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% Dimensional Hausdorff Properties of Singular Continuous Spectra %
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% By S. Jitomirskaya and Y. Last %
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\begin{titlepage}
\Large
\title{Dimensional Hausdorff Properties\\
of Singular Continuous Spectra}
\date{January 28, 1996}
\large
\author{Svetlana Ya. Jitomirskaya\\
Department of Mathematics\\
University of California\\
Irvine, California 92717\\
\\ and \\
\\
Yoram Last\\
Division of Physics, Mathematics, and Astronomy\\
California Institute of Technology\\
Pasadena, California 91125}
\end {titlepage}
\maketitle
\begin{abstract}
\normalsize
We present an extension of the Gilbert-Pearson theory of subordinacy,
which relates dimensional Hausdorff spectral properties of one-dimensional
Schr\"odinger operators to the behavior of solutions of the corresponding
Schr\"odinger equation. We use this theory to analyze these
properties for several examples having singular-continuous
spectrum, including sparse barrier potentials, the almost Mathieu operator
and the Fibonacci Hamiltonian.
\end{abstract}
\vfill
PACS: 02.30.Sa, 71.25.-s, 71.50.+t, 72.15.Rn
\newpage\clearpage
\noindent
Singular continuous spectra have been extensively studied recently.
Our interest here is in the classification
and decomposition of such spectra with respect to dimensional Hausdorff
measures. The measure-theoretical aspect of this point of view goes back
to Rogers-Taylor \cite{rtab},
and it has been studied recently within spectral
theory by Last \cite{qdavl} and by del-Rio et al.\ \cite{djls} who have shown
that the singular-continuous spectrum which is produced by localized
rank-one perturbations of Anderson-model Hamiltonians in the localized
regime \cite{gdms} must be purely zero-dimensional --- in the sense that the
associated spectral measures are supported on a set of zero Hausdorff
dimension.
The main purpose of this paper is to report a general method for spectral
analysis of one-dimensional Schr\"odinger operators from this point of
view. It is a natural extension of the Gilbert-Pearson theory of
subordinacy \cite{gilpear,gilb}, and it allows us to analyze the dimensional
Hausdorff properties for a number of examples with singular-continuous
spectrum.
Below we describe the main ideas of our study and some of the main
results. Mathematically complete proofs of these results will be given
elsewhere \cite{jitlast}.
Most of our discussion will be restricted to one-dimensional
discrete (tight-binding) Schr\"o\-dinger operators of the form
\begin{equation}
(H\psi)(n) =\psi(n+1) + \psi(n-1)+ V(n)\psi(n)\;.
\end{equation}
We shall consider two kinds of such
operators: ``line'' operators acting on $\ell^2({\Bbb Z})$
($-\infty 0$), which are considered with a phase boundary
condition of the form:
\begin{equation}
\psi(0)\cos\theta + \psi(1)\sin\theta \;=\; 0\;,
\end{equation}
where $-\pi/2<\theta<\pi/2$.
Before formulating our main result, which would require some
definitions, we would like to describe some of its applications.
We stress at this point that the dimensional Hausdorff properties which
we study are those which are associated with the spectral measures
of the corresponding operators. The spectra themselves, as sets,
are closed sets, and their dimensions may be larger than those which
are associated with the spectral measures. A description of the precise
spectral-theoretic scheme which underlies our study is given below.
We start with a somewhat artificial example of ``half-line'' operators
with sparse barrier potentials. More specifically, we consider potentials
which vanish for all $n$'s outside a sparse (fastly growing) sequence of
points $\{L_n\}_{n=1}^\infty$ where $|V(L_n)|\to\infty$ as $n\to\infty$.
