It should be pointed out, however, that all the results
we describe in this section, and much more, can be found in the original
Rogers-Taylor papers [\rta,\rtb].
We start by recalling some standard terminology:
\gap
{\bf Definition 4.1. } {\it Let $\mu$ be a Borel measure on $R$.
\item{\rm (i)} $\mu$ is called $\sigma$-finite if
$R=\bigcup_{j=1}^\infty S_j$ and $\mu(S_j)<\infty$ for each $j$.
Similarly, we say that $S\subseteq R$ has $\sigma$-finite $\mu$
measure, if $S=\bigcup_{j=1}^\infty S_j$ and $\mu(S_j)<\infty$
for each $j$.
\item{\rm (ii)} We say that $\mu$ is supported on $S$, $S\subseteq R$,
if $\mu(R\setminus S) = 0$.}
\gap
{\bf Definition 4.2. } {\it Let $S\subseteq R$. A countable collection
of intervals $\{b_\nu\}_{\nu=1}^\infty$ is called a $\delta -cover$ of
$S$ if $S\subset\bigcup_{\nu=1}^\infty b_\nu$ and $|b_\nu|<\delta$ for
all $\nu$'s.}
\gap
{\bf Definition 4.3. } {\it Let $\alpha\in [0,\,1]$.
For any subset $S\subseteq R$, the
$\alpha$-dimensional Hausdorff measure of $S$ is
$$h^\alpha(S)\;\equiv\;\lim_{\delta\to 0}\,\inf_{\delta -covers}\,
\sum_{\nu=1}^\infty |b_\nu|^\alpha\;.$$
\gap
Remarks. }
\item{(i) } It is well known (see, e.g., [\fal]), and not hard to
verify, that the above limit exists for any $S\subseteq R$
(possibly being $\infty$), and that $h^\alpha$ is an outer measure
on $R$. Moreover, $h^\alpha$ restricted to Borel subsets of $R$ is
a Borel measure. Below, we shall be mostly interested in Borel
subsets, and we shall relate to $h^\alpha$ as a Borel measure on $R$.
\item{(ii) } $h^1$ coincides with Lebesgue measure; $h^0$ is the
counting measure.
\item{(iii) } $h^\alpha$ can be defined also for $\alpha<0$ or
$\alpha>1$. For $\alpha<0$, however, $h^\alpha(S)=\infty$ for any
$S\not=\emptyset$; while for $\alpha>1$, $h^\alpha(R)=0$. Thus, there
is no real interest in such $h^\alpha$'s. For the purpose of our
discussion below (namely, for the strict correctness of some of our
statements), however, it is useful to think of $h^\alpha$ as being
defined also for $\alpha$ outside $[0,\,1]$.
\item{(iv) } Given any right-continuous monotonely increasing
function $h:[0,\infty)\to[0,\infty)$, which obeys $h(x)>0$ for
$x>0$, one can define the corresponding $h$-Hausdorff measure on $R$,
via
$$h^h(S)\;\equiv\;\lim_{\delta\to 0}\,\inf_{\delta -covers}\,
\sum_{\nu=1}^\infty h(|b_\nu|)\;.$$
Moreover, essentially all of the decomposition theory that we describe
below can be extended to such general Hausdorff measures and to suitable
families of such measures. We refer to Rogers [\rodg] and
Rogers-Taylor [\rta,\rtb] for such more general treatments.
\item{(v) } For $\alpha<1$, $h^\alpha$ is {\it not} $\sigma$-finite.
Moreover, it is not regular in the usual sense of being outer regular
by open sets and inner regular by compacts. $h^\alpha$ is, however,
$G_\delta$ outer regular and $F_\sigma$ inner regular in the following
sense [\rodg]: For any Borel set $S\subseteq R$ there exists a
$G_\delta$ set $S_1$ such that $S\subseteq S_1$ and $h^\alpha(S)=
h^\alpha(S_1)$. Moreover, if $h^\alpha(S)<\infty$, then there exists
an $F_\sigma$ set $S_2$ such that $S_2\subseteq S$ and $h^\alpha(S)=
h^\alpha(S_2)$.
\gap
Given any $\emptyset\not= S\subseteq R$, there exists a unique
$\alpha(S)\in [0,\,1]$ such that $h^\alpha(S)=0$ for any
$\alpha>\alpha(S)$, and $h^\alpha(S)=\infty$ for any
$\alpha<\alpha(S)$. $h^{\alpha(S)}(S)$ may be 0, finite, or
infinite. This unique $\alpha(S)$ is called the {\it Hausdorff
dimension} of $S$, and is denoted by $\dimh(S)$.
The above notions of Hausdorff measures and dimensions lead
to a rich collection of notions of continuity and singularity,
as given by the following definitions:
\gap
{\bf Definition 4.4. } {\it Let $\mu$ be a Borel measure on $R$
and $\alpha\in [0,\,1]$.
\item{\rm (i)} $\mu$ is called $\alpha$-continuous (denoted $\alpha c$)
if $\mu(S)=0$ for any set $S$ with $h^\alpha(S)=0$.
\item{\rm (ii)} $\mu$ is called strongly $\alpha$-continuous
(denoted $s\alpha c$) if $\mu(S)=0$
for any set $S$ which has $\sigma$-finite $h^\alpha$ measure.
\item{\rm (iii)} $\mu$ is called $\alpha$-singular (denoted $\alpha s$)
if it is supported on a set $S$ with $h^\alpha(S)=0$.
\item{\rm (iv)} $\mu$ is called almost $\alpha$-singular
(denoted $a\alpha s$) if it is supported on a set $S$ which
has $\sigma$-finite $h^\alpha$ measure.
\item{\rm (v)} $\mu$ is called absolutely continuous with respect
to $h^\alpha$ (denoted $\alpha ac$) if $d\mu=f(x)\,dh^\alpha$ for
some Borel function $f$.
\gap
Remark. } If $\mu$ is $\sigma$-finite, then it follows from the
Radon-Nikodym theorem that $\mu$ is absolutely continuous with
respect to $h^\alpha$ if and only if it is both $\alpha$-continuous
and almost $\alpha$-singular.
\gap
{\bf Definition 4.5. } {\it Let $\mu$ be a Borel measure on $R$
and $\alpha\in [0,\,1]$.
\item{\rm (i)} $\mu$ is called $\alpha$-dimension continuous
(denoted $\alpha dc$)
if $\mu(S)=0$ for any set $S$ with $\dimh(S)<\alpha$.
\item{\rm (ii)} $\mu$ is called strongly $\alpha$-dimension continuous
(denoted $s\alpha dc$) if $\mu(S)=0$
for any set $S$ with $\dimh(S)\leq\alpha$.
\item{\rm (iii)} $\mu$ is called $\alpha$-dimension singular
(denoted $\alpha ds$)
if it is supported on $S=\bigcup_{j=1}^\infty S_j$, where
$\dimh(S_j)<\alpha$ for each $j$.
\item{\rm (iv)} $\mu$ is called almost $\alpha$-dimension singular
(denoted $a\alpha ds$) if it is supported on a set $S$
with $\dimh(S)\leq\alpha$.
\item{\rm (v)} We say that $\mu$ has exact dimension $\alpha$
(denoted $ed\alpha$)
if $\mu$ is both $\alpha$-dimension continuous and almost
$\alpha$-dimension singular.
\gap}
Given a finite Borel measure $\mu$ and $\alpha\in [0,\,1]$, we define
the upper $\alpha$-derivative of $\mu$ by
$$D_\mu^\alpha(x)\;\equiv\;\limsup_{\epsilon\to 0}
{{\mu((x-\epsilon,\,x+\epsilon))}\over {(2\epsilon)^\alpha}}\;,
\eqno (4.1)$$
and denote
$$\eqalign{T_0\;\equiv&\;T_0(\alpha,\,\mu)\;
\equiv\;\bigl\{ x\;\big|\; D_\mu^\alpha(x)=0\, \bigr\}\cr
T_+\;\equiv&\;T_+(\alpha,\,\mu)\;
\equiv\;\bigl\{ x\;\big|\; 0< D_\mu^\alpha(x)<\infty \,\bigr\}\cr
T_\infty\;\equiv&\;T_\infty(\alpha,\,\mu)\;
\equiv\;\bigl\{ x\;\big|\; D_\mu^\alpha(x)=\infty \,\bigr\}\;.\cr}
\eqno (4.2)$$
\gap
{\bf Theorem 4.1. } (Rogers and Taylor [\rodg,\rta])
{\it If $\mu$ is a finite Borel measure on $R$,
then $T_0$,$T_+$, and $T_\infty$ are Borel sets, and
\item{\rm (i)} $h^\alpha(T_\infty) = 0$.
\item{\rm (ii)} $T_+$ has $\sigma$-finite $h^\alpha$ measure.
\item{\rm (iii)} $\mu(S\cap T_+)=0$ for any $S$ with $h^\alpha(S)=0$.
\item{\rm (iv)} $\mu(S\cap T_0)=0$ for any $S$ which has
$\sigma$-finite $h^\alpha$ measure.}
\gap
Since the sets $T_0$,$T_+$, and $T_\infty$ are disjoint and
$T_0\cup T_+\cup T_\infty = R$, one can decompose $\mu$ as
$d\mu=\chi_{T_\infty}d\mu + \chi_{T_+}d\mu + \chi_{T_0}d\mu$, where
$\chi_{(\cdot)}$ denotes characteristic function. Thus, Theorem 4.1
implies the following:
\gap
{\bf Corollary 4.1.1. } {\it
Let $\mu$ be a finite Borel measure on $R$, then
for every $\alpha\in [0,\,1]$, $\mu$ has a unique decomposition
$$d\mu\;=\;d\mu_{\alpha s} + d\mu_{\alpha ac} + d\mu_{s\alpha c}\;,$$
where $d\mu_{\alpha s}$ is $\alpha$-singular, $d\mu_{\alpha ac}$ is
absolutely continuous with respect to $h^\alpha$ (on a set of
$\sigma$-finite $h^\alpha$ measure), and $d\mu_{s\alpha c}$ is
strongly $\alpha$-continuous.
