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\begin{document}
\begin{titlepage}
\Large
\title{Anderson Localization for the Almost Mathieu Equation, \\ II:
Point Spectrum for $\lambda >2$}
\large
\author{Svetlana Ya. Jitomirskaya \thanks{Permanent address: International
Institute of Earthquake Prediction Theory and Mathematical Geophysics.
Moscow, Russia .} \\
Department of Mathematics \\
University of California\\
Irvine, California 92717 }
\date {}
\end {titlepage}
\maketitle
\begin{abstract}
\normalsize
We prove that for any $\lambda>2$ and a.e. $\omega,\theta$
the point spectrum of the almost Mathieu operator $(H(\theta)\Psi )_n =
\Psi_{n-1} +\Psi_{n+1} + \lambda
\cos(2\pi(\theta+n \omega))\Psi_n$ contains the essential closure
$\sigma_{ess}$ of the spectrum. Corresponding eigenfunctions decay
exponentially. The singular continuous
component, if it exists,
is concentrated on a zero measure set which is
nowhere dense in $\sigma_{ess}.$
\end{abstract}
\noindent{\bf 1. INTRODUCTION}
\vskip .25in
This paper is another attack on the almost-Mathieu
operator on $\ell ^2({\bf Z}):$\\
$$
(H(\theta)\Psi )_n = \Psi_{n-1} +\Psi_{n+1} + \lambda
\cos(2\pi(\theta+n \omega))\Psi_n
$$
This simple-looking operator has been studied extensively for many years.
We refer the reader to [1,2] for - still incomplete - list
of references.
The critical (and physical) value of the coupling constant $\lambda$ is
$\lambda=2 $ (from now on we assume without loss of generality that
$\lambda \ge 0$); it is believed that at $\lambda=2$ there occurs a transition
from pure absolutely continuous to pure point spectrum. The $\omega$
here is supposed to be "irrational enough" since for rational $\omega$ the
potential is periodic and the spectrum is absolutely continuous for all
$\lambda$;
for Liouville $\omega$ (abnormally well approximated by
rationals) and $\lambda>2 $
the spectrum of $H(\theta)$ is purely singular continuous [3,4].
Up to recently the only
rigorous reason for this belief was that for $\lambda >2 $ and
irrational $\omega$ the
Lyapunov exponents are positive;
that proves the absence of the absolutely continuous part of the
spectrum [5,6]. By Aubry duality
there is no pure point spectrum for $\lambda <2 $
[7]. The latest development for any $\lambda<2 $ is the proof of existence
of the absolutely continuous spectrum
that was given by Last
[8] for a.e. $ \omega,\theta $ and by Gesztesy and Simon [13] for all $
\omega,\theta $;
Last [8] also proved that for a.e. $ \omega $
the absolutely continuous spectrum, $\sigma_{ac},$
coincides with
the spectrum, $\sigma$, up to a set of zero Lebesque measure.
In the present paper we study $\lambda'$s above the critical value.
Localization, i.e., pure point spectrum with
exponentially decaying eigenfunctions,
was proved by Sinai [9] and Fr\"{o}hlich, Spencer and Wittwer [10] in the
perturbative regime:
$\lambda $ "big enough." In [11] we developed a nonperturbative way of
proving localization, which worked for
$\lambda\ge 15$. The method also
allowed to prove localization in the middle of the spectrum for $\lambda\ge
5.4.$ Those restrictions on $\lambda$ were caused exclusively by the roughness
of the
a priory estimates from above on the growth of the formal solution $\Psi_E,$ i.e.,
on the Lyapunov exponents. In this paper we
notice that the results of [8,13] together with Aubry duality and certain
regularity of the Lyapunov exponent [12] give a much nicer estimate for
"most of" the Lyapunov exponents.
We combine this soft argument with the method of [11] to prove the existence of
"a lot of" pure point spectrum for a.e. $\omega.$ The result holds for any
$\lambda>2$ but we have to pay for that by not being able to rule out the
possibility of some singular continuous spectrum.
