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% Anderson Localization for the Almost Mathieu Equation, %
% III. Semi-Uniform Localization, Continuity of Gaps, and Measure of %
% the Spectrum %
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% By: Svetlana Ya. Jitomirskaya and Yoram Last %
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\begin{document}
\begin{titlepage}
\Large
\title{ Anderson Localization for the Almost Mathieu Equation, III.
Semi-Uniform Localization,
Continuity of Gaps, and Measure of the Spectrum}
\large
\author{Svetlana Ya. Jitomirskaya\thanks{Alfred P. Sloan Research
Fellow.
The author was supported in part by NSF Grants DMS-9208029
and DMS-9501265.} \\
Department of Mathematics \\
University of California\\
Irvine, California 92697\\\\ \and
Yoram Last\\
Division of Physics, Mathematics, and Astronomy\\
California Institute of Technology\\
Pasadena, California 91125\\}
\date{July 7, 1997}
\end{titlepage}
\maketitle
\begin{abstract}
\normalsize
We show that the almost Mathieu operator,
$(H_{\omega,\lambda,\theta}\Psi)(n)=\Psi(n+1) + \Psi(n-1) +
\lambda\cos(\pi\omega n +\theta)\Psi(n)$, has semi-uniform (and thus
dynamical)
localization for $\lambda > 15$ and a.e.\ $\omega,\theta$. We also
obtain a new
estimate on gap continuity (in $\omega$) for this operator with $\lambda
> 29$
(or $\lambda < 4/29$), and use it to prove that the measure of its
spectrum is
equal to $|4-2|\lambda||$ for $\lambda$ in this range and {\it all}
irrational
$\omega$'s.
\end{abstract}
\newpage\clearpage
\section{Introduction}
In this paper we study localization for the almost Mathieu operator
$H_{\om,\lb,\th}$
acting on $\ell^2(\bbz)$:
\be
(H_{\om,\lb,\th}\Psi)(n)=\Psi(n+1)+\Psi(n-1)+v_n\Psi(n),
\ee
where the potential $v_n$ is given by
\be
v_n=\lambda \cos(\pi\omega n +\theta).
\ee
For background and some recent results on the almost Mathieu operator
see
\ci{congrl,congrj,jlprl,dual}.
An often used notion of localization is that of pure point spectrum with
exponentially decaying eigenfunctions. It can be expressed by the
following definition:
\noindent{\bf Definition 1.}
$H$ exhibits localization if there exists a constant $\gamma >0$ such
that for any
eigenfunction $\psi_s $ one can find a constant $C(s)>0$ and a site
$n(s) \in {\bbz}$
(center of localization) so that $|\psi_s(k)| \le
C(s)e^{-\gamma|n(s)-k|}$ for any
$k \in {\bbz}.$
Localization, in this sense, for the almost Mathieu operator has been
proven
for $\lambda>15$ and $\omega, \theta$ satisfying certain arithmetic
conditions
\cite{sin,fsw,j1}.
At the same time, the physical understanding of localization is
connected with what is
often called dynamical localization---the non-spreading of initially
localized wave-packets
under the Schr\"odinger time evolution. It can be expressed by, for
example, boundedness in the
time $t$ of $\|xe^{itH}\delta_{0}\|^{2}=(e^{-itH}\delta_0,
x^2e^{-itH}\delta_0)$. An even
stronger condition that we prefer to adapt here is the following:
\noindent{\bf Definition 2.}
$H$ exhibits {\it dynamical localization} if there exists a
constant $\tilde\gamma >0$ such that for any $\ell \in \bbz$,
there exists $\tilde C(\ell) > 0$ so that
\begin{equation}
\sup\limits_{t}|e^{-itH}(n,\ell)|\leq\tilde
C(\ell)e^{-\tilde{\ga}|n-\ell|}.
\end{equation}
Both of the above definitions (as well as Definitions 3 and 4 below),
with $\bbz$ replaced
by $\bbz^d,$ work for operators $H$ acting on $\ell^2(\bbz^d).$
Dynamical localization has been established for the $d$-dimensional
Anderson model in
\cite{ai,dks} (and, in a restricted form, \ci{marsco}); it remained an
open question
for the almost Mathieu operator.
While dynamical localization implies pure point spectrum \ci{ks,cfks},
the converse is
not true in general. There exist operators $H$ with localization, but,
nevertheless,
with\hfil\break
$\limsup_{t\to\infty} \|xe^{itH}\delta_{0}\|^{2}/t^{\alpha} = \infty$
for any $\alpha <2$ \cite{ne} (also see \ci{al}).
In fact, $t^{\alpha}$ can be replaced by an arbitrary function $f(t)=o
(t^2)$.
This example shows that Simon's theorem on the absence
of ballistic motion for operators with pure point spectrum \cite{bal} is
optimal, and that mere
``exponential localization'' of eigenfunctions is not sufficient to
determine the dynamics.