Simon-Spencer \cite{simsp} have
shown that the Schr\"odinger operators corresponding to such potentials
have no absolutely-continuous spectrum, and
Gordon \cite{gkmp} has
shown that if the $|V(L_n)|$'s grow sufficiently fast (compared to
the growth of the $L_n$'s) then for (Lebesgue) a.e.\ boundary phase
$\theta$ the corresponding operators have pure-point spectrum with
exponentially decaying eigenfunctions. It is easy to see \cite{simst},
however, that if the $L_n$'s grow sufficiently fast
(compared to the growth of the
$|V(L_n)|$'s) then, for every boundary phase $\theta$, the spectrum in
$(-2,2)$ is purely singular-continuous,
and Simon \cite{sbtd} has recently shown that if
the growth is even faster then the spectrum in $(-2,2)$ is purely
one-dimensional, in the sense that the spectral measure does not give
weight to sets of Hausdorff dimension less than 1. By applying Theorem 1
below, we have shown:
\bigskip\noindent
{\bf Theorem 2. } {\it
Let $\alpha\in (0,1)$.
Let $L_n = 2^{(n^n)}$ and define a potential
$V(k)$ for $k > 0$ by $V(L_n) = L_n^{(1-\alpha)/(2\alpha)}\,$;
$V(k) = 0$ if $k\notin\{L_n\}_{n=1}^\infty$. Then:
\begin{itemize}
\item[{\rm (i)} ] For every boundary phase $\theta$,
the spectrum of the corresponding ``half-line'' discrete
Schr\"odinger operator consists of the
interval $[-2,2]$ (which is the essential spectrum) along with some
discrete point spectrum outside this interval.
\item[{\rm (ii)} ] For every $\theta$, the Hausdorff dimensionality
of the spectrum in
$(-2,2)$ is bounded between dimensions $\alpha$ and
$\beta\equiv 2\alpha/(1+\alpha)$, in the sense that the restriction of the
spectral measure to $(-2,2)$ is supported on a set of Hausdorff dimension
$\beta$ and does not give weight to sets of Hausdorff dimension less than
$\alpha$.
\item[{\rm (iii)} ] For Lebesgue a.e.\ $\theta$, the spectrum in
$[-2,2]$
is of exact dimension $\alpha$, namely, the restriction of the spectral
measure to $[-2,2]$ is supported on a set of Hausdorff dimension $\alpha$
and does not give weight to sets of Hausdorff dimension less than $\alpha$.
\end{itemize}}
\bigskip\noindent
{\bf Remark. } The result only requires the $L_n$'s to be sufficiently
sparse (namely, to grow sufficiently fast). $L_n = 2^{(n^n)}$ is
a particular choice for which the sufficient sparseness is easy to
show.
\bigskip
Next we consider two examples of ``line'' operators with quasiperiodic
potentials. The first is the almost Mathieu (also called Harper)
operator $H_{\beta, \lambda, \theta}$,
which is the operator of the form (1) on $\ell^2({\Bbb Z})$ with potential
$V(n) = V_{\beta, \lambda, \theta}(n) =
\lambda\cos(2\pi\beta n + \theta)$, where
$\lambda,\theta$ are any real numbers, and $\beta$ is an irrational.
Aubry and Andre \cite{aaa} have conjectured that
$H_{\beta, \lambda, \theta}$ has purely
absolutely-continuous spectrum whenever $|\lambda|<2$, and purely point
spectrum (with exponentially localized eigenfunctions) whenever
$|\lambda|>2$. While the $|\lambda|<2$ part of this conjecture may be
correct (so far, the existence of
absolutely-continuous spectrum \cite{lags} and absence of point spectrum
\cite{delyon} have been established rigorously), the
$|\lambda|>2$ case turned out to be more delicate: Absolutely-continuous
spectrum is absent \cite{lasi},
but both pure-point and singular-continuous
spectra occur, depending on arithmetical properties
of both $\beta$ and $\theta$ \cite{asjjs}. It turns out, though,
that if we concentrate on the dimensional Hausdorff properties
of the spectral measures, rather than distinguishing between pure-point
and singular-continuous spectra, the situation becomes simpler:
\bigskip\noindent
{\bf Theorem 3. } {\it
For $|\lambda|>2$, every irrational $\beta$, and
every $\theta$, $H_{\beta, \lambda, \theta}$ has purely zero-dimensional
spectrum, in the sense that its spectral measures are all supported on
a set of zero Hausdorff dimension.}
\bigskip\noindent
{\bf Remarks. } (i) The spectrum of $H_{\beta, \lambda, \theta}$,
as a set, is known in this case
($|\lambda|>2$) to have positive Lebesgue measure \cite{tavsl}.