\gap
Remarks. }
\item{(i) } If $\alpha=0$, then
$d\mu_{\alpha s}=0$, and the decomposition
$d\mu=d\mu_{\alpha ac} + d\mu_{s\alpha c}$ coincides with the
decomposition of $\mu$ into a pure point part and a continuous part.
If $\alpha=1$, then $d\mu_{s\alpha c}=0$, and the decomposition
$d\mu=d\mu_{\alpha s} + d\mu_{\alpha ac}$ coincides with the
decomposition of $\mu$ into a singular part and an
absolutely continuous (with respect to Lebesgue measure) part.
\item{(ii) } $d\mu_{\alpha c}\equiv d\mu_{\alpha ac} + d\mu_{s\alpha c}$
is the $\alpha$-continuous part of $d\mu$;
$d\mu_{a\alpha s}\equiv d\mu_{\alpha s} + d\mu_{\alpha ac}$
is the almost $\alpha$-singular part of $d\mu$.
\gap
The decomposition in Corollary 4.1.1 is with respect to the Hausdorff
measure $h^\alpha$. One can obtain further decompositions by also
considering Hausdorff dimensions. For a given $\alpha\in [0,\,1]$, define
$$\eqalign{T_{d\infty}\;\equiv&\;T_{d\infty}(\alpha,\,\mu)\;
\equiv\; \bigcup_{n=1}^\infty T_\infty(\alpha-n^{-1},\,\mu)\;,\cr
T_{d\infty +}\;\equiv&\;T_{d\infty +}(\alpha,\,\mu)\;
\equiv\; T_\infty(\alpha,\,\mu)\,\setminus\,T_{d\infty}(\alpha,\,\mu)\;,\cr
T_{d0}\;\equiv&\;T_{d0}(\alpha,\,\mu)\;
\equiv\; \bigcup_{n=1}^\infty T_0(\alpha+n^{-1},\,\mu)\;,\cr
T_{d+0}\;\equiv&\;T_{d+0}(\alpha,\,\mu)\;
\equiv\; T_0(\alpha,\,\mu)\,\setminus\,T_{d0}(\alpha,\,\mu)\;,\cr
T_{d+}\;\equiv&\;T_{d+}(\alpha,\,\mu)\;
\equiv\; T_{d\infty +}(\alpha,\,\mu)\cup T_+(\alpha,\,\mu)\cup
T_{d+0}(\alpha,\,\mu)\;.\cr} \eqno (4.3)$$
\gap
{\bf Lemma 4.1. }
{\it If $\mu$ is a finite Borel measure on $R$,
then $T_{d\infty}$, $T_{d\infty +}$, $T_{d+0}$, $T_{d0}$,
and $T_{d+}$ are Borel sets, and
\item{\rm (i)} $\dimh(T_{d+0})\leq\alpha$.
\item{\rm (ii)} $\dimh(T_{d+})\leq\alpha$.
\item{\rm (iii)} $\mu(S\cap T_{d0})=0$ for any $S$
with $\dimh(S)\leq\alpha$.
\item{\rm (iv)} $\mu(S\cap T_{d\infty +})=0$ for any $S$
with $\dimh(S)<\alpha$.
\item{\rm (v)} $\mu(S\cap T_{d+})=0$ for any $S$
with $\dimh(S)<\alpha$.
\gap
Proof.} That the various $T$'s in (4.3) are Borel sets is clear
from their construction. Assertions (i)--(v) follow very easily
from Theorem 4.1. It is helpful to note that the dimension of a
countable union of sets is the supremum of their dimensions, and
that the sets $T_{d\infty +}$ and $T_{d+0}$ obey
$$\eqalign{T_{d\infty +}\;=&\;
T_\infty(\alpha,\,\mu)\cap\left(
\bigcap_{n=1}^\infty T_0(\alpha-n^{-1},\,\mu)\right)\;,\cr
T_{d+0}\;=&\;
T_0(\alpha,\,\mu)\cap\left(
\bigcap_{n=1}^\infty T_\infty(\alpha+n^{-1},\,\mu)\right)\;.
\quad\bigcirc\cr} \eqno (4.4)$$
\gap
The sets $T_{d0}$,$T_{d+}$, and $T_{d\infty}$ are easily seen to be
disjoint, and obey
$T_{d0}\cup T_{d+}\cup T_{d\infty} = R$. Thus, one can also
decompose $\mu$ as
$d\mu=\chi_{T_{d\infty}}d\mu + \chi_{T_{d+}}d\mu
+ \chi_{T_{d0}}d\mu$ to obtain the following:
\gap
{\bf Corollary 4.1.2. } {\it
Let $\mu$ be a finite Borel measure on $R$, then
for every $\alpha\in [0,\,1]$, $\mu$ has a unique decomposition
$$d\mu\;=\;d\mu_{\alpha ds} + d\mu_{ed\alpha} + d\mu_{s\alpha dc}\;,$$
where $d\mu_{\alpha ds}$ is $\alpha$-dimension singular,
$d\mu_{ed\alpha}$ has exact dimension $\alpha$, and
$d\mu_{s\alpha dc}$ is strongly $\alpha$-dimension continuous.
\gap
Remark. } $d\mu_{\alpha dc}\equiv d\mu_{ed\alpha} + d\mu_{s\alpha dc}$
is the $\alpha$-dimension continuous part of $d\mu$;
$d\mu_{a\alpha ds}\equiv d\mu_{\alpha ds} + d\mu_{ed\alpha}$
is the almost $\alpha$-dimension singular part of $d\mu$.
\gap
So, while Corollary 4.1.1 decomposes $\mu$ with respect to a Hausdorff
measure, Corollary 4.1.2 gives an analogous decomposition with respect
to the corresponding Hausdorff dimension. Note that at the edges
$\alpha=0$ and $\alpha=1$, where the decomposition of Corollary 4.1.1
coincides with the decompositions into pure point$+$continuous and
singular$+$absolutely continuous; the decomposition of Corollary 4.1.2
gives something quite different. For $\alpha=1$, we have
$d\mu\;=\;d\mu_{1ds} + d\mu_{ed1}$, where $\mu_{ed1}$ combines the
absolutely continuous part of $\mu$ along with some of its singular
part. Similarly, for $\alpha=0$ we have
$d\mu\;=\;d\mu_{ed0} + d\mu_{s0dc}$, where $\mu_{ed0}$ combines the
pure point part of $\mu$ along with some of its continuous part.
One can also combine the two different decompositions to obtain
a 5-part decomposition for each $\alpha$. Namely,
\gap
{\bf Corollary 4.1.3. } {\it
Let $\mu$ be a finite Borel measure on $R$, then
for every $\alpha\in [0,\,1]$, $\mu$ has a unique decomposition
$$d\mu\;=\;d\mu_{\alpha ds} + d\mu_{ed\alpha/\alpha s} +
d\mu_{\alpha ac} + d\mu_{ed\alpha/s\alpha c} + d\mu_{s\alpha dc}\;,$$
where $d\mu_{\alpha ds}$ is $\alpha$-dimension singular,
$d\mu_{ed\alpha/\alpha s}$ is $\alpha$-singular and has exact dimension
$\alpha$, $d\mu_{\alpha ac}$ is absolutely continuous with respect
to $h^\alpha$ (on a set of $\sigma$-finite $h^\alpha$ measure),
$d\mu_{ed\alpha/s\alpha c}$ is strongly $\alpha$-continuous
and has exact dimension $\alpha$, and
$d\mu_{s\alpha dc}$ is strongly $\alpha$-dimension continuous.
\gap
Remarks. }
\item{(i) } Note that the extremal parts of the decomposition in
Corollary 4.1.3 are the same as those of Corollary 4.1.2.
\item{(ii) } Corollary 4.1.3 can be obtained by employing Corollary
4.1.1 to decompose $d\mu_{ed\alpha}$. One can also use the
decompositions $d\mu = d\mu_{\alpha s} + d\mu_{\alpha c}$ and
$d\mu = d\mu_{a\alpha s} + d\mu_{s\alpha c}$ to decompose
$d\mu_{ed\alpha}$, such that
$d\mu_{ed\alpha} = d\mu_{ed\alpha/\alpha s} + d\mu_{ed\alpha/\alpha c}$
and
$d\mu_{ed\alpha} = d\mu_{ed\alpha/a\alpha s} + d\mu_{ed\alpha/s\alpha c}$.
\gap
Continuity properties with respect to $h^\alpha$ are related to the
H\"older continuity, discussed in previous sections (and defined in
Definition 2.1), by the following:
\gap
{\bf Theorem 4.2. } (Rogers and Taylor [\rodg,\rtb]) {\it
Let $\mu$ be a finite Borel measure on $R$ and let
$\alpha\in [0,\,1]$; then
\item{\rm (i)} $\mu$ is $\alpha$-continuous if and only
if for each $\epsilon>0$ there are mutually singular Borel measures
$\mu_1^\epsilon$, $\mu_2^\epsilon$, such that
$d\mu = d\mu_1^\epsilon + d\mu_2^\epsilon$, $\mu_1^\epsilon$ is
{\rm U$\alpha$H}, and $\mu_2^\epsilon(R)<\epsilon$.
\item{\rm (ii)} $\mu$ is strongly $\alpha$-continuous if and only
if for each $\epsilon>0$ there are mutually singular Borel measures
$\mu_1^\epsilon$, $\mu_2^\epsilon$, such that
$d\mu = d\mu_1^\epsilon + d\mu_2^\epsilon$, $\mu_1^\epsilon$ is
{\rm US$\alpha$H}, and $\mu_2^\epsilon(R)<\epsilon$.
\gap
Remark. } Theorem 4.2 says that $\alpha$-continuous measures are
precisely limits of U$\alpha$H measures in an appropriate topology.
Similarly, strongly $\alpha$-continuous measures are precisely
limits of US$\alpha$H measures in the same topology.