We already denoted the spectrum of $H(\theta)$ by $\sigma ,$ and now denote the
singular continuous
part of the spectrum
by $\sigma_{sc}, $
and the pure
point part of the spectrum
by $\sigma_{pp} .$ It is well known that for irrational $\omega$ the sets
$\sigma,\sigma_{sc} $ and
$\sigma_{pp} $ (understood as the
closure of the set of eigenvalues) do not depend on the phase
$\theta$ for a.e. $\theta.$ Though for $\lambda < 2$ the results on the
absolutely continuous spectrum probably hold for every $\theta$ (see [13]),
the pure
point spectrum for $\lambda > 2$ should be an essentially "a.e." result, since
for generic $\theta$ the spectrum is purely singular continuous in this case
[14].\\
We put
$$\Theta=\{ \theta :\mbox{for every}\; s>1 \;\mbox{the relation}\;
|\sin (\theta +{k \over 2}\omega )|2 $ the arithmetic nature of $\omega$
plays a major role.
Let ${p_n\over q_n }$ be the $n^{th}$ continuous fractions approximant of
$\omega.$ Throughout the rest of the paper we assume that $\omega$ is
Diophantine, i.e., an
irrational such that for some $r > 1$ and $C>0$ we have
$|q_n\omega -p_n|>C q_n^{-r}.$
In Theorem 1 we will also use another, rather technical, restriction on $\omega $
\begin{equation}
{C \over q_n^{r}}<|q_n\omega -p_n| = {o(1)\over q_n}\;\;\mbox{as}\;
n\rightarrow \infty
\end{equation}
The set of $\omega'$s described in (1) has full Lebesque measure [15].\\
For $\lambda>2$ the spectrum $\sigma$ is a set of positive Lebesque measure
[13]:$|\sigma|\ge 2\lambda-4$ . We put
$$\sigma_{ess}=\{E\in \sigma: \;\mbox{for any}\; \epsilon >0 \; |(E-\epsilon, E+
\epsilon)\cap \sigma |>0\}.$$
We will prove
\vskip .25in
\noindent{\bf Theorem 1.} {\it Suppose $\omega$ satisfies (1). Then
for any $2<\lambda < 15$ we have
\begin{enumerate}\item $|\sigma_{pp}|=|\sigma|,$
\item $\sigma_{sc} ,$ if it exists, has measure zero and is nowhere dense in
$\sigma_{ess}
$ (in the relative topology).
\item For $\theta \notin \Theta$ the set of eigenvalues of the operator
$H(\theta)$ is dense in $\sigma_{pp}$ and the corresponding eigenfunctions are
exponentially
decaying .
\end{enumerate}}
\vskip .25in
The zero measure set of Diophantine $\omega$'s not satisfying (1) happens
to include the golden mean - the most popular object for numerical
studies. Without technical assumption (1) we can prove a slightly weaker
result:
\vskip .25in
\noindent{\bf Theorem 1'.} {\it For any Diophantine $\omega$ and any $2<\lambda
< 15$
we have
\begin{enumerate}\item $|\sigma_{pp}|\ge 2\lambda-4.$
\item There exists a closed set $A\subset \sigma_{pp}, \; |A|\ge 2\lambda-4,$ such
that $\sigma_{sc}$ is nowhere dense in $A.$
\item For $\theta \notin \Theta$ the set of eigenvalues of the operator
$H(\theta)$ is dense in $A$ and the corresponding eigenfunctions are
exponentially
decaying .\end{enumerate}}\vskip .25in
\noindent{\bf Remarks.}
\begin{enumerate}\item As can be seen from the proof, in order to to prove the
complete localization for $\lambda >2$ it suffices to
prove the continuity of the Lyapunov exponent $\gamma(E).$ For $\lambda$ large
enough the continuity of $\gamma(E)$ follows from the proof in [9]. This
continuity would also be enough to rule out the singular continuous spectrum
for $\lambda <2$ [16].
\item An estimate on the rate of the exponential decay of the eigenfunctions
can be easily obtained from the proof and is given by
$|\Psi(x)|\le const
\left({\lambda\over 2}
\right)^{(-{1\over 4}+\epsilon )|x|}$ for any $\epsilon>0.$
This decay is slower than what is
suggested by the Lyapunov exponent.