Indeed, localization might not have much physical meaning if there is no
control on
the dependence of $C$ on $n$ (or, equivalently, on the eigenenergy
$E_n)$. In particular, if
the $C(m)$'s are allowed to grow arbitrarily fast with $m$, then
eigenvectors may be
``extended'' over arbitrarily large length-scales and one cannot
effectively define a ``localization length'' corresponding to a typical
size of the ``essential support" of the eigenfunction. An appropriate
level of control
over the $C(m)$'s is given by the following definition, introduced in
\cite{ne}:
\noindent{\bf Definition 3.}
$H$ has SULE ({\it semi-uniformly localized eigenvectors}) if
there exists a constant $\gamma >0$ such that for any $b>0$, there
exists a constant
$C(b) >0$ such that for any eigenfunction $\psi_s $ one can find $n(s)
\in {\bbz}$
so that $|\psi_s(k)| \le C(b)e^{b|n(s)|-\gamma|k-n(s)|}$ for any $k \in
{\bbz}.$
The notion of SULE, which strengthens mere localization, also has a
dynamical counterpart,
which strengthens mere dynamical localization:
\noindent{\bf Definition 4.}
$H$ has SUDL ({\it semi-uniform dynamical localization}) if
there exists a constant $\tilde\gamma >0$ such that for any $b>0$, there
exists a constant
$\tilde C(b) > 0$ so that
\begin{equation}
\sup\limits_{t}|e^{-itH}(n,\ell)|\leq\tilde
C(b)e^{b|\ell|-\tilde{\ga}|n-\ell|}.
\end{equation}
It is shown in \ci{ne} that SULE implies SUDL (and thus dynamical
localization), and
that SUDL, along with a simple spectrum, implies SULE.
There is another direction from which the notion of localization had
been challenged
recently, when Gordon \ci{g} and del Rio-Makarov-Simon \cite{dms} have
shown that
localization can be destroyed by infinitesimally small localized
rank-one perturbations.
However, SULE appears to be a condition that implies certain
semistability
(or physical stability) of localization \cite{ne}. Specifically, if $H$
has SULE and $H'$
is obtained from $H$ by adding a localized rank-one perturbation, then
all the spectral
measures of $H'$
are supported on a set of zero Hausdorff dimension; and, more
importantly,
while $\|xe^{-itH'}\delta_{0}\|$ may be unbounded, it never
grows faster than $C\ln^2(t)$ \ci{ne}.
That makes it particularly interesting to establish SULE for systems
with localization.
For the $d$-dimensional Anderson model, SULE had been derived in \ci{ne}
from the dynamical
estimates of Aizenman \ci{ai} (\cite{dks} in the one-dimensional case).
More precisely,
the dynamical estimates imply SUDL, from which SULE follows by the SULE
$\Leftrightarrow$
SUDL relation. In the present paper, we obtain SULE for the almost
Mathieu operator by direct
analysis of eigenfunctions, and so we deduce SUDL (and, in particular,
dynamical localization)
for this operator by using the SULE $\Leftrightarrow$ SUDL relation in
the other direction.
(It should be pointed out that we do not know any other way to prove
dynamical localization
for the almost Mathieu operator.)
Throughout the paper we will often assume $\om$ to be Diophantine; that
is, that
there exist $c(\om) > 0$ and $1< r(\om) < \infty$ such that
\be |\sin \pi j \om | > {c(\om) \over |j|^{r(\om)}}
\ee
for all $j \not= 0.$
Our main result is the following:
\bt
The almost Mathieu operator has SULE for any Diophantine $\omega$,
$\lambda > 15$,
and a.e.\ $\theta$.
\et
As discussed above this immediately implies:
\bc
The almost Mathieu operator with $\om,\lb,\th$ as in Theorem 1 has
dynamical localization.
\ec
\noindent{\bf Remarks.}
\begin{enumerate}
\item The set of parameters $\om,\lb$ in Theorem 1.1 is exactly the set
for which
localization has been proven \ci{j1}.
\item We will, in fact, obtain a polynomial bound on $C(n(s))$. See
(2.5).
\item The set of $\th$'s for which we show SULE is smaller than the set
of $\th$'s for which
localization has been proven. One can show that there is a zero measure
set of $\th$'s
for which there is localization but not SULE.
\item Similar techniques can be applied to non-Diophantine $\om$'s with
exponential rate
of approximation by rationals, for which localization is proven for
large $\lb$ \ci{j3}.
\item One can study local SULE, that is, semi-uniform localization of
eigenfunctions
with corresponding eigenvalues belonging to a certain interval. Such
local SULE
can be shown to imply dynamical localization for the spectral projection
of $H$ onto
this interval.
Our method allows us to establish local SULE for all energy intervals
where localization
has been shown so far. That includes certain intervals in the center of
the spectrum for
$\lb > 5.6$ \ci{j1}.
\item One can think that a more natural control on the $C(n)$'s is given
by
{\it uniform localization}, UL, defined as localization with a uniform
bound
$C(n) 29$ or $|\lb| < 4/29$,
$$|\sigma(\om,\lb)|= |4-2|\lb||,$$
where $|\,\cdot\,|$ denotes Lebesgue measure.