(ii) The result extends to potentials of the form
$V(n) = f(2\pi\beta n + \theta)$, where
$f(x)\equiv\sum_{k=1}^N\lambda_k\cos(kx)$,
in which case we prove that the spectrum is purely
zero-dimensional whenever$|\lambda_N|>2$.
\bigskip\noindent
Our second ``line'' example is the Fibonacci Hamiltonian
$H_\lambda$, which is the operator of the form (1) on $\ell^2({\Bbb Z})$
with potential
$V(n) =\lambda([(n+1)\omega] - [n\omega])$, where
$\omega = (\sqrt{5}-1)/2$ is the golden mean,
and $[x]\equiv\max\{m\in{\Bbb Z}\,|\,m\leq x\}$. $H_\lambda$
is the most studied of all one-dimensional quasicrystal models.
It is known \cite{suto} that, for every $\lambda\not= 0$,
it has purely singular-continuous spectrum,
and, moreover, its spectrum (as a set) is
a Cantor set of zero Lebesgue measure.
We have shown:
\bigskip\noindent
{\bf Theorem 4. } {\it
For every $\lambda$ there exists an $\alpha>0$ such that $H_\lambda$
has purely $\alpha$-continuous spectrum, namely,
its spectral measures do not give weight to sets of Hausdorff dimension
less than $\alpha$.}
\bigskip\noindent
{\bf Remark. } There exists strong numerical evidence \cite{hirkoh} that
the spectrum of $H_\lambda$ (as a set) has Hausdorff dimension
strictly less than 1 (for every $\lambda\not= 0$), and this would
imply that its spectrum must also be $\beta$-singular (see below) for some
$\beta<1$.
\bigskip
Let us now describe the spectral-theoretic scheme in the context of
which the above results should be understood.
Consider a separable Hilbert space ${\cal H}$, and a self adjoint operator
$H$. Recall \cite{rsi} that for each $\psi\in{\cal H}$, the spectral measure
$\mu_\psi$ (also known to physicists as the local spectral density)
is the unique Borel measure obeying
$\langle\psi ,\, f(H)\psi\rangle = \int f(x)\,d\mu_\psi(x)$ for any
measurable function $f$.
By Lebesgue's decomposition theorem, every Borel
measure $\mu$ decomposes uniquely as: $\mu = \mu_{ac}+\mu_{sc}+\mu_{pp}$.
The absolutely-continuous part, $\mu_{ac}$, gives zero weight to sets of
zero Lebesgue measure. The pure-point part, $\mu_{pp}$, is a countable sum
of atomic (Dirac) measures. The singular-continuous part, $\mu_{sc}$,
gives zero weight to countable sets and is supported on some set of
zero Lebesgue measure (we say that a measure $\mu$ is supported on a set
$S$ if $\mu({\Bbb R}\setminus S) = 0$).
Letting
${\cal H}_{ac} \equiv \{\psi\,|\, \mu_\psi\; \hbox{\rm is purely
absolutely-continuous}\}$,
${\cal H}_{sc} \equiv \{\psi\,|\, \mu_\psi\; \hbox{\rm is purely
singular-continuous}\}$, and
${\cal H}_{pp} \equiv \{\psi\,|\, \mu_\psi\; \hbox{\rm is purely
pure-point}\}$,
one obtains a decomposition:
${\cal H} = {\cal H}_{ac}\oplus{\cal H}_{sc}\oplus{\cal H}_{pp}$.
${\cal H}_{ac}$, ${\cal H}_{sc}$, and ${\cal H}_{pp}$ are closed (in norm),
mutually orthogonal subspaces, which are invariant under $H$.
The absolutely-continuous spectrum ($\sigma_{ac}$), singular-continuous
spectrum ($\sigma_{sc}$), and pure-point spectrum ($\sigma_{pp}$) are
defined as the spectra of the restrictions of $H$ to the corresponding
subspaces, and
${\rm Spec\,}(H)\equiv\sigma =
\sigma_{ac}\cup\sigma_{sc}\cup\sigma_{pp}$.