\gap
The richness of possible decompositions, described in Corollaries
4.1.1, 4.1.2, and 4.1.3 above, seems too great for most practical
purposes; and suggests that it may be useful to concentrate on some
partial set of decompositions. Theorem 4.2 suggests that the
decomposition $d\mu = d\mu_{\alpha s} + d\mu_{\alpha c}$ (which
decomposes $\mu$ into an $\alpha$-singular part and an
$\alpha$-continuous part) should be of particular interest in our
context. Indeed, we shall adopt it below as ``the canonical
decomposition within dimension $\alpha$,'' in the sense that, for
$\alpha\in (0,\,1)$, we shall concentrate most of our attention on
this particular decomposition. The situation is somewhat different
at the edges $\alpha=0$ and $\alpha=1$. For $\alpha=0$,
the $\alpha$-singular part always vanishes, while for $\alpha=1$,
$d\mu = d\mu_{\alpha s} + d\mu_{\alpha c}$ coincides with the
decomposition into a singular part and an absolutely continuous part.
Thus, at these edges, we shall also be very much interested in the
dimensional decompositions of Corollary 4.1.2. In particular, it seems
useful to have compact terminology for the dimensional continuity
and singularity properties at these edges. For this reason we define:
\gap
{\bf Definition 4.6. } {\it Let $\mu$ be a Borel measure on $R$.
\item{\rm (i)} $\mu$ is called zero-dimensional (denoted $zd$)
if it is supported on a set $S$ with \break
$\dimh(S)=0$.
\item{\rm (ii)} $\mu$ is called positive-dimensional
(denoted $pd$) if $\mu(S)=0$ for any set $S$ with \break
$\dimh(S)=0$.
\item{\rm (iii)} $\mu$ is called one-dimensional (denoted $od$)
if $\mu(S)=0$ for any set $S$ with $\dimh(S)<1$.
\item{\rm (iv)} $\mu$ is called sub-one-dimensional
(denoted $sod$) if it is supported on $S=\bigcup_{j=1}^\infty S_j$,
where $\dimh(S_j)<1$ for each $j$.
\gap
Remark. } The terminology of Definition 4.6 overlaps Definition 4.5,
such that a zero-dimensional measure is the same as a measure of
exact dimension 0, a positive-dimensional measure is the same as a
strongly 0-dimension continuous measure, etc. This should not lead,
however, to any real confusion.
\gap
In the language of Definition 4.6, Corollary 4.1.2 implies
\gap
{\bf Corollary 4.1.4. } {\it
Let $\mu$ be a finite Borel measure on $R$, then
$\mu$ has unique decompositions:
\item{\rm (i) } $d\mu = d\mu_{zd} + d\mu_{pd}$, where $d\mu_{zd}$
is zero-dimensional and $d\mu_{pd}$ is positive-dimensional.
\item{\rm (ii) } $d\mu = d\mu_{sod} + d\mu_{od}$, where $d\mu_{sod}$
is sub-one-dimensional and $d\mu_{od}$ is one-dimensional.}
\gap
\gap\gap\gap
{\bf 5. Decompositions of $\hils$}
\medskip\no
Given any $\alpha\in [0,\,1]$, we define:
$$\eqalign{\hils_{\alpha c}&\;\equiv\;
\{\psi\,|\, \mu_\psi\; \hbox{\rm is $\alpha$-continuous}\}\;,\cr
\hils_{\alpha s}&\;\equiv\;
\{\psi\,|\, \mu_\psi\; \hbox{\rm is $\alpha$-singular}\}\;.\cr}
\eqno (5.1)$$
\gap
{\bf Theorem 5.1. } {\it $\hils_{\alpha c}$ and $\hils_{\alpha s}$
are closed (in norm), mutually orthogonal subspaces, which are
invariant under $H$, and $\hils =
\hils_{\alpha s}\oplus\hils_{\alpha c}$.
\gap
Remarks. }
\item{(i) } We call $\hils_{\alpha s}$ the $\alpha$-singular subspace
and $\hils_{\alpha c}$ the $\alpha$-continuous subspace. The spectra
$\sigma_{\alpha s}$ and $\sigma_{\alpha c}$ are defined as the spectra
of $H$ restricted to the corresponding subspaces, and they are called
$\alpha$-singular spectrum and $\alpha$-continuous spectrum accordingly.
Clearly, $\sigma = \sigma_{\alpha s}\cup\sigma_{\alpha c}$.
\item{(ii) } For $\alpha=0$, $\hils_{\alpha s}=0$ and
$\hils_{\alpha c}=\hils$. For $\alpha>0$, $\hils_{\alpha c}$ is
a subspace of $\hils_c$, and $\hils_{pp}$ is a subspace of
$\hils_{\alpha s}$.
\gap
{\it Proof. } Let $\{\psi_n\}_{n=1}^\infty$ be an orthonormal basis
of $\hils$. Denote
$$T_\infty^\hils\;=\;
\bigcup_{n=1}^\infty T_\infty(\alpha,\,\mu_{\psi_n})\;, \eqno (5.2)$$
and let $P_{T_\infty^\hils}$ be the spectral projection on
$T_\infty^\hils$. One easily sees that
$\hils_{\alpha s}=P_{T_\infty^\hils}\hils$ and
$\hils_{\alpha c}=(1-P_{T_\infty^\hils})\hils$, so Theorem 5.1
follows from known facts about spectral projections [\sri].
$\quad\bigcirc$
\gap
We denote
$$\hils_{uh}(\alpha)\;\equiv\;
\{\psi\,|\, \mu_\psi\; \hbox{\rm is U$\alpha$H}\}\;. \eqno (5.3)$$
\gap
{\bf Theorem 5.2. } {\it
For every $\alpha\in [0,\,1]$, $\hils_{uh}(\alpha)$
is a vector space and
$$\overline{\hils_{uh}(\alpha)}\;=\;\hils_{\alpha c}\;,$$
where $\overline{\,\cdot\,}$ denotes norm closure in $\hils$.
\gap
Proof. } Let $\psi_1,\,\psi_2\in \hils_{uh}(\alpha)$ and let $\varphi=
a\psi_1+b\psi_2$. By assumption, there are constants $C_1,\,C_2$
such that $\mu_{\psi_1}(I)0$ be given.
By Theorem 4.2 we have
$d\mu_\varphi = d\mu_{\varphi,1}^\epsilon + d\mu_{\varphi,2}^\epsilon$
such that $\mu_{\varphi,1}^\epsilon$ is U$\alpha$H,
$\mu_{\varphi,2}^\epsilon(R)<\epsilon$, and
$\mu_{\varphi,1}^\epsilon,\,\mu_{\varphi,2}^\epsilon$ are mutually
singular. Let $S_{\varphi,\epsilon}\subset R$ be such that
$\mu_{\varphi,2}^\epsilon$ is supported on $S_{\varphi,\epsilon}$,
$\mu_{\varphi,1}^\epsilon(S_{\varphi,\epsilon})=0$, and let
$P_{S_{\varphi,\epsilon}}$ be the spectral projection on
$S_{\varphi,\epsilon}$. We have
$\varphi=P_{S_{\varphi,\epsilon}}\varphi +
(1-P_{S_{\varphi,\epsilon}})\varphi$,
$(1-P_{S_{\varphi,\epsilon}})\varphi\in\hils_{uh}(\alpha)$, and
$\|P_{S_{\varphi,\epsilon}}\varphi\|^2<\epsilon$. Thus, $\varphi$
is a norm limit of vectors in $\hils_{uh}(\alpha)$, and we obtain
$\hils_{\alpha c}\subseteq\overline{\hils_{uh}(\alpha)}$, which
completes the proof. $\quad\bigcirc$
Theorem 5.2 and the dynamical results of Section 3 about U$\alpha$H
measures are the main reasons that we consider the decomposition
$\hils=\hils_{\alpha s}\oplus\hils_{\alpha c}$ to be of special
interest. Nevertheless, it's easy to verify that each of the measure
decompositions discussed in Section 4 leads to a corresponding
decomposition of $\hils$. For each $\alpha\in [0,\,1]$, the subspaces
$\hils_{\alpha ac}$, $\hils_{s\alpha c}$, $\hils_{a\alpha s}$,
$\hils_{\alpha ds}$, $\hils_{ed\alpha}$, $\hils_{s\alpha dc}$,
$\hils_{a\alpha ds}$, $\hils_{\alpha dc}$, $\hils_{ed\alpha/\alpha s}$,
$\hils_{ed\alpha/\alpha c}$, $\hils_{ed\alpha/s\alpha c}$, and
$\hils_{ed\alpha/a\alpha s}$ are all well defined analogously to
(5.1), and they are all closed and invariant under $H$. They give rise
to a total of 14 different possible decompositions of $\hils$, such
as $\hils=\hils_{\alpha ds}\oplus\hils_{\alpha dc}$,
$\hils=\hils_{\alpha ds}\oplus\hils_{ed\alpha}\oplus\hils_{s\alpha dc}$,
$\hils=\hils_{\alpha s}\oplus\hils_{\alpha ac}
\oplus\hils_{ed\alpha/s\alpha c}\oplus\hils_{s\alpha dc}$, etc.
For each of these subspaces there is an associated spectrum, defined
as the spectrum of $H$ restricted to the corresponding subspace.
At the edges $\alpha=0$ and $\alpha=1$, we shall use the terminology
introduced in Definition 4.6 and the dimensional decomposition of
Corollary 4.1.4. We have $\hils=\hils_{zd}\oplus\hils_{pd}$, where
we call $\hils_{zd}$ the zero-dimensional subspace and $\hils_{pd}$
the positive-dimensional subspace. They are defined in the obvious
way, analogously to (5.1). The corresponding spectra obey
$\sigma=\sigma_{zd}\cup\sigma_{pd}$. Similarly, we have
$\hils=\hils_{sod}\oplus\hils_{od}$, where we call $\hils_{sod}$ the
sub-one-dimensional subspace and $\hils_{od}$ the one-dimensional
subspace. The corresponding spectra obey
$\sigma=\sigma_{sod}\cup\sigma_{od}$.
For $\alpha>0$, Theorem 5.2 along with Theorem 3.1 relate
$\hils_{\alpha c}$ to vectors with power-law decaying survival
probabilities, in the sense that $\hils_{\alpha c}$ must have a
dense subset of vectors for which
$\sup_{T}T^\alpha\langle|\hat\mu_\psi|^2\rangle_T < \infty$.