\et
\noindent{\bf Remark.}
The equality $|\sigma(\om,\lb)|= |4-2|\lb||$ was conjectured by Aubry
and Andre
\ci{aa} to hold for every $\lb$ and irrational $\omega$, and was studied
by Thouless
\ci{t} and by Avron, van Mouche, and Simon \ci{ams}, who proved the
lower bound.
Last \ci{l1} obtained the equality for every $\lb$ and a.e.\ $\om$.
In Section 2 we prove Theorem 1.1 up to some lemmas that we prove in
Sections 3 and 4.
In Section 5 we prove a result about continuity of gaps, which we use in
Section 6 to
prove Theorem 1.3. The Appendix provides a proof for a somewhat
technical lemma that we
use in Section 4.
\section{Proof of Theorem 1.1}
In this section we prove Theorem 1.1 up to some lemmas that will be
proven later. Our
proof consists of two parts:
\begin{enumerate}
\item Obtaining uniform estimates in the proof of localization for the
nonresonant regime.
\item Studying the statistics of resonances, and, particularly, the
dependence
of the number of resonances on the position of the center of
localization.
\end{enumerate}
We introduce the sets of resonant phases:
$$\Theta_{j,k}^s =\left\{\,\theta\, :\, (k+1)^{-s} <
\left|\sin 2\pi \left(\theta +{j \over 2}\omega
\right)\right|r(\om)} \Th^s.$$
Note that
$$\Theta = \{ \theta :\mbox{for every}\; s>r(\om) \;\mbox{the
relation}\;
|\sin 2\pi (\theta +(j/2)\omega )|r(\om),$ has zero Lebesgue measure and so does $\Th$. For
$\theta \notin \Th$,
we define the resonant rate as $s(\theta )\equiv\inf \{s> r(\om):
\theta \notin\Th^s \}+1\ge r(\om)+1.$ For $s> s(\theta )-1$ we define
the $s$-resonant number of $\theta $,
$k(\theta , s) = \#\{m \in \bbn : \theta \in \Th_m^s\}.$
Let $n_1(\theta , s)<\cdots < n_{k(\theta , s)} (\theta , s)$ be the
positions
of resonances: all $m\in \bbn$ such that $ \theta \in \Th_m^s.$
% Let $n_{k(\theta , s)}=\max \{m \in \bbn : \theta \in \Th_m^s\}.$
For a fixed $\theta$, the numbers $k(\theta , s)$ and $n_{k(\theta ,
s)}(\theta , s)$
decrease with $s.$ Also, $s(\theta )$ is an invariant function:
$s(\theta + \om) =s(\theta ),$ and $\Th$ is an invariant set. Let $k
(\theta ) =
k(\theta , s(\theta )).$
We put $n_i(\theta )=n_i(\theta , s(\theta )), \; i=1,\ldots,k(\th),$
and will sometimes write $n_{k(\theta )}$ for $n_{k(\theta )}(\theta
).$
We first obtain some elementary information on the sparseness of the
sequence $n_i(\theta ).$
\bl
Suppose $\theta \in \Th_n^s$, $\om$ satisfies {\rm (1.5)}, and $s >
r(\om)$. Then there
exists a positive constant $c_1(\om)$
such that $\theta \notin \Th_m^s, n |j|,$ imply $|\sin \pi (\ell-j
)\om)|<2n^{-s},$
and so, by (1.5), we have $m \ge |\ell|\ge |\ell -j| -|j|\ge
(c(\om)n^s/2)^{{1 \over r(\om)}}-n.$ \qed
For $\theta \notin \Th, \lambda >15$ and $\om$ satisfying (1.5)
the localization has been proven in \ci{j1}. Thus every eigenfunction
$\Psi_E$
with eigenvalue $E$, attains its maximal value at no more than finitely
many
points. We define $n(E)$ to be the position of the leftmost maximum of
$\Psi_E.$
Our key technical result is
\bl Let $\theta \notin \Th, \lambda >15$ and $\om$ satisfies {\rm
(1.5)}.
Then there exist $C=C(\theta,\om, \lb) <\infty$ and $\ga =\ga(\lb)>0$
such that for
any eigenfunction
$\Psi_E$ of $H_{\theta }$, we have $|\Psi_E(m)| < 2 |\Psi_E(n(E))|
e^{-\ga(\lb)|m-n(E)|}$
for $|m-n(E)| > C(\theta, \om,\lb)\ln n_{k(\theta + n(E)\om)}.$
\el
Lemma 2.2 will be proven in Section 4. For the continuity of gaps and
measure of the
spectrum parts, we will also need a similar, slightly more detailed
statement, Lemma 5.3,
asserting the exponential decay between the resonances.
In order to relate the number of resonances to the position of $n(E)$,
we will
need the following:
\bl
Fix $0< r < \infty.$ Then for a.e.\ $\theta\notin\Th^s$ with $s>r+1$,
there exists $q(\theta,r,s)<\infty$ such that
for every eigenvalue $E$
of $H_{\theta }$, we have $n(E) > n_{k(\theta + n(E)\om)}^r$
if $n_{k(\theta + n(E)\om)}>q(\theta,r,s).$
\el
This lemma will be proven in Section 3.