The above standard scheme of spectral theory can be extended, to further
decompose the singular-continuous subspace, by using Hausdorff measures.
Recall that for any subset $S$ of ${\Bbb R}$ and $\alpha\in [0,1]$, the
$\alpha$-dimensional Hausdorff measure, $h^\alpha$, is given by
\begin{equation}
h^\alpha(S)\;\equiv\;\lim_{\delta\to 0}\,\inf_{\delta -covers}\,
\sum_{\nu=1}^\infty |b_\nu|^\alpha\;,
\end{equation}
where a $\delta -cover$ is a cover of $S$ by a countable collection of
intervals, $S\subset\bigcup_{\nu=1}^\infty b_\nu$,
such that for each $\nu$ the length of $b_\nu$ is at most $\delta$.
$h^1$ coincides with Lebesgue measure, and $h^0$ is the counting measure
(assigning to each set the number of points in it).
Given any $\emptyset\not= S\subseteq {\Bbb R}$, 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
$\alpha<\alpha(S)$.
This unique $\alpha(S)$ is called the Hausdorff dimension of $S$.
A rich theory of decomposing measures with respect to Hausdorff
measures has been developed by Rogers and Taylor \cite{rtab}. Below
we discuss only a small part of it. A much more detailed description
has been given by Last \cite{qdavl}.
Given $\alpha\in [0,1]$, a measure $\mu$ is called $\alpha$-continuous
($\alpha c$) if $\mu(S)=0$ for every set $S$ with $h^\alpha(S)=0$.
It is called $\alpha$-singular ($\alpha s$) if it is supported on some
set $S$ with $h^\alpha(S)=0$.
We say that $\mu$ is one-dimensional ($od$) if it is
$\alpha$-continuous for every $\alpha < 1$.
We say that it is zero-dimensional ($zd$) if it is $\alpha$-singular for
every $\alpha > 0$. $\mu$ is said to have exact dimension $\alpha$
if, for every $\epsilon > 0$, it is both
$(\alpha-\epsilon)$-continuous and $(\alpha+\epsilon)$-singular.
Given a (positive, finite) measure $\mu$ and $\alpha\in [0,1]$, we define
\begin{equation}
D_\mu^\alpha(x) \;\equiv\; \limsup_{\epsilon\to 0}
{{\mu((x-\epsilon,\,x+\epsilon))}\over {(2\epsilon)^\alpha}}
\end{equation}
and $T_\infty \equiv \{ x\, |\, D_\mu^\alpha(x)=\infty \}$.
The restriction
$\mu(T_\infty\cap\,\cdot\,)\equiv\mu_{\alpha s}$ is $\alpha$-singular,
and $\mu(({\Bbb R}\setminus T_\infty)\cap\,\cdot\,)\equiv\mu_{\alpha c}$
is $\alpha$-continuous.
Thus, each measure decomposes uniquely into
an $\alpha$-continuous part and an $\alpha$-singular part:
$\mu = \mu_{\alpha c} + \mu_{\alpha s}$.
Moreover, an $\alpha$-singular measure must have $D_\mu^\alpha(x)=\infty$
a.e.\ (with respect to it) and an $\alpha$-continuous measure must have
$D_\mu^\alpha(x)<\infty$ a.e..
It is important to note that $D_\mu^\alpha(x)$ is defined with a
$\limsup$. The corresponding limit need not exist.
We let
${\cal H}_{\alpha c} \equiv \{\psi\,|\, \mu_\psi\; \hbox{\rm is
$\alpha$-continuous}\}$ and
${\cal H}_{\alpha s} \equiv \{\psi\,|\, \mu_\psi\; \hbox{\rm is
$\alpha$-singular}\}$.
${\cal H}_{\alpha c}$ and ${\cal H}_{\alpha s}$ are mutually orthogonal
closed subspaces which are invariant under $H$, and
${\cal H}$ decomposes
as ${\cal H} = {\cal H}_{\alpha c}\oplus{\cal H}_{\alpha s}$.