The relation is actually more extensive:
Given a finite measure $\mu$ and
$\alpha\in (0,\,1)$, consider the integral
$$J_\alpha(\mu)\;\equiv\;\int_0^\infty
|\hat\mu(t)|^2\,t^{\alpha-1}\,dt\;. \eqno (5.7)$$
An elementary calculation (also see, e.g., [\fal]) shows that
$$\eqalign{J_\alpha(\mu)\;=&\;
\int_0^\infty dt\,t^{\alpha-1}\,\int d\mu(x)\,d\mu(y)\,e^{-i(x-y)t}\cr
=&\;\int d\mu(x)\,d\mu(y)\,\int_0^\infty dt\,t^{\alpha-1}
\,e^{-i(x-y)t}\cr
=&\;\int d\mu(x)\,d\mu(y)\,\int_0^\infty dt\,t^{\alpha-1}
\,\cos(|x-y|t)\cr
=&\;\int {{d\mu(x)\,d\mu(y)}\over {|x-y|^\alpha}}\,
\int_0^\infty du\,u^{\alpha-1}\,\cos(u)\cr
=&\;C\int {{d\mu(x)\,d\mu(y)}\over {|x-y|^\alpha}}\;,\cr} \eqno (5.8)$$
where $C=\int_0^\infty du\,u^{\alpha-1}\,\cos(u)$.
While (5.8) is only a formal calculation, it can be easily justified
rigorously by considering the integration over $t$ from $0$ to $T$ and
then taking $T\to\infty$.
The following lemma is classical:
\gap
{\bf Lemma 5.1. } (See, e.g., Falconer [\fal]) {\it
If $J_\alpha(\mu)<\infty$, then $\mu$ is $\alpha$-continuous.
\gap
Proof. }
If $J_\alpha(\mu)<\infty$, then, by (5.8),
$\int d\mu(y)/|x-y|^\alpha < \infty$ for a.e.\ $x$ with respect to $\mu$.
Since, for every $x$ and $\epsilon >0$,
$$\mu((x-\epsilon,\,x+\epsilon))\;\leq\;\epsilon^\alpha
\int {{d\mu(y)}\over {|x-y|^\alpha}}\;, \eqno (5.9)$$
we have $D_\mu^\alpha(x)<\infty$ for a.e.\ $x$ with respect to $\mu$.
$\quad\bigcirc$
\gap
We also have:
\gap
{\bf Lemma 5.2. } {\it If
$\sup_{T}T^\alpha\langle|\hat\mu|^2\rangle_T < \infty$,
then, for any $0<\epsilon<\alpha$, $J_{\alpha-\epsilon}(\mu)<\infty$.
\gap
Remark. } Note that for every $T>0$,
$T^\alpha\langle|\hat\mu|^2\rangle_T \leq J_\alpha (\mu)$, and
so $J_\alpha (\mu)<\infty$ implies
$\sup_{T}T^\alpha\langle|\hat\mu|^2\rangle_T < \infty$.
\gap
{\it Proof. } Since $|\hat\mu(t)|^2\leq 1$, we clearly have
$$\int_0^1 |\hat\mu(t)|^2\,
t^{\alpha-\epsilon-1}\,dt\;<\;\infty\;. \eqno (5.10)$$
Since, by assumption, $\langle|\hat\mu|^2\rangle_T < CT^{-\alpha}$,
we also have
$$\eqalign{
\int_1^\infty |\hat\mu(t)|^2\,t^{\alpha-\epsilon-1}\,dt\;&=\;
\sum_{n=0}^\infty
\int_{2^n}^{2^{n+1}} |\hat\mu(t)|^2\,t^{\alpha-\epsilon-1}\,dt\cr
&\leq\;\sum_{n=0}^\infty (2^n)^{\alpha-\epsilon-1}
\int_0^{2^{n+1}} |\hat\mu(t)|^2\,dt\cr
&\leq\;\sum_{n=0}^\infty (2^n)^{\alpha-\epsilon-1}
C (2^{n+1})^{1-\alpha}\cr
&=\; C 2^{1-\alpha} \sum_{n=0}^\infty 2^{-\epsilon n}\;<\;\infty\;.
\quad\bigcirc\cr} \eqno (5.11)$$
\gap
By combining Lemmas 5.1 and 5.2 with Theorems 3.1 and 5.2, we obtain
the following (weak) analog of Theorem 2.4:
\gap
{\bf Theorem 5.3. } {\it
If $\alpha\in (0,\,1)$, then, for any
$0<\epsilon<\min\{\alpha,1-\alpha\}$,
$$\overline{\bigl\{\psi\,\big|\,\sup_{T}T^{\alpha+\epsilon}
\langle|\hat\mu_\psi|^2\rangle_T < \infty\,\bigr\}}\;\subseteq\;
\overline{\bigl\{\psi\,\big|\,J_\alpha(\mu_\psi)<\infty\,\bigr\}}
\;\subseteq\;\hils_{\alpha c}\;,$$
$$\hils_{\alpha c}\;\subseteq\;
\overline{\bigl\{\psi\,\big|\,\sup_{T}T^\alpha
\langle|\hat\mu_\psi|^2\rangle_T < \infty\,\bigr\}}\;\subseteq\;
\overline{\bigl\{\psi\,\big|\,J_{\alpha-\epsilon}(\mu_\psi)<\infty
\,\bigr\}}\;.$$
\gap
Remarks. }
\item{ (i)} Note that $J_\alpha(\mu_\psi)<\infty$ is the same as
$\hat\mu_\psi\in L^2(R,\,t^{\alpha-1}\,dt)$.
\item{ (ii)} Note that $\hat\mu \in L^2(R,\, dt)$ if and only if
$\sup_{T}T\langle|\hat\mu|^2\rangle_T < \infty$, and so Theorem 2.4
is equivalent to
$\hils_{ac}\;=\;\overline{\{\psi\,|\,\sup_{T}T
\langle|\hat\mu_\psi|^2\rangle_T < \infty\,\}}$.
\gap
{\it Proof. }
$\overline{\{\psi\,|\,J_\alpha(\mu_\psi)<\infty\,\}}\;\subseteq\;
\hils_{\alpha c}$ follows from Lemma 5.1, along with the fact that
$\hils_{\alpha c}$ is closed.
$\hils_{\alpha c}\;\subseteq\;
\overline{\{\psi\,|\,\sup_{T}T^\alpha
\langle|\hat\mu_\psi|^2\rangle_T < \infty\,\}}$ follows from Theorems
3.1 and 5.2. The other two inclusions follow from Lemma 5.2.
$\quad\bigcirc$
\gap
\vfil\eject
\gap
{\bf 6. Lower bounds on $\langle\langle|X|^m\rangle\rangle_T$}
\medskip\no
Our main purpose in this section is to prove the following
strengthened version of Theorem 2.6:
\gap
{\bf Theorem 6.1. } {\it If $H$ is self-adjoint on $\ell^2(Z^d)$
and $P_{\alpha c}\psi\not= 0$, where
$P_{\alpha c}$ is the orthogonal projection on $\hils_{\alpha c}$;
then, for each $m>0$, there exists a constant $C_{\psi,m}$, depending
on $\psi$ and $m$, such that for every $T>0$
$$\langle\langle|X|^m\rangle\rangle_T > C_{\psi,m} T^{m\alpha/d}\;.$$
\gap
Remarks. }
\item{(i) } By Theorem 4.1, the condition $P_{\alpha c}\psi\not= 0$
is equivalent to the condition that $\mu_\psi$ is {\it not}
supported on a set of zero $h^\alpha$ measure.
\item{(ii) } As the proof below will show, Theorem 6.1 follows from
Theorem 3.2 in a similar way to that by which Corollary 2.3.1 follows
from Theorem 2.3.
\gap
{\it Proof. } Let $\psi_{\alpha c}\equiv P_{\alpha c}\psi$,
$\psi_{\alpha s}\equiv P_{\alpha s}\psi = (1-P_{\alpha c})\psi$.
By Theorem 4.2, there exist mutually singular Borel measures
$\mu_{\psi_{\alpha c},1}$ and $\mu_{\psi_{\alpha c},2}$, such that
$d\mu_{\psi_{\alpha c}} = d\mu_{\psi_{\alpha c},1} +
d\mu_{\psi_{\alpha c},2}$, where $\mu_{\psi_{\alpha c},1}$ is
U$\alpha$H and
$\mu_{\psi_{\alpha c},2}(R)<{1\over 2}\|\psi_{\alpha c}\|^2$.
Let $S_1\subseteq R$ be a Borel set which supports
$\mu_{\psi_{\alpha c},1}$, and has $\mu_{\psi_{\alpha c},2}(S_1)=0$.