\noindent{\bf Proof of Theorem 1.1:} Suppose $ \lambda >15$
and $\om$ satisfies (1.5). For a.e.\ $\theta \notin \Th,$ as in Lemma
2.3, we
obtain, using Lemma 2.2, that for any eigenfunction
$\Psi_E$ of $H_{\om,\lb,\th}$ and any $m \in \bbz$,
\be
|\Psi_E(m)| < 2|\Psi_E(n_E)|(n_{k(\theta + n(E)\om)})^{C(\theta,
\om,\lb)\ga(\lb)}
e^{-\ga(\lb)|m-n(E)|)}.
\ee
Thus, if we normalize $\Psi$ by $\Psi_E(n(E))=1$ and fix $0 q(\theta,r,s)$, we have
$p^s_n(\theta ) > n^r$.
\el
\proof Since $| \Th_n^s| = O(n^{-s})$, we have $|\{\theta :
\; p^s_n(\theta )< n^r\}| \le 2n^{r+1-s}.$ Thus, the
Borel-Cantelli lemma implies the result. \qed
\noindent{\bf Proof of Lemma 2.3:} Since
$\theta + n(E)\om \in \Th_{n_{k(\theta + n(E)\om)}}^{s}$,
we have by the definition of $p^s_n(\theta )$ that
$n(E)\ge p_{n_{k(\theta + n(E)\om)}}^s(\theta ).$
So, by Lemma 3.1, we obtain the needed statement.\qed
\section {Uniform decay: Proof of Lemma 2.2}
\sss
Throughout this section we assume that $\om$ satisfies (1.5).
We start with recalling the main definitions and lemmas from the proof
of
localization in \ci{j1}.
We will use the notation $G_{[x_1,x_2]}(E)$ for the Green's function
$(H-E)^{-1}$ of the operator
$H_{\om,\lb,\th}$ restricted to the interval $[x_1,x_2]$ with zero
boundary
conditions at $x_1-1$ and $x_2+1$.
Let us denote
$$P_k(\theta,E)=\det \left[ (H(\theta)-E)\bigg|_{[0,k-1]}\right] .$$
We now fix $E \in R; \;1 < m_1 < {\lambda \over 2},{1\over m_1}< m_2<1,
\eps >1.$
We will need the following quantities:
$$M(E,\lambda ) = {1\over \sqrt{3}}|E+i+\sqrt {(E+i)^2-\lambda ^2}|$$
where by $\sqrt {(E+i)^2-\lambda ^2}$ we understand the value with
positive imaginary part;
$$C(E,\lambda ) = {\ln{\lambda \over 2}\over
\ln M(E,\lambda )}-{3\over 4}
$$ and $c_{\lambda,\epsilon } = {\ln(m_1m_2)\over \ln(\epsilon
M(E,\lambda ))}.$
Given $k>0$, let us denote
$$
A_k(E) = \{x \in \bbz: |P_k(\theta + x\omega, E )| > m_1^k \}.
$$
We will sometimes drop the $E$-dependence, assuming $E$ is fixed.
\vskip .25in
\noindent{\bf Definition.} Fix $E \in \bbr.$ A point $y \in {\bbz}$ will
be called
$(m_2,k)$-regular if there exists an interval $[x_1,x_2]$
containing $y$ such that
$$|G_{[x_1,x_2]}(y,x_i)| k(E,m_2,\theta ,y).$ Of course, in general there is no uniform bound
on
$k(E,m_2,\theta ,y).$ It turns out though that if we pick
$H_{\om,\lb,\th}$ with
an eigenvalue $E$ and take $y$ to be $n(E),$ such a bound becomes
immediate.
\bl
$n(E)$ is $(m,k)$-singular for $k> -{\ln 2\over\ln m}.$
\el
\proof Obvious from (4.1).\qed
We will need to recall several statements from the proof of localization
in \ci{j1}.
\bl {\rm (Proposition 1 of \ci{j1})}
For any $\epsilon > 1$, there exists $k(\epsilon, E )$
such that for $k>k(\epsilon, E )$ and all $\theta$ we have
$$|P_k(\theta ,E)| < (\epsilon M(E,\lambda ))^k.$$
\el
\bl {\rm (Proposition 2 of \ci{j1})} {\it Suppose $y\;\in \; {\bbz}$ is
$(m_2,k)$-singular. Then for any $x$ such that
$k(1-c_{\lambda,\epsilon } )\le y-x \le kc_{\lambda,\epsilon }$,
we have that $x$ does not belong to $A_k$.}
\el
The following lemma is proven in the proof of Lemma 3 in \ci{j1}
\bl
Let ${2m_1\over \lambda} **k_1(b)$, if the two points $x_1,x_2 \in \bbz$ are such that \\
{\rm 1)} $x_i, x_i+1,...,x_i+[{k+1\over 2}]\notin A_k$, $i=1,2,$\\
{\rm 2)} ${\rm dist}(x_1,x_2)>[{k+1\over 2}]$,\\
then
\begin{equation}
\left|\cos\left(2\pi\left(\theta + \left({k-1\over 2}+x_1 +
j_1\right)\omega\right)\right)
-\cos\left(2\pi\left(\theta + \left({k-1\over 2} + x_1 +
j_2\right)\omega\right)\right)\right|
\leq b^{k\over 4}
\end{equation}
for some $j_1,\,j_2 \in [0,[{k+1\over 2}]] \cup
[x_2-x_1,x_2-x_1+k-1-[{k+1\over 2}]]$.