The $\alpha$-continuous spectrum ($\sigma_{\alpha c}$) and
$\alpha$-singular spectrum ($\sigma_{\alpha s}$) are defined as the
spectra of the restrictions of $H$ to the corresponding subspaces, and
$\sigma = \sigma_{\alpha c}\cup\sigma_{\alpha s}$. Note, in particular,
that when we
classify spectra as being $\alpha$-singular, zero-dimensional,
of exact dimension $\alpha$ etc., we always relate to the corresponding
properties of the spectral measures.
The above scheme for spectral classification can be related to the
dynamics of the underlying quantum systems. A detailed account of such
relations has been given by Last \cite{qdavl}.
It should be pointed out that certain fractal
and multifractal studies of some operators with singular-continuous
spectrum (including some of the examples we discussed above) have been
carried out by several authors \cite{hirkoh,gkpwapbj}.
While such studies
are related to the above decomposition theory, the
relations are generally far from trivial, and we believe that
they are only partial. One should exercise extreme care when attempting
to interpret the results of such studies within the framework of the
scheme discussed above.
>From here on we shall restrict our discussion to one-dimensional
tight-binding Schr\"odinger operators of the form (1).
While we discuss discrete operators, the
subordinacy results we describe are equally valid for continuous
Schr\"odinger operators of the form $-{{d^2}\over {dx^2}}+V$.
Consider first ``half-line'' operators, defined with a phase boundary
condition of the form (2). For such operators, it is well known that
the spectral measures for lattice-point vectors $\delta_n$,
where $\delta_n(m)=\delta_{nm}$,
are all mutually equivalent
(namely, they have the same sets of zero measure).
Thus, the spectral problem reduces to analyzing a single
spectral measure, which we choose to be $\mu=\mu_{\delta_1}$. The
Gilbert-Pearson theory of subordinacy \cite{gilpear} relates the
pointwise
behavior of the spectral measure $\mu$ at some energy $E$ to the
behavior of solutions of the corresponding Schr\"odinger equation
\begin{equation}
\psi(n+1)+\psi(n-1)+V(n)\psi(n)\;=\;E\psi(n)\;.
\end{equation}
Given a solution of (5), we let $\|\psi\|_L$ denote the norm of the
solution $\psi$ over length $L.$
It is useful to consider the length $L$
as a continuous variable (allowed to take any positive real value),
and so we define:
\begin{equation}
\|\psi\|_L\;=\;\left[\sum_{n=1}^{[L]}|\psi(n)|^2\;+\;
(L-[L])|\psi([L]+1)|^2\right]^{1/2}\;,
\end{equation}
where $[L]$ denotes the integer part of $L$.
A (non-trivial) solution
$\psi$ of (5) is called a subordinate solution if for any other
solution $\varphi$ of (5), which is not a constant multiple of $\psi$,
$\lim_{L\to\infty}{{\|\psi\|_L }\over {\|\varphi\|_L}} = 0$. Note that
a subordinate solution must be unique (up to constant multiples).
The Gilbert-Pearson theory relates the decomposition of the spectral
measure $\mu$ to subordinacy of solutions as follows:
The absolutely-continuous part of $\mu$ is supported on the set of
energies for which (5) has no subordinate solutions. (In fact, this set
of energies is, up to a set of both Lebesgue and spectral measure zero,
the set where $\mu$ has a finite non-vanishing derivative.) The singular
part of $\mu$ is supported on the set of energies for which the
solutions which obey the appropriate boundary condition (2) are
subordinate.
Let us now denote by $\psi_1$ the solution of (5) which obeys
the boundary condition (2) and has normalization
$|\psi_1(0)|^2 + |\psi_1(1)|^2 = 1$. Let us denote by $\psi_2$ the
solution of (5) which obeys the orthogonal boundary condition to
(2), namely, $\psi_2(0)\sin\theta - \psi_2(1)\cos\theta = 0$, and has
normalization $|\psi_2(0)|^2 + |\psi_2(1)|^2 = 1$. Our main result is
the following:
\bigskip\noindent
{\bf Theorem 1. } {\it
For any $\alpha\in (0,1)$ and every $E\in{\Bbb R}$,
$D_\mu^\alpha(E)=\infty$ if and only if
$$\liminf_{L\to\infty}
{{\|\psi_1\|_L}\over {\|\psi_2\|_L^\beta}}\;=\; 0\;,$$
where $\beta = {\alpha/(2-\alpha)}$.}
\bigskip\noindent
{\bf Remark. } Theorem 1 is proven with the same ideas used by
Gilbert-Pearson, but it requires some optimization of their analysis.