Let $P_{S_1}$ be the spectral projection on $S_1$, and denote
$\psi_1\equiv P_{S_1}\psi_{\alpha c}=P_{S_1}P_{\alpha c}\psi$,
$\psi_2\equiv (1-P_{S_1})\psi_{\alpha c} + \psi_{\alpha s} =
(1-P_{S_1}P_{\alpha c})\psi$. Clearly,
$\mu_{\psi_1} = \mu_{\psi_{\alpha c},1}$, such that $\mu_{\psi_1}$
is U$\alpha$H, and we have
$$\|\psi_1\|^2\;=\;\int d\mu_{\psi_1}\;=\;
\int d\mu_{\psi_{\alpha c}} - \int d\mu_{\psi_{\alpha c},2}
\;>\;{1\over 2}\|\psi_{\alpha c}\|^2\;, \eqno (6.1)$$
$$\psi\;=\;\psi_1+\psi_2\;,\qquad\;\|\psi_1\|^2 + \|\psi_2\|^2\;=\;
\|\psi\|^2\;=\;1\;. \eqno (6.2)$$
Let $P_N$ be the projection on a sphere of radius $N$, defined by
(2.5) (we consider $N$ as a continuous variable, allowed to take
any positive value). There exists a constant $C_d$, depending only
on the space dimension $d$, such that:
$$\|P_N\|_1\;=\;{\rm Tr}(P_N)\;<\;C_d N^d\;. \eqno (6.3)$$
Thus, it follows from Theorem 3.2 that there exists a constant
$C_{\psi_1}$, such that for any $T,\,N>0$
$$\langle\|P_N\psi_1(t)\|^2\rangle_T\;=\;
\langle\langle\psi_1(t),\,P_N\psi_1(t)\rangle\rangle_T\;<\;
C_{\psi_1}\|P_N\|_1 T^{-\alpha}\;<\;
C_{\psi_1}C_d N^d T^{-\alpha}\;. \eqno (6.4)$$
For each $T>0$, we define
$$N_T\;\equiv\;\left({{\|\psi_1\|^4\,T^\alpha}\over
{64\,C_{\psi_1}C_d}}\right)^{1/d}\;. \eqno (6.5)$$
By (6.4) we have
$$\langle\|P_{N_T}\psi_1(t)\|^2\rangle_T\;<\;
{{\|\psi_1\|^4}\over {64}}\;, \eqno (6.6)$$
and thus
$$\eqalign{\langle\|P_{N_T}\psi(t)\|^2\rangle_T\;\leq&\;
\left\langle\left(\|P_{N_T}\psi_1(t)\|+\|P_{N_T}\psi_2(t)\|
\right)^2\right\rangle_T\cr
\leq&\;\left\langle\left(\|P_{N_T}\psi_1(t)\|+\|\psi_2\|
\right)^2\right\rangle_T\cr
\leq&\;\left(\sqrt{\langle\|P_{N_T}\psi_1(t)\|^2\rangle_T}+\|\psi_2\|
\right)^2\cr
<&\;\left({{\|\psi_1\|^2}\over {8}}+\|\psi_2\|
\right)^2\;=\;{{\|\psi_1\|^4}\over {64}}+\|\psi_2\|^2+{1\over 4}
\|\psi_2\| \|\psi_1\|^2\cr
<&\;\|\psi_2\|^2+{1\over 2}\|\psi_1\|^2\;=\;1-{1\over 2}\|\psi_1\|^2
\;.\cr} \eqno (6.7)$$
Since
$$\langle\|P_{N_T}\psi(t)\|^2\rangle_T +
\langle\|(1-P_{N_T})\psi(t)\|^2\rangle_T\;=\;1\;, \eqno (6.8)$$
we obtain
$$\langle\|(1-P_{N_T})\psi(t)\|^2\rangle_T\;>\;
{1\over 2}\|\psi_1\|^2\;, \eqno (6.9)$$
which implies
$$\langle\langle|X|^m\rangle\rangle_T\;>\;
{1\over 2}\|\psi_1\|^2 N_T^m\;=\;
{{\|\psi_1\|^2}\over 2} \left({{\|\psi_1\|^4}\over
{64\,C_{\psi_1}C_d}}\right)^{m/d}\,T^{m\alpha/d}\;.
\quad\bigcirc\eqno (6.10)$$
\gap
Theorem 6.1 also has a continuum analog, involving Schr\"odinger
operators on $L^2(R^d)$, for which the moments of the position
operator are defined by
$$(|X|^m\psi)(x)\;=\;|x|^m\psi(x)\;. \eqno (6.11)$$
We need the following notions about classes of potentials on $R^d$:
\gap
{\bf Definition 6.1. } {\it Let $V$ be a real-valued measurable
(Borel) function on $R^d$.
\item{\rm (i)} $V$ is said to lie in the class $K_d$ if and only if:
\itemitem{\rm (a)}
$\displaystyle{\qquad\lim_{\epsilon\to 0}\left[\sup_x
\int_{|x-y|\leq\epsilon}|x-y|^{2-d}|V(y)|\,d^dy\right]\;=\; 0\;,
\qquad}$ if $d\geq 3$.
\itemitem{\rm (b)}
$\displaystyle{\qquad\lim_{\epsilon\to 0}\left[\sup_x
\int_{|x-y|\leq\epsilon}\ln(|x-y|^{-1})|V(y)|\,d^2y\right]\;=\; 0\;,
\qquad}$ if $d = 2$.
\itemitem{\rm (c)}
$\displaystyle{\qquad\sup_x\int_{|x-y|\leq 1}|V(y)|\,dy
\;<\; \infty\;,\qquad}$ if $d = 1$.
\medskip\no
\item{\rm (ii)} $V$ is said to lie in the class $K_d^{loc}$ if
and only if $\chi_{B_r}V\in K_d$ for every $r<\infty$, where
$\chi_{B_r}$ is the characteristic function of
$B_r\equiv\{x\,|\,|x|\leq r\,\}$.}
\gap
For every potential $V$, we define its positive and negative parts
$V_+,\,V_-$ by
$$V_+(x)\;\equiv\;\max\{V(x),\,0\}\;,\qquad
V_-(x)\;\equiv\;\max\{-V(x),\,0\}\;. \eqno (6.12)$$
\gap
{\bf Theorem 6.2. } {\it Let $H=-\Delta + V$ on $L^2(R^d)$, where
$V_+\in K_d^{loc}$ and $V_-\in K_d$,
and suppose that $P_{\alpha c}\psi\not= 0$, where
$P_{\alpha c}$ is the orthogonal projection on $\hils_{\alpha c}$;
then, for each $m>0$, there exists a constant $C_{\psi,m}$, depending
on $\psi$ and $m$, such that for every $T>0$
$$\langle\langle|X|^m\rangle\rangle_T > C_{\psi,m} T^{m\alpha/d}\;.$$}
\gap
The following Lemma is a consequence of semi-group kernel
inequalities, and is essentially due to Simon [\sssg]:
\gap
{\bf Lemma 6.1. } {\it Let $H=-\Delta + V$ on $L^2(R^d)$, where
$V_+\in K_d^{loc}$ and $V_-\in K_d$. Let $S$ be a compact subset of
$R$, and denote by $P_S$ the spectral projection on $S$, and let
$Q_r$ be the operator of multiplication by $\chi_{B_r}$, the
characteristic function of $B_r\equiv\{x\,|\,|x|\leq r\,\}$;
then the operator $P_SQ_rP_S$ is trace class and its trace-norm
obeys
$$\|P_SQ_rP_S\|_1\;\leq\;Cr^d\;,$$
where $C$ is independent of $r$.
\gap
Proof. } It follows from Theorem B.9.1 of [\sssg] (as a special
case), that $P_SQ_r$ is Hilbert-Schmidt; and while it is not
explicitly stated, it's easy to see from the proof in [\sssg]
that $\|P_SQ_r\|^2_2\leq Cr^d$. Since
$P_SQ_rP_S=P_SQ_rQ_rP_S=P_SQ_r(P_SQ_r)^*$, we have
$$\|P_SQ_rP_S\|_1\;=\;\|P_SQ_r\|^2_2\;\leq\;Cr^d\;.
\quad\bigcirc\eqno (6.13)$$
\gap
{\it Proof of Theorem 6.2. } This is a variant of the proof of
Theorem 6.1: Clearly, we can repeat the initial
steps of the proof of Theorem 6.1 and choose $S_1$ to be compact.
(6.1) and (6.2) hold as before. Since $P_{S_1}$ is a projection
which commutes with $e^{-iHt}$, we have for every $r$
$$\|Q_r\psi_1(t)\|^2\;=\;
\langle\psi_1(t),\,Q_r\psi_1(t)\rangle\;=\;
\langle\psi_1(t),\,P_{S_1}Q_rP_{S_1}\psi_1(t)\rangle\;. \eqno (6.14)$$
Thus, replacing (6.3) by Lemma 6.1, we obtain
an analog of (6.4) with $Q_r$ replacing $P_N$, and $r$ replacing $N$.
The rest of the proof is essentially identical. $\quad\bigcirc$
\gap
\gap\gap\gap
{\bf 7. A pathological example}
\medskip\no
In this section, we study the Almost Mathieu operator $\irrh$ on
$\ell^2(Z)$, defined by:
$$\eqalign{\irrh\;=&\;\Delta+V_{\beta,\lambda,\theta}\;, \qquad
(\Delta\psi)(n)\;=\;\psi(n+1)+\psi(n-1)\;,\cr
&\quad (V_{\beta,\lambda,\theta}\psi)(n)\;=\;
\lambda\cos(2\pi\beta n + \theta)\,\psi(n)\;.\cr} \eqno (7.1)$$
We call (following Avron-Simon [\ids]) an irrational number $\beta$
a Liouville number, if there exists a sequence of rationals
$\{p_n/q_n\}_{n=1}^\infty$ with $q_n\to\infty$ as $n\to\infty$,
and a constant $C$, such that
$$\left|\beta - {{p_n}\over {q_n}}\right|\;<\;Cn^{-q_n} \eqno (7.2)$$
for each $n$.
\gap
{\bf Theorem 7.1. } {\it If $\beta$ is a Liouville number, then
for any $|\lambda|>2$ and $\theta\in R$, $\irrh$ has purely
zero-dimensional spectrum, namely, all its spectral measures are
supported on a set of zero Hausdorff dimension.
\gap
Remarks. }
\item{(i) }
A result of Avron-Simon [\ids], based on Gordon's theorem, says
that under the conditions of Theorem 7.1 $\irrh$ has purely continuous
spectrum. Thus, Theorem 7.1 implies that for Liouville $\beta$,
$|\lambda|>2$, and any real $\theta$, $\irrh$ has purely
zero-dimensional singular continuous spectrum.
\item{(ii) } Avron-Simon [\ids] have already shown that for Liouville
$\beta$ and $|\lambda|>2$, the spectrum of $\irrh$ is purely singular
(and thus singular-continuous) for a.e.\ $\theta$.
\item{(iii) } The spectrum of $\irrh$, under the conditions of
Theorem 7.1, is known to be a Cantor set [\cey] of positive Lebesgue
measure [\avs,\yortp,\tha].
\gap
Our proof of Theorem 7.1 is based on the following:
\gap
{\bf Proposition 7.1. } (Avron, van Mouche, and Simon [\avs])
{\it Let $\sigma(\beta,\lambda,\theta)$ denote the spectrum of
$\irrh$, then for any $\theta$, $|\lambda|\geq 2$, and a pair of
relatively prime integers $p,q$:
$$|\sigma(p/q,\lambda,\theta)|\;\leq\;
4\pi\left({2\over {|\lambda|}}\right)^{{q\over 2}-1}\;.$$
\gap
Remark. } Proposition 7.1 is not explicitly given in [\avs], but it
follows immediately from their analysis. Essentially, it follows by
duality from the main estimate in their ``Proof of Theorem 2.''