\el
Let $E(\theta)$ be a generalized eigenvalue of $H_{\om,\lb,\th}$, $\Psi
(x)$
the corresponding generalized eigenfunction.
To finish the proof of Lemma 2.2 we will need $E$-independent bounds on
how
large the scale $k$ should be in Lemmas 4.1--4.4. Since Lemmas 4.1,
4.3, 4.4 are
already $E$-independent, we will only have to take care of Lemma 4.2.
It turns out that Lemma 4.2, although formulated and proven in \ci{j1}
in a
non-uniform way, is, in fact, a uniform statement:
\bl
There exists $k(\eps) <\infty$ such that $k(\eps , E) \le k(\eps)$
for any $E \in [-\lb -2, \lb + 2].$
\el
Lemma 4.5 will be proven in the appendix.
\noindent{\bf Proof of Lemma 2.2:} Let $\lambda > 15.$ It was shown in
\ci{j1}
that in such case, $C(E,\lambda)>C(\lb +2, \lb)>0$ for any $E\in
[\lambda -2, \lambda +2].$ Thus, there exist
($E$-independent) $1< m_1<{\lambda \over
2},\; m_2<1$ and $\epsilon >1$ such that
$ 2c_{\lambda,\epsilon }-1 > {1 \over 2}$.
Fix ${{2m_1}\over\lambda} < b< 1.$ Let
$k = |x-n(E)| > \max [k(\eps), k_1(b), -{\ln 2 \over \ln m_2}].$
Suppose $x$ is $(m_2,k)$-singular. Since, by Lemma 4.1, $n(E)$ is also
$(m_2, k)$-singular,
we can, by Lemma 4.3, apply Lemma 4.4 with
$x_1=x-[c_{\lambda,\epsilon }|x-n(E)|]$ and
$x_2=n(E)-[c_{\lambda,\epsilon }|x-n(E)|].$ We then obtain, by (1.5),
(4.2),
$$\left|\sin \left(2\pi \left(\theta +\left({k-1 \over 2}+x_1\right)\om
+
{j_1+j_2 \over 2}\om\right)\right)\right| <
{b^{k/4}(3k/2)^{r(\om)} \over {2c(\om)}} < b^{k/5},\; k> \hat k (\om,
b)$$
with some $j_1,\,j_2 \in [0,[{k+1\over 2}]] \cup
[x_2-x_1,x_2-x_1+k-1-[{k+1\over 2}]].$
Noting that $\theta +n(E)\om \notin \Th_n^{s(\th)}$ for
$n > n_{k(\theta +n(E)\om)}$, we obtain that either
$k < {-5s(\th)\ln n_{k(\theta +n(E)\om)} \over \ln b}$ or
$|k-1-2c_{\lb,\epsilon}k+j_1+j_2|>
b^{{-k \over 5s(\th)}} .$ Since for any allowed $j_1,j_2$ we have
$|k-1-2c_{\lb,\epsilon}k+j_1+j_2|< 5/2 k,$ the second inequality is
contradictory
for $k>\hat k_1(b, s(\th)).$ Thus any $x$ with
\begin{eqnarray*}
k=|x-n(E)| &>& k_0(\epsilon, m_2, b, \om, \th)\\
&=& \max\left(k(\epsilon), k_1(b),
-{\ln 2 \over \ln m_2}, \hat k (\om, b),\hat k_1(b, s(\th)),
{-5s(\th)\ln n_{k(\theta +n(E)\om)} \over \ln b}\right)
\end{eqnarray*}
is $(m_2,|x-n(E)|)$-regular. Repeating the argument of \ci{j1},
we have that there exists an interval $[y_1,y_2]$ containing $x$ such
that
$$|y_i-x| \le |x-n(E)|,\;\; |G_{[y_1,y_2]}(x,y_i)|\le
m_2^{|x-n(E)|},\;i=1,2.$$
Using (4.1), we obtain the estimate:
$|\Psi (x)| \le 2|\Psi(n(E))|e^{-\ga(\lb)|x-n(E)|}, \; \ga(\lb)=-\ln
m_2,$
for any $x$ with $|x-n(E)| > C_1(\om,\th,\lb)\ln n_{k(\theta
+n(E)\om)},$
since our choice of $\epsilon, m_2, b$ was dependent on $\lb$ only. \qed
\section{Continuity of gaps}
\sss
Let $\sigma(\om,\lb,\th)$ denote the spectrum of $H_{\om,\lb,\th}$ which
depends on $\th$ only if $\om$ is a rational.