\item Theorems 1 and 1' can be proved with $\theta$ satisfying a weaker condition.
Namely, let us fix $1<\mu<\left({\lambda\over 2}\right)^{{1\over 16}}$
and put $\Theta_{\mu}=\{ \theta : |\sin(\theta +{k \over 2}\omega) |<\mu^{-k}$
holds for
infinitely many $k\} .$ Then for $\theta \notin \Theta_{\mu}$ the same
result holds, but the estimate on the decay of the eigenfunctions will be:
$|\Psi(x)|\le const
\left(\left({\lambda\over 2}
\right)^{-{1\over 4}}\mu^4\right)^{|x|}.$ It is quite
clear that the rate of decay should depend on $\mu .$ This result can be
compared with the fact that for $\theta \in \Theta_{\mu},\;\mu$ sufficiently
large, the operator $H(\theta)$ has no pure point component in the spectrum [14].
\item The Diophantine property can be made weaker (see the comment in [11]) but
not too weak since for
Liouville $\omega$ the spectrum of $H(\theta)$ is purely singular continuous.
The upper bound in (1) appears here only because of the same bound in [8] and is
presumably an artifact of the Last's (and, of course, of the present) proof.
\item All the results certainly hold for $\lambda \ge 15$ as well [11].
\end{enumerate}
In Section 2 we
describe the method for proving localization which is a certain
modification of the method of [17] (see also [10], [18]). In Section 3
we formulate our main technical result, Theorem 2, which is very similar to
Theorem 2 in [11], and present the soft arguments which prove Theorems 1 and 1'
from Theorem 2. In Section 4 we prove Theorem 2.
\vskip .50in
\noindent{\bf 2. THE GENERAL SETUP.}
\vskip .25in
We start with some definitions\\
\noindent{\bf Definition.} A formal solution $\Psi _E(x) $ of the equation
$H(\theta)\Psi _E = E\Psi _E$ will
be called {\bf a generalized eigenfunction} if $ \Psi_E(x) \le C(1+|x|)$
for some $C=C(\Psi _E)<\infty $. The energy $E$ for which such a solution
exists will be called {\bf a generalized eigenvalue}. \vskip .25in
It is well known that to prove pure point spectrum one only needs to prove
that generalized eigenfunctions belong to $\ell ^2$ (see [17]).\\
We denote the Green's function
$(H-E)^{-1}$ of the operator
$H(\theta )$ restricted to the interval $[x_1,x_2]$ with zero boundary
conditions at $x_1-1$ and $x_2+1$ by $G_{[x_1,x_2]}(E).$\\
Let us fix a number $m<1.$
\vskip .25in
\noindent{\bf Definition.} A point $y\;\in \;{\bf Z}$ will be called
$(m,k)$-regular if \underline {there exists} an interval $[x_1,x_2]$
containing $y$ such that
$$|G_{[x_1,x_2]}(y,x_i)|k_0$ the
point $x$ is $(m,k)$-singular.}\vskip .25in
Lemma 1 is the same kind of statement as Lemma 3.1 in [10]
and so is the proof.\hfill $\Box$ \\
Suppose one can prove that $(m,k)$-singular points are "far apart" which we
formulate as the following\\
\noindent{\bf Quasilemma.}{ \it Suppose the points $x_1,x_2$ are
$(m,k)$-singular, k is large enough and
$dist(x_1,x_2)>{k\over 2},\;$
then
$dist(x_1,x_2) > k.$}\vskip .25in
Then the rest of the proof can be organized as follows.
Assume without loss of generality that $\Psi (0) \not= 0$.
Let $|x|$ be bigger than $k_0(0,m,\theta ,E)$ and sufficiently large so that
we can apply Quasilemma with $k=|x|.$
Suppose $x$ is $(m,|x|)$-singular. Since 0 is $(m,|x|)$-singular, the
Quasilemma asserts that $dist(0,x)=|x| > |x|.$
The contradiction implies that $x$ is $(m,|x|)$-regular. Thus we have that
there exists an interval $[x_1,x_2]$ containing $x$ such that
$$|x_i-x| \le |x|,\;\; |G_{[x_1,x_2]}(x,x_i)|\le m^{|x|},\;i=1,2.$$
We now can use the formula
$$\Psi (x)=G_{[x_1,x_2]}(x,x_1)\Psi (x_1-1)+
G_{[x_1,x_2]}(x,x_2)\Psi (x_2+1)$$
to obtain the estimate:
$$|\Psi (x)| \le 2C(1+2|x|)m^{|x|}.$$
This argument shows that the localization follows immediately if we make a Lemma
out of the Quasilemma, i.e. provide it with hypotheses and, of course, with a
proof.