\gap
Consider now a periodic Jacobi matrix of period $q$, namely,
$H = \Delta + V$, where $V(n+q)=V(n)\;\;\forall n$. Recall
(or see, e.g., [\toda]) that
the spectrum $\sigma(H)$ of $H$ is the union of $q$ bands (closed intervals),
and it is precisely the set of energies for which the discriminant
$D(E)$ obeys $|D(E)|\leq 2$. $D(E)$ is defined as the trace of the
one-period transfer matrix, namely,
$D(E)\equiv {\rm Tr}(\Phi_q(E))$, where
$\Phi_q(E)\equiv T_q(E) T_{q-1}(E)\;\cdots\; T_1(E)$, and
$$T_n(E)\;\equiv\;\pmatrix{E-V(n)&-1\cr
\noalign{\medskip} 1&0\cr}\;. \eqno (7.3)$$
$D(E)$ is a polynomial of order $q$ in $E$, with leading coefficient 1, and
$q$ real simple zeroes (one in each band). It is monotone on each band,
and obeys $|D(E)|\geq 2$ in each of its ($q-1$) local extremum points
(which occur either
inside gaps in $\sigma(H)$, or at points where two bands touch).
We need the following technical lemma:
\gap
{\bf Lemma 7.1. } {\it Let $H=\Delta+V$ be a periodic Jacobi matrix
of period $q$. Consider a band $b=[E_1,E_2]$, of $\sigma(H)$,
and let $E_{g1}$ and $E_{g2}$ be the (local)
extremum points of the discriminant $D(E)$
just below and above $b$ (one of the $E_{gi}$'s may be infinite if $b$
is an extremal band). Let $E_m$ be the zero of $D(E)$ inside $b$,
then:
\item{\rm (i) } For $E\in (E_2, E_{g2})$,
$\;\;\displaystyle{{{|D(E)|}\over 2}\;>\;{E-E_m\over e(E_2-E_m)}}\;$.
\item{\rm (ii) } For $E\in (E_{g1}, E_1)$,
$\;\;\displaystyle{{{|D(E)|}\over 2}\;>\;{E_m-E\over e(E_m-E_1)}}\;$.
\gap
Proof. } Assume that the band $b$ is a nonextremal band of $\sigma(H)$,
and that $D(E)$ is increasing on $b$, such that $D(E_1) = -2$ and
$D(E_2) = 2$. (Otherwise, a simple variant of the proof below would yield
the result.) We shall prove statement (i). (ii) is essentially the same.
Define:
$$f(E)\;\equiv\;{d\over {dE}}(\ln (D(E))) \; . \eqno (7.4)$$
Since $D(E)$ can be expressed as:
$$D(E)\; = \;\prod_{j=1}^q (E-E_j) \; , \eqno (7.5)$$
$f(E)$ can be written as:
$$ f(E)\; = \;\sum_{j=1}^q {1\over{E-E_j}} \; , \eqno (7.6)$$
and we have:
$$f'(E) \;\equiv\; {d\over {dE}} f(E)\; = \;
- \sum_{j=1}^q {1\over {(E-E_j)^2}} \; . \eqno (7.7)$$
>From (7.7) we see that:
$$f'(E)\; < \;{{-1}\over {(E-E_m)^2}} \; , \eqno (7.8)$$
and since $E_{g2}$ is a zero of $f(E)$ , we have for every $E \in (E_m ,
E_{g2} )$:
$$f(E)\;=\; - \int_E^{E_{g2}} f'(E')\, dE'
\;>\; \int_E^{E_{g2}} {{dE'}\over {(E'-E_m)^2}}
\;=\; {1\over {E-E_m}} - {1\over {E_{g2} - E_m}} \; .
\eqno (7.9)$$
Thus, we obtain for any $E\in (E_2,E_{g2})$:
$$\ln {D(E)\over {2}}\; = \;
\ln D(E) - \ln D(E_2)\; > \;\ln\left({{E-E_m}\over
{E_2-E_m}}
\right) - 1 \; , \eqno (7.10)$$
from which follows:
$${D(E)\over {2}}\;>\;{E-E_m\over e(E_2-E_m)}\;.\quad\bigcirc \eqno (7.11)$$
\gap
{\it Proof of Theorem 7.1. }
Let $\{p_n/q_n\}_{n=1}^\infty$ be a sequence of rationals obeying (7.2).
Clearly, we can assume that $p_n,q_n$ are relatively prime. For
each $n$, $H_{p_n/q_n,\lambda,\theta}$ is a periodic Jacobi matrix of
period $q_n$, and its spectrum $\sigma(p_n/q_n,\lambda,\theta)$ is a union
of $q_n$ bands:
$$\sigma(p_n/q_n,\lambda,\theta)\;=\;\bigcup_{m=1}^{q_n}b_m^n\;,
\eqno (7.12)$$
where $b_m^n=[E_{m,1}^n,E_{m,2}^n]$.
Let $D_{p_n/q_n,\lambda,\theta}(E)$ be the discriminant of
$H_{p_n/q_n,\lambda,\theta}$, and let $\{E_m^n\}_{m=1}^{q_n}$ be
its zeroes (such that $E_m^n$ is inside the band $b_m^n$).
We define the intervals $\{B_m^n\}_{m=1}^{q_n}$ by
$$B_m^n\;\equiv\;[E_m^n - {q_n}^2(E_m^n - E_{m,1}^n)\,,\;
E_m^n + {q_n}^2(E_{m,2}^n - E_m^n)]\;, \eqno (7.13)$$
and let
$$S_n\;\equiv\;\bigcup_{m=1}^{q_n}B_m^n\;. \eqno (7.14)$$
We define the set $S$ by
$$S\;\equiv\;\liminf_{n\to\infty}S_n\;\equiv\;
\bigcup_{k=1}^\infty\bigcap_{n=k}^\infty S_n\;. \eqno (7.15)$$
The proof of Theorem 7.1 would now follow from showing:
\item{(i) } All the spectral measures of $\irrh$ are supported on
$S$.
\item{(ii) } $\dimh(S)=0$.
\gap
{\it Proof of} (i). By general principles (see, e.g., [\cala]), for a.e.\
$E$ with respect to any spectral measure of $\irrh$ there exist polynomially
bounded solutions of the Schr\"odinger equation $(\irrh-E)\psi=0$
(generalized eigenfunctions), obeying, in particular,
$$|\psi(k)|\;{{{q_n}^2}\over e}\;.
\eqno (7.17)$$
Let $\Phi_{q_n}(E)$ be the one period transfer matrix for
$H_{p_n/q_n,\lambda,\theta}$. Since
${\rm Tr}(\Phi_{q_n}(E)) = D_{p_n/q_n,\lambda,\theta}(E) > 2$,
$\Phi_{q_n}(E)$ has two real eigenvalues of the form
$e^{\pm\gamma q_n}$, or $-e^{\pm\gamma q_n}$, and
$|D_{p_n/q_n,\lambda,\theta}(E)|=2\cosh (\gamma q_n)$. We assume that
the eigenvalues of $\Phi_{q_n}(E)$ are positive (otherwise
the proof is essentially the same), and denote the corresponding normalized
eigenvectors by $\vec u_\pm$. Let $\varphi$ be a solution of
$(H_{p_n/q_n,\lambda,\theta}-E)\varphi=0$, obeying
$|\varphi(0)|^2+|\varphi(1)|^2=1$, and denote, for each $k$,
$\vec\varphi_k\equiv (\varphi(k+1), \varphi(k))^T$.
There exist constants $a,b$ such that
$\vec\varphi_0 = a\vec u_+ + b\vec u_-$, and thus
$$\vec\varphi_{q_n}\;=\;
ae^{\gamma q_n}\vec u_+ + be^{-\gamma q_n}\vec u_-\;, \eqno (7.18)$$
$$\vec\varphi_{-q_n}\;=\;
ae^{-\gamma q_n}\vec u_+ + be^{\gamma q_n}\vec u_-\;. \eqno (7.19)$$
Since the $\vec u_\pm$ are normalized, and since $\|\vec\varphi_0\|=1$,
we have $|a|+|b|\geq 1$. Suppose that $|a|\geq |b|$, then (7.18) implies
$\|\vec\varphi_{q_n}\|\geq 2|a|\sinh(\gamma q_n)\geq\sinh(\gamma q_n)$.
Similarly, if $|b|\geq |a|$, then (7.19) implies
$\|\vec\varphi_{-q_n}\|\geq\sinh(\gamma q_n)$,
such that in either case
$$\max\{\|\vec\varphi_{-q_n}\|,\|\vec\varphi_{q_n}\|\}\;\geq\;
\sinh(\gamma q_n)\;=\;
\sinh\left(\cosh^{-1}(D_{p_n/q_n,\lambda,\theta}(E)/2)\right)\;.
\eqno (7.20)$$
Consider now a solution $\psi$ of $(\irrh-E)\psi=0$, obeying
$\vec\psi_0=\vec\varphi_0$, and let
$$T_k^\beta(E)\;\equiv\;\pmatrix{E-V_{\beta,\lambda,\theta}(k)&-1\cr
\noalign{\medskip} 1&0\cr}\;. \eqno (7.21)$$
By the same telescoping estimate used in proving Gordon's theorem [\srv],
we have
$$\eqalign{\|\vec\psi_{q_n}-\vec\varphi_{q_n}\|\;\leq&\;
\|T_{q_n}^\beta\;\cdots\;T_1^\beta-
T_{q_n}^{p_n/q_n}\;\cdots\;T_1^{p_n/q_n}\|\;\cr\leq&\;
q_n\left[\sup_k\max\{\|T_k^\beta\|,\|T_k^{p_n/q_n}\|\}\right]^{q_n-1}
\max_{1\leq k\leq q_n} \|T_k^\beta - T_k^{p_n/q_n}\|\;\cr\leq&\;
q_n(2+|\lambda|+|E|)^{q_n}|\lambda|2\pi q_n |\beta-p_n/q_n|\;\cr\leq&\;
2\pi|\lambda|C{q_n}^2 (2+|\lambda|+|E|)^{q_n} n^{-q_n}\;,\cr}
\eqno (7.22)$$
and similarly
$$\|\vec\psi_{-q_n}-\vec\varphi_{-q_n}\|\;\leq\;
2\pi|\lambda|C{q_n}^2 (2+|\lambda|+|E|)^{q_n} n^{-q_n}\;. \eqno (7.23)$$
Thus, (7.20) implies
$$\max\{\|\vec\psi_{-q_n}\|,\|\vec\psi_{q_n}\|\}\;\geq\;
\sinh\left(\cosh^{-1}(D_{p_n/q_n,\lambda,\theta}(E)/2)\right)
-2\pi|\lambda|C{q_n}^2 (2+|\lambda|+|E|)^{q_n} n^{-q_n}\;,
\eqno (7.24)$$
for any solution $\psi$ of $(\irrh-E)\psi=0$ obeying $\|\vec\psi_0\|=1$.