We denote $S(\om,\lb) = \cup_{\th}\sigma(\om,\lb,\th).$
In this section we establish the following continuity property of the
set
$S(\om,\lb)$:
\bt
For every $\lb > 29$, there are constants $C(\lb),D(\lb) > 0$, such that
if $\om,\om'\in\bbr$
satisfy $|\om -\om'|< C(\lb)$,
then for every $E \in S(\om,\lb)$ there is $E' \in S(\om',\lb)$ with
$$|E-E'|0$ by
Avron, van Mouche, and Simon \ci{ams}.
\item The constant $D(\lb)$ can be effectively estimated. Namely, as can
be seen from the
proof, it can be shown that $D(\lb) < {264 \over 3\ln {\lb \over 2M(\lb
+2,\lb)^{5/6}}}\lb.$
This estimate explodes for $\lb$ approaching the root of $\lb = 2M(\lb
+2,\lb)^{5/6}$ which is
slightly smaller than 29. However, as $\lambda$ grows, the estimate
becomes increasingly better.
\end{enumerate}
Theorem 5.1 immediately implies the following corollary:
\bc
{\rm (i)} For every $\lb > 29$, there are constants $C(\lb),D(\lb) > 0$,
such that if
$|\om -\om'| n^{-s},
\; |j| < |j(n)|.$ By the proof of Lemma 2.1, we have, for $m>n$:
\be
|j(m)| > c_1(\om)|j(n)|^{{s \over r(\om)}}.
\ee
\bl
Let $\lb > 29$, $\om$ satisfies {\rm (1.5)}, $\theta \notin \Th$, $E$ is
an eigenvalue
of $H_{\theta}$ and $\theta+ n(E)\om \in \Th_n^s \cap \Th_m^s, \; nk(\lb, \om)$ we have $$|\Psi_E(m)| < 2|\Psi_E(n(E))|
e^{-\ga(\lb)|m-n(E)|}$$ whenever $3j(n)<|m-n(E)| < 3/8j(m),$ where
$\ga(\lb)$ is the same as in Lemma 2.2.
\el
For the proof of Theorem 5.1, we will need the following elementary
lemma:
\bl
For any Borel set $S \subset [0,1]$ with $|S| > 0$, there exist $\om \in
S$ for which
{\rm (1.5)} holds with $r(\om)=3$ and $c(\om) > {|S|\over 3}$.
\el
\noindent{\bf Proof:}
If not, then for every $\om \in S$ with $r(\om)=3$ (such $\om$'s form a
set of full measure)
there would exist $j \not=0$ such that
\be
|\sin \pi j \om| < {|S| \over {3|j|^3}}.
\ee
If we denote by $A_j$ the set of $\om \in S$ for which (5.2) is
satisfied, we obtain
$|S|\le\sum |A_j| < |S|$, and the contradiction proves the lemma.\qed
\noindent{\bf Proof of Theorem 5.1:}
Fix $\lb > 29.$ By Lemma 5.4, we can find $\om_1 \in (\om,\om')$
satisfying (1.5)
with $r(\om_1)=3,$ such that $c(\om)
> {|\om -\om'|\over 3}.$
Take $E \in
\sigma(\om_1, \lb)$ and $L > k(\lb, \om_1)$ (from Lemma 5.3).
Pick $\theta \notin \Th$ so that $H_{\om_1,\lb,\th}$
has pure point spectrum. Pick $E_1$, an eigenvalue of
$H_{\om_1,\lb,\th}$ with $|E_1 -E| < e^{-3/4\ga(\om_1)L}.$
Let $\Psi_{E_1}$ be the corresponding eigenfunction.
Let $j(n_i)$ be a sequence of resonance positions for the phase
$\theta +n({E_1})\om_1.$ Let $i$ be such that $j(n_i) 29.$
\qed
\noindent{\bf Proof of Lemma 5.3:} The constant 29 was chosen so that
${\ln {\lb\over 2}\over \ln M(31,29)}>5/6.$ This implies that for
$\lb>29$, one can choose
$11$ so that $2c_{\lb, \eps} -1 >2/3.$
That means, by Lemma 4.3, that every $(m_2,k)$-singular point
``produces"
$[2k/3]$ points not belonging to $A_k$.
We will now formulate a version of Lemma 4.4:
\bl
Let ${2m_1\over \lambda} ****k_2(b)$, if the two points $x_1,x_2 \in \bbz$ are such that\\
{\rm 1)} $x_i, x_i+1,...,x_i+2[{k+1\over 3}]\notin A_k$, $i=1,2$,\\
{\rm 2)} ${\rm dist}(x_1,x_2)>[{k+1\over 2}]$,\\
then
\begin{equation}
\left|\cos\left(2\pi\left(\theta + \left({k-1\over 2}+x_1 +
j_1\right)\omega\right)\right)
-\cos\left(2\pi\left(\theta + \left({k-1\over 2} +x_1+
j_2\right)\omega\right)\right)\right|
\leq b^{k\over 4}
\end{equation}
for some $j_1,\,j_2 \in [0,2[{k+1\over 3}]] \cup [x_2-x_1,x_2-x_1+
[{k+1\over 3}]]$.