As a by-product, we also get a simplified proof of their original results.
A key observation is to assign to each
$\epsilon > 0$ a length $L(\epsilon)$ via the equality
$\|\psi_1\|_{L(\epsilon)}\|\psi_2\|_{L(\epsilon)} = 1/(2\epsilon)$,
for which we prove the explicit inequality
$${{5-\sqrt{24}}\over {|m(E+i\epsilon)|}} \;<\;
{{\|\psi_1\|_{L(\epsilon)}}\over {\|\psi_2\|_{L(\epsilon)}}} \;<\;
{{5+\sqrt{24}}\over {|m(E+i\epsilon)|}}\;,$$
where $m(z)$ is the Weyl-Titchmarsh function \cite{cala}.
\bigskip
For spectral analysis, Theorem 1 can be combined with the existence
of generalized eigenfunctions
\cite{cala}, from which one can show that for a.e.\ $E$ with respect to
the spectral measure $\mu$, the solution $\psi_1$ must obey
$\limsup_{L\to\infty}{{\|\psi_1\|_L}\over {L^{1/2}\ln L}} < \infty\,$
and $\,\liminf_{L\to\infty}{{\|\psi_1\|_L}\over {L^{1/2}}} < \infty$.
Another useful fact is the constancy of the Wronskian
$\psi_1(n+1)\psi_2(n) - \psi_2(n+1)\psi_1(n)$,
which implies
$\|\psi_1\|_L \|\psi_2\|_L \geq (L-1)/2$.
We now discuss briefly ``line'' operators.
The spectral measures of a ``line'' operator can be
constructed from those of corresponding two ``half-line'' operators
(a left and a right), and while the relations are not completely
trivial, they do allow an extension of the subordinacy theory to
this case. Gilbert \cite{gilb} has shown that the
absolutely-continuous part
of the spectral measures of a ``line'' operator is supported on
the set of energies for which at least one of the ``half-line''
problems has no subordinate solution. The singular part is supported
on the set of energies for which (5) has a solution which is
subordinate both to the right and to the left. The probing of
dimensional Hausdorff properties is somewhat more delicate in this case
since it involves a $\liminf$ rather than a limit. Nevertheless,
in concrete settings, such as the
ones discussed in Theorems 3 and 4, the required control can be
obtained.
In conclusion we would like to remark the following: The classification
of spectra with respect to dimensional Hausdorff measures extends the
usual spectral classification in a natural way, and provides a useful
way of distinguishing between different kinds of singular-continuous
spectra. The subordinacy theory extends to this point of view in a
natural way, and allows to answer the relevant spectral questions
whenever the nature of the solutions of the corresponding Schr\"odinger
equation is sufficiently well understood. We note, in particular, that
singular-continuous spectrum which occurs in ``close neighborhood'' to
Anderson localization (as in the case of the strongly coupled almost
Mathieu operator or the rank-one perturbed Anderson model) tends to be
purely zero-dimensional; while the singular-continuous spectrum of the
Fibonacci Hamiltonian, which has been identified as having ``critical
states'' in physics literature \cite{hirkoh}, is $\alpha$-continuous
for some positive $\alpha$.
As we were completing this paper we learned of a preprint by
Remling \cite{reml} which obtains a restricted version of Theorem 1.
\bigskip
\centerline{\bf Acknowledgments}
We would like to thank J. Avron and B. Simon for useful discussions.
This research was supported in part by the Institute for Mathematics
and its Applications with funds provided by the National Science
Foundation, and by the Erwin Schr\"odinger Institute (Vienna) where
part of this work was done. The work of S.J. was supported in part by
NSF Grants DMS-9208029 and DMS-9501265.
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\end{document}