The negative term in the r.h.s.\ of (7.24) clearly vanishes as
$n\to\infty$, and the positive term can be written as
$\tanh\left(\cosh^{-1}(D_{p_n/q_n,\lambda,\theta}(E)/2)\right)
(D_{p_n/q_n,\lambda,\theta}(E)/2)$. Thus, (7.17) and (7.24) imply
that for sufficiently large $n$
$$\max\{\|\vec\psi_{-q_n}\|,\|\vec\psi_{q_n}\|\}\;\geq\;
{{{q_n}^2}\over {2e}}\;. \eqno (7.25)$$
If $E\notin S$, then there are infinitely many $n$'s for which
$E\notin S_n$ and (7.25) holds. Thus, for such $E$, there can be
no solution of $(\irrh-E)\psi=0$ which obeys (7.16), and this proves
(i).
\gap
{\it Proof of} (ii). Since a countable union of sets with zero
Hausdorff dimension has zero Hausdorff dimension, we see from
(7.15) that it is sufficient to prove:
$$\dimh\left(\bigcap_{n=k}^\infty S_n\right)\;=\;0\;, \eqno (7.26)$$
for each $k$.
Since, for each $m$, $|B_m^n|={q_n}^2|b_m^n|$, we have, by Proposition 7.1,
$$|B_m^n|\;\leq\;{q_n}^2|\sigma(p_n/q_n,\lambda,\theta)|\;\leq\;
4\pi{q_n}^2\left({2\over {|\lambda|}}\right)^{{{q_n}\over 2}-1}\;.
\eqno (7.27)$$
Thus, $|B_m^n|\to 0$, uniformly in $m$, as $n\to\infty$. Moreover, for any
positive $\alpha$ we have
$$\sum_{m=1}^{q_n}|B_m^n|^\alpha\;\leq\;
(4\pi)^\alpha{q_n}^{1+2\alpha}
\left({2\over {|\lambda|}}\right)^{\alpha({{q_n}\over 2}-1)}\;,
\eqno (7.28)$$
and the r.h.s.\ of (7.28) clearly vanishes as $n\to\infty$.
Since each $S_n=\bigcup_{m=1}^{q_n}B_m^n$ with $n\geq k$ covers
$\bigcap_{n=k}^\infty S_n$, this implies (7.26) and completes
the proof. $\quad\bigcirc$
\gap
Theorem 7.1 indicates that spectral questions arising from the
decomposition theory we have presented earlier are answerable.
Our interest in it, in the context of the current paper, however,
is mainly due to the following:
\gap
{\bf Theorem 7.2. } {\it Let $\psi=\psi(0)=\delta_0$ and fix
$\lambda,\theta\in R$; then there exists a Liouville number
$\beta$ and a sequence $\{T_n\}_{n=1}^\infty$, where $T_n\to\infty$
as $n\to\infty$, such that for $H=\irrh$
$$\langle\langle |X|^2\rangle\rangle_{T_n}\;>\;
{{T_n^2}\over {\log T_n}}\;,$$
for every $n$.
\gap
Remarks. }
\item{(i) } $\log T_n$ can be replaced by $F(T_n)$, where $F$ is
any positive valued monotonely increasing function, obeying
$F(T)\to\infty$ as $T\to\infty$. The number
$\beta$ and the sequence $\{T_n\}_{n=1}^\infty$ will depend
also on $F$ in such case. Thus, Theorem 7.2 shows that the growth
rate of $\langle\langle |X|^2\rangle\rangle_T$ can be arbitrarily
close to ``ballistic'' (at least for some time scales), even
when $H$ has purely zero-dimensional spectrum. This indicates
that Theorem 6.1 is a strictly one-sided inequality, as far as
singularity or continuity with respect to Hausdorff measures is
concerned.
\item{(ii) } Del Rio et al.\ [\djls] use a strengthened
variant of Theorem 7.2, which obtains the dynamical estimate uniformly
in $\theta$ and in some range of suitable rank-one perturbations of
$\irrh$, to show that the growth rate of
$\langle\langle |X|^2\rangle\rangle_T$ can be arbitrarily close to
``ballistic,'' even in cases where $H$ has pure-point spectrum
with exponentially localized eigenvectors. This shows that Simon's
result on the absence of ballistic motion [\sabm] is optimal.
\gap
\gap
{\bf Lemma 7.2. } {\it Let $H_1=\Delta+V_1$ and $H_2=\Delta+V_2$
on $\ell^2(Z)$, such that $|V_1(k)|,|V_2(k)|0$ and $\epsilon>0$ be given, then there exist
$L,\delta>0$ such that if $|V_1(k)-V_2(k)|<\delta$ for all $|k|L$, we see that the r.h.s.\ of
(7.30) is bounded by
$$2(2L+1)L^2\left[\delta T e^{T(2+C)} +
2\sum_{k>L}{{(T(2+C))^k}\over {k!}}\right]\;+\;4\sum_{|n|>L}n^2
\sum_{k\geq |n|}{{(T(2+C))^k}\over {k!}}\;. \eqno (7.32)$$
The terms that are independent of $\delta$ in (7.32) are easily seen
to vanish as $L\to\infty$; so choose $L$ such that their sum is less
than $\epsilon/2$, then choose $\delta$ such that the remaining term
is also less than $\epsilon/2$. $\quad\bigcirc$
\gap
{\it Proof of Theorem 7.2. }
We shall prove the existence of an appropriate Liouville number $\beta$
by constructing its continued fraction expansion [\nmt]:
$$\beta\;=\;[m_1,\,m_2,\,m_3,\,\dots\;]\;=\;
{\strut\displaystyle 1\over {\displaystyle m_1+{\strut 1\over
{\displaystyle m_2 +
{\strut 1\over {\displaystyle m_3+\;\cdots\;}}}}}}\;\;. \eqno (7.33)$$
We will show that an infinite sequence of quotients,
$\{m_n\}_{n=1}^\infty$, can be chosen inductively, such that the resulting
$\beta$, given by (7.33), is a Liouville number, and the dynamical estimate
of Theorem 7.2 is obeyed for an appropriate sequence $\{T_n\}_{n=1}^\infty$.
Let $\beta$ be given by (7.33) and denote
$$\beta_n\;\equiv\;[m_1,\,m_2,\,\dots\,,\,m_n\;]\;. \eqno (7.34)$$
The $\beta_n$'s are rational, and for each $n$ there are unique,
relatively prime, $p_n,q_n$, such that $\beta_n=p_n/q_n$. It is well
known [\nmt] (and not hard to verify) that the $q_n$'s obey
$$q_{n+1}\;=\;m_{n+1}q_n+q_{n-1}\;, \eqno (7.35)$$
and that for each $n$:
$$|\beta-\beta_n|\;\leq\;{1\over {q_n q_{n+1}}}\;. \eqno (7.36)$$
Thus, if $m_1,\,m_2,\,\dots\,,\,m_n$ (and thus $\beta_n$) are given,
we can always make $\beta$ arbitrarily close to $\beta_n$ by choosing
$m_{n+1}$ to be sufficiently large.
Suppose now that $m_1,\,m_2,\,\dots\,,\,m_n$ are given, and consider
the time evolution of $\psi=\psi(0)=\delta_0$ with
$H=H_{\beta_n,\lambda,\theta}$. Since $H_{\beta_n,\lambda,\theta}$ is
a periodic Jacobi matrix, its spectrum is purely absolutely continuous.
Thus, by Theorem 6.1, there exists a constant $C_n$ such that
$\langle\langle |X|^2\rangle\rangle_T>C_nT^2$. In particular, we can
choose $T_n$ large enough (and as large as we want) such that
$$\langle\langle |X|^2\rangle\rangle_{T_n}\;>\;
{{2T_n^2}\over {\log T_n}}\;. \eqno (7.37)$$
By Lemma 7.2, there exists $\delta_n$ such that if
$|\beta-\beta_n|<\delta_n$, then for $\psi=\psi(0)=\delta_0$ and
$H=\irrh$ we have
$$\langle\langle |X|^2\rangle\rangle_{T_n}\;>\;
{{T_n^2}\over {\log T_n}}\;. \eqno (7.38)$$
Choose $m_{n+1}\geq\max\{n^{q_n},\,1/\delta_n\}$, then clearly
$|\beta-\beta_n|<\delta_n$, and thus (7.38) is obeyed. Moreover,
$$|\beta-\beta_n|\;<\;n^{-q_n}\;. \eqno (7.39)$$
By continuing this construction indefinitely, (7.39) implies that
we obtain a Liouville number, and the dynamical estimate (7.38) is
obeyed for the sequence $\{T_n\}_{n=1}^\infty$. $\quad\bigcirc$
\gap
\gap\gap\gap
{\bf 8. Ergodic Schr\"odinger operators}
\medskip\no
In this section, we study ergodic Schr\"odinger operators on
$\ell^2(Z^d)$. That is, we consider families of potentials of the
form $\{V_\omega\}_{\omega\in\Omega}$, along with a probability
measure $dp$ on $\Omega$, and assume that $dp$ is stationary
and ergodic with respect to all of the shift
operators $\{T_j\}_{j\in Z^d}$, given by
$(T_jV_\omega)(n)=V_\omega(n-j)$. Random, almost periodic, and
periodic potentials can all be realized in the context of such
families. We are interested in the spectral
properties of the family $\{H_\omega\}_{\omega\in\Omega}$, where
$H_\omega=\Delta + V_\omega$. An excellent introduction to the
spectral theory of such families is given by Cycon et al.\ [\cyc]
(also see [\cala]).