\el
The proof of this lemma is identical to that of Lemma 4.4 and is given
in \ci{j1}.
Assume now that
$\theta+ n(E)\om \in \Th_n^s \cap \Th_m^s, \; nn(E)>n(E)+j(n)$, we apply Lemma 5.5 with $x_1= x-5/6k$ and
$x_2=n(E)-{k\over 2}.$ We get a resonant condition (5.4) with
$j=4k/3-1+j_1+j_2,$
with $j_1+j_2 \in [-4k/3,4k/3].$ By the same argument we obtain a
contradiction from $k<3/8 j(m)$ if $k$ obeys (5.5).
\item If $x>n(E)+j(n) >n(E)$, we apply Lemma 5.5 with $x_1= x-5/6k$ and
$x_2=n(E)-5/6k.$ The possible values for $j$ in (5.4) are now $j \in
[-2k/3,0]\cup
[k/3,8k/3].$ For the contradiction, we need $3j(n) \max (k(\epsilon),k_2(b), {- \ln 2 \over \ln m_2}, {20 \ln (2c(\om)) \over \ln b} + 3/2)$
and satisfying
$3j(n)< |x-n(E)|<3/8j(m)$ we have that $x$ is $(m_2,|x-n(E)|)$-regular
which,
as before, proves the statement of Lemma 5.3. Here, in estimating how
large $k$ should
be to satisfy (5.5), (5.6), we used that $c(\om) \le 1$ for all $\om$
obeying (1.5).
\qed
\section{Measure of the spectrum: Proof of Theorem 1.3}
\sss
Once we have established the strong version of continuity of gaps,
Theorem 5.1, the
proof of Theorem 1.3 simply follows the lines of the
measure-of-the-spectrum theorem in
\ci{l1,l2}. We present the argument here for the reader's convenience.
Let $G(\om,\lb)$ be the union of the gaps in $S(\om,\lb)$, so that
\be
|S(\om,\lb)|=\max S(\om,\lb) - \min S(\om,\lb) - |G(\om,\lb)|.
\ee
If $\om =p/q$ is a rational, then $S(\om,\lb)$ consists of no more than
$q$ bands,
and $G(\om,\lb)$ of no more than $q-1$ intervals.
It is well known that for any irrational $\om$, there exists a sequence
of rationals
$p_n/q_n\to\om$ such that
\be
|\om - p_n/q_n|< {1 \over {q_n^2}}.
\ee
Avron, van Mouche, and Simon \ci{ams} had proven that for every $\lb$
and every sequence
$\{p_n/q_n\}$ with $p_n$ and $q_n$ relatively prime and $q_n \to
\infty$,
\be
\displaystyle\lim_{n \to \infty} |S(p_n/q_n, \lb)|=|4-2|\lb||.
\ee
(6.3) along with (any) gap continuity implies (see \ci{ams,l2}) the
upper bound
\be
|\sigma(\om,\lb)|\ge |4-2|\lb||
\ee
for any irrational $\om$.
We now obtain from (i) of Corollary 5.2:
\be
|G(\om,\lb)| > |G(p_n/q_n,\lb)| - 2D(\lb)(q_n-1)\lb|\om -p_n/q_n|\ln
|\om -p_n/q_n|.
\ee
By (6.1) and (ii) of Corollary 5.2, this implies:
$$ |S(\om,\lb)| < |S(p_n/q_n,\lb)|+ 2D(\lb)q_n
\lb|\om -p_n/q_n|\ln |\om -p_n/q_n|.$$
By (6.2) and (6.3), we obtain $|\sigma(\om,\lb)|=|S(\om,\lb)|\le
|4-2|\lb||,$
which together with (6.4), completes the proof of Theorem 1.3 for $\lb >
29$.
Since $S(\om,\lb)$ is independent of the sign of $\lb$, it is enough to
have
$|\lb| > 29$. The result for $|\lb| < 4/29$ follows from duality:
$S(\om,\lb) = (\lb/2)S(\om,4/\lb)$. \qed
\sss
\section{Appendix: Proof of Lemma 4.5}
We denote
$$B(\theta ,E) =\left( \matrix{ E-\lambda \cos\theta &1\cr
-1&0\cr}\right)\,,
\quad B_k(\theta ,E)= B(\theta +k\pi\omega ,E).$$
It was shown in the proof of Proposition 1 in \ci{j1} that for any $k >
0$, we
have
$$
|P_k(\theta,E)|\le
\left(\frac{2}{\sqrt{3}}\right)^{k+1}\prod^{k}_{i=0}\|B_j(\theta,E)\|,
$$
where for $A= \left(\matrix{a&b\cr c&d \cr}\right)$,
we use $||A||=\max(\sqrt{a^2+c^2},\sqrt{b^2+d^2}).$
We now want to find $k(\epsilon)$ (not dependent on $E$!)
such that for any $k\ge k(\epsilon)$,
\be
\left(\frac{2}{\sqrt{3}}\right)^{k+1}\prod^{k}_{j=0}\|B_j(\theta,E)\|\le
\epsilon^k\left(\frac{2}{\sqrt{3}}\right)^ke^{{\frac{k}{2\pi}}
\int^{2\pi}_{0}\ln\|B(\theta,E)\|d\theta}
\ee
which can be rewritten as:
\be
\ln\left(\frac{2}{\sqrt{3}}\right) +
\sum^{k}_{j=0}\ln\|B_j(\theta,E)\|\le k\ln \epsilon +
\frac{k}{2\pi}\int^{2\pi}_{0}\ln\|B(\theta,E)\|d\theta.