It is well known that, for such a family, there exist subsets of $R$:
$\sigma$, $\sigma_{ac}$, $\sigma_{sc}$, and $\sigma_{pp}$, such that
for a.e.\ $\omega$ (w.r.t.\ $dp$), $\sigma$ is the spectrum of
$H_\omega$, and $\sigma_{ac}$, $\sigma_{sc}$, and $\sigma_{pp}$
are, respectively, the absolutely continuous, singular continuous,
and pure point spectra of $H_\omega$. Our result in this section is
a generalization of this to the various spectra introduced in
Section 5, namely:
\gap
{\bf Theorem 8.1. } {\it For any $\alpha\in [0,\,1]$, there exist
subsets of $R$: $\sigma_{\alpha ds}$, $\sigma_{ed\alpha/\alpha s}$,
$\sigma_{\alpha ac}$, $\sigma_{ed\alpha/s\alpha c}$, and
$\sigma_{s\alpha dc}$, such that for a.e.\ $\omega$ they are,
respectively, the $\alpha$-dimension singular, $\alpha$-singular of exact
dimension $\alpha$, absolutely continuous with respect to $h^\alpha$,
strongly $\alpha$-continuous of exact dimension $\alpha$,
and strongly $\alpha$-dimension continuous spectra of $H_\omega$.
\gap
Remark. } Theorem 8.1 corresponds to the measure
decomposition given by Corollary 4.1.3.
\gap
{\it Proof. }
We shall rely on the proof of the corresponding theorem for the standard
decomposition ($\sigma_{ac}$, $\sigma_{sc}$, $\sigma_{pp}$), as given by
Cycon et al. [\cyc]. It's easy to see, from the analysis given in [\cyc],
that it is sufficient to prove the weak measurability (in $\omega$) of
certain spectral projections. More precisely, it is sufficient to show
that for each spectral measure, $\mu^\omega=\mu_\psi^\omega$, which is
the spectral measure for $H_\omega$ and some fixed vector $\psi$, and
for each Borel set $A$, $\mu^\omega_{\alpha ds}(A)$,
$\mu^\omega_{ed\alpha/\alpha s}(A)$, $\mu^\omega_{\alpha ac}(A)$,
$\mu^\omega_{ed\alpha/s\alpha c}(A)$, and $\mu^\omega_{s\alpha dc}(A)$
are measurable functions of $\omega$.
Since, for any finite Borel measure $\mu$,
$\mu_{ed\alpha/\alpha s}=\mu_{\alpha s}-\mu_{\alpha ds}$,
$\mu_{\alpha ac}=\mu_{a\alpha s}-\mu_{\alpha s}$,
$\mu_{ed\alpha/s\alpha c}=\mu_{a\alpha ds}-\mu_{a\alpha s}$, and
$\mu_{s\alpha dc}=\mu -\mu_{a\alpha ds}$; it is clearly sufficient
to show that $\mu^\omega(A)$, $\mu^\omega_{\alpha ds}(A)$,
$\mu^\omega_{\alpha s}(A)$, $\mu^\omega_{a\alpha s}(A)$, and
$\mu^\omega_{a\alpha ds}(A)$ are measurable. Cycon et al.\ [\cyc]
already show that $\mu^\omega(S)$ is measurable for any fixed Borel
set $S$. The theorem would thus follow if we can show that
$\mu^\omega_{\alpha ds}(A)$, $\mu^\omega_{\alpha s}(A)$,
$\mu^\omega_{a\alpha s}(A)$, and $\mu^\omega_{a\alpha ds}(A)$ can be
obtained by suitable countable operations from the restrictions of
$\mu^\omega$ to some fixed countable family of Borel sets. (That is,
we need appropriate analogs of the lemma in the bottom of page
170 of [\cyc].)
Let ${\cal J}$ be the family of finite unions of intervals, each of
which has rational endpoints (note that ${\cal J}$ is countable).
For each ${\cal J}\ni J=\bigcup_{\nu=1}^N I_\nu$ (where each $I_\nu$
is an interval with rational endpoints), we denote $\delta(J)\equiv
\max_\nu |I_\nu|$, $M_\alpha(J)\equiv\sum_{\nu=1}^N |I_\nu|^\alpha$.
The theorem is now implied by the following equalities, which hold for
any Borel set $A\subseteq R$ and a finite Borel measure $\mu$ on $R$:
$$\mu_{\alpha s}(A)\;=\;
\lim_{n\to\infty}\sup_{J\in{\cal J},\, M_\alpha(J)<1/n}\mu(A\cap J)\;,
\eqno (8.1)$$
$$\mu_{a\alpha s}(A)\;=\;
\lim_{m\to\infty}\lim_{n\to\infty}
\sup_{J\in{\cal J},\, M_\alpha(J)0$, we have, by statement
(i) of Theorem 4.2, $\mu_{\alpha c}=\mu_1^\epsilon+\mu_2^\epsilon$
with $\mu_2^\epsilon(R)<\epsilon$, and $\mu_1^\epsilon$ U$\alpha$H,
such that $\mu_1^\epsilon(I)0$,
there exists $J\in{\cal J}$ with $M_\alpha(J)<1/n$, and
$\mu(A\cap J)>\mu_{\alpha s}(A)-\epsilon$. This implies the inverse
inequality to (8.6), and thus completes the proof of (8.1).
\gap
{\it Proof of} (8.2). For each $\epsilon>0$, we have, by statement
(ii) of Theorem 4.2, $\mu_{s\alpha c}=\mu_1^\epsilon+\mu_2^\epsilon$
with $\mu_2^\epsilon(R)<\epsilon$, and $\mu_1^\epsilon$ US$\alpha$H,
such that for each $\eta>0$ there is $\delta>0$ so that
$\mu_1^\epsilon(I)<\eta|I|^\alpha$ if $|I|<\delta$.
Thus, for sufficiently large $n$, we have for any
$J\in{\cal J}$ with $M_\alpha(J)0$,
a set $S_\epsilon$, such that $h^\alpha(S_\epsilon)<\infty$ and
$\mu_{a\alpha s}(R\setminus S_\epsilon)<\epsilon$. $S_\epsilon$ has
covers by disjoint intervals:
$S_\epsilon\subseteq\cup_{\nu=1}^\infty b_\nu$, such that $\sup_\nu|b_\nu|$
is arbitrarily small, and $\sum_{\nu=1}^\infty |b_\nu|^\alpha$ is
arbitrarily close to $h^\alpha(S_\epsilon)$. Thus, there exists sufficiently
large $m$, such that for any $n$ there is $J\in{\cal J}$ with
$M_\alpha(J)\mu_{a\alpha s}(A)-2\epsilon$. This implies the inverse
inequality to (8.8), and thus completes the proof of (8.2). $\quad\bigcirc$
\gap\gap\gap
{\bf 9. Discussion and open problems}
\medskip\no
The spectral decomposition theory arising from the measure
decomposition theory of Rogers and Taylor extends the usual spectral
decomposition theory in a natural way, and provides a rich collection
of hierarchies, that can be used for spectral classification. Theorems
3.1 and 3.2 along with Theorem 5.2, Theorem 5.3, and Theorems 6.1 and 6.2
indicate that this classification contains some significant information
about the dynamical behavior of the underlying quantum systems.
Nevertheless, as Theorems 7.1 and 7.2, along with their variants by
del Rio et al.\ [\djls,\djlsp], indicate, the long time behavior of
dynamical quantities like $\langle|X|^2\rangle$ is not completely
determined by spectral properties such as singularity or continuity
with respect to Hausdorff measures. Del Rio et al.\ [\djls,\djlsp],
who study quantum dynamics in the Anderson localization regime,
show that dynamical localization (namely,
$\sup_t\langle|X|^2\rangle<\infty$) is related to pure point spectrum
which is accompanied by additional information on the spatial behavior
of the eigenfunctions. It is an interesting open problem to obtain some
relations of this kind for continuous spectrum. Another
interesting open problem is: Are there any purely spectral (i.e., not
involving information about spatial behavior of eigenfunctions)
properties which imply meaningful (e.g., sub-ballistic by a power law)
upper bounds on $\langle|X|^2\rangle$?
In particular, are there such bounds if the spectrum (as a set) is
sufficiently ``fractal''? For example, while $\dimh(\sigma(H))$
might not imply any bound on $\langle|X|^2\rangle$, what happens if
$\overline{\rm dim}_{\rm B}(\sigma(H))$, the upper box-counting
dimension of the spectrum (see [\fal]), is small?
Some recent results [\gumab,\wiau] suggest that
even this need not bound $\langle|X|^2\rangle$.
The above mentioned difficulties need not diminish the interest in
classifying singular continuous spectra according to singularity or
continuity with respect to Hausdorff measures. Such spectral
classification allows one to distinguish between various
operators with singular
continuous spectrum, and we believe that this should help in improving our
understanding of such operators. For example:
Simon [\sbtd] has recently shown that barrier potentials in one-dimension
with sufficiently sparse barriers lead to purely one-dimensional
(but singular continuous) spectrum.
Del Rio et al.\ [\djls,\djlsp] show that, while point spectrum can
turn into one-dimensional spectrum under rank-one perturbations,
the Anderson model (in any space dimension) has semi-stability,
in the sense that the singular continuous spectrum arising from localized
rank-one perturbations of typical realizations [\delms,\gord] is always
purely zero-dimensional.
Another aspect of the results in [\djls] involves the genericity of
singular continuous spectrum that has been recently discovered in a
variety of contexts (largely by Simon and co-workers [\delms,\jisi,\ssci];
also see Gordon [\gord]). In the context of cyclic rank-one perturbations,
where singular continuous spectrum has been shown to be generic (i.e.,
to occur for a dense $G_\delta$ of couplings) within every sub-interval
of the spectrum in which there is no absolutely continuous spectrum, the
results of [\djls] indicate that there is no specific type (in the sense
of classification with respect to Hausdorff measures) of singular
continuous spectrum which is generic
\gap\gap\gap
{\bf Acknowledgments}
\medskip\no
We would like to thank J. Avron, who should have been a rightful
coauthor of this paper, and S. Fishman, for useful discussions that
led to this work. We also benefited from discussions with R. del Rio,
P. Hislop, S. Jitomirskaya, and B. Simon.
\gap
\vfill\eject
\gap
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\end