\ee
This will prove Lemma 4.5 since
$\int^{2\pi}_{0}\ln\|B(\theta,E)\|d\theta=\ln ({\sqrt{3} \over
2}M(E,\lb)).$
Let ${p_n/q_n}$ be the sequence of continued fraction approximants of
$\om.$
Let $n(k)$ be such that $q_{n(k)}\le k < q_{n(k)+1}$. We will write $r$
for
$r(\om)$ and $c$ for $c(\om)$.
\vskip .25in
\noindent{\bf PROPOSITION 1.} {\it For any $f\in C[0,2\pi),k > 0$ we
have}:
\be
\left|\sum^{k-1}_{j=0}f(\theta + j\pi\om)-\frac{k}{2\pi}\int^{2\pi}_{0}
f(\th)d\theta\right| \le k(c^{-1/r}{\frac{n(k)+1}
{k^{1/r}}}){\rm Var}(f).
\ee
\vskip .125in
\proof Writing $k = b_nq_n + b_{n-1}q_{n-1}+\cdots +b_1q_1+b_0$ and
using
the Denjoy-Koksma inequality (see, e.g., Lemma 4.1, Ch.~3 \ci{ksf}), we
get
\be
\left|\sum^{k-1}_{j=0}f(\theta + j\alpha)-\frac{k}{2\pi}\int^{2\pi}_{0}
f(\th)\|d\theta\right| \le (b_0+\cdots +b_n){\rm Var}(f)\le
\left(\sum^{n}_{
i=0}\left[\frac{q_{i+1}}{q_i}\right]\right){\rm Var}(f).
\ee
Since (1.5) implies $q_{i+1} < \frac{q^{r}_{i}}{c},$ we have
$\frac{q_{i+1}}{q_i} < \frac{q_{i+1} }{ (cq_{i+1})^{1/r} } =
\frac{q^{1-1/r}_{i+1} }{ c^{1/r} }$.
The right-hand side of (7.4) can now be estimated as
$$
\le
\left(c^{-1/r}\sum^{n(k)}_{i=1}q^{1-1/r}_{i}+\frac{k}{q_{n(k)}}\right){\rm
Var}(f) <
\left(c^{-1/r}n(k)q^{1-1/r}_{n(k)}+\frac{k}{q_{n(k)}}\right){\rm
Var}(f).
$$
Since $k < q_{n(k)+1} \le \frac{ q^{r}_{n(k)}}{c}$, we have
$q_{n(k)}\ge (ck)^{1/r}$, and $\frac{k}{q_{n(k)}}\le c^{-1/r}k^{1-1/r}$.
Thus we can continue our estimate as
$$\le(c^{-1/r}(n(k)+1)k^{1-1/r}){\rm Var}(f).\qed$$
Proposition 1 implies that
\begin{eqnarray*}
&&\left|\sum^{k}_{j=0}\ln\|B_k(\theta,E)\|
-\frac{k}{2\pi}\int^{2\pi}_{0}\ln\|B(\theta,E)\|d\theta\right|\\
&&\qquad\qquad\qquad\qquad\qquad\qquad
\le\; k\cdot\frac{c^{-1/r}n(k)}{k^{1/r}}\max_{-\lambda -2 < E < \lambda
+ 2}
{\rm Var}(\ln((E-\lambda\cos \theta)^2+1)).
\end{eqnarray*}
Denote $\displaystyle\max_{-\lambda -2 < E < \lambda + 2}{\rm Var}
(\ln((E-\lambda\cos \theta)^2+1)) = A(\lambda)$.
Since $q_n \ge (\sqrt {2})^n$, $n \ge 2$, we have that $n(k)\le
\frac{\ln k}{\ln\sqrt{2}}$,
and we can find $k(\epsilon)$ such that for any $k > k(\epsilon)$ we
have
$$
\frac{c^{-1/r}n(k)}{k^{1/r}}A(\lambda) + \frac{\ln(2/\sqrt{3})}{k}
\le\ln\epsilon
$$
which implies (7.2). This completes the proof of Lemma 4.5. \qed
\section{ Acknowledgments}
S.J. would like to thank J.~Avron for the hospitality of the ITP at the
Technion, and B.~Simon
for the hospitality of Caltech; where parts of this work were done. We
are also grateful to
Ya.~Sinai and to B.~Simon for enlightening conversations.
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