\vskip .25in
\noindent{\bf 3. PROOF OF THEOREM 1.}
\vskip .25in
We define
$$B(\theta ,E,\lambda) =\left( \matrix{ E-\lambda \cos2\pi\theta &-1\cr
1&0\cr}\right) ,
B_k(\theta ,E,\lambda)= B(\theta +k\omega ,E,\lambda),$$
$$M_k(\theta ,E,\lambda)= B_k(\theta ,E,\lambda)... B_0(\theta ,E,\lambda).$$
The Lyapunov exponent $\gamma(E,\lambda)$ is defined by $\gamma(E,\lambda)
=\displaystyle
{\inf }_k \int _0^
{2\pi }|k|^{-1}\ln ||M_k(\theta ,E,\lambda)||d\theta.$\\
Theorem 1 will follow from\\
\noindent{\bf Theorem 2.} {\it Let $\theta, \omega $ be as in Theorem 1 and
suppose $E,\lambda$ are such that ${\ln{\lambda \over 2} \over \gamma(E,\lambda)}
>{3\over 4}$ and $E$ is a generalized eigenvalue of $H(\theta).$ Then the
corresponding generalized
eigenfunction $\Psi(x)$ is exponentially decaying.} \\
\noindent{\bf Proof of Theorem 1.} Take any $\lambda >2.$ The Aubry duality and
the Thouless formula yield the famous relation [19,6,7]
\begin{equation}
\gamma(E,\lambda)= \ln{\lambda \over 2} + \gamma\left({2E \over \lambda} ,
{4 \over \lambda}\right)
\end{equation}
Since ${4 \over \lambda} <2,$ we can usethe result of Last [8] that for a.e.
$E\in \sigma(H(\theta ,{4 \over \lambda})) $ the
Lyapunov exponent $\gamma(E,{4 \over \lambda})$ is equal to $0.$\\
Craig and Simon have proven [12] that $\gamma(E)$ is continuous at points with
$\gamma(E)=0.$
We set
$$G= \{E: \gamma\left( E,{4 \over \lambda}\right) < {1\over 3}\ln{\lambda
\over 2}\}.$$
The set $2 \lambda^{-1}G \cap \sigma$ contains an open (in $\sigma_{ess}$)
dense
(in $\sigma_{ess}$) set of measure $|\sigma|$, thus
the set $\sigma \backslash 2 \lambda^{-1}G$ is a nowhere
dense in $\sigma_{ess}$ set of zero measure.
For $E\in 2 \lambda^{-1}G$ relation (2) implies that $\gamma(E,\lambda) <
{4\over 3}\ln {\lambda \over 2}$ and we can apply Theorem 2 to obtain that
$\sigma _c \cap 2 \lambda^{-1}G=\emptyset $ which gives the statement of the
theorem.\hfill $\Box$\\
\noindent{\bf Proof of Theorem 1'.} Gesztesy and Simon [13] have
proven
that for any irrational $ \omega$ and any $\lambda$ the inequality $|\sigma_{ac}
|\ge 4-2\lambda$ holds. This plus Ishii-Pastur-Kotani theorem (see [1]) and the
same argument as above implies the result.\hfill $\Box$
\vskip .50in
\noindent{\bf 4. PROOF OF THEOREM 2.}
\vskip .25in
As we learned from Section 2 it suffices to prove the Quasilemma under the
conditions of Theorem 2.\\
Let us put
$$P_k(\theta,E)=\det \left[ (H(\theta)-E)\bigg|_{[0,k-1]}\right] .$$
$P_k(\theta,E)$ is an even function of the argument
$\theta+{k-1 \over 2} \omega$ and can be written as a polynomial of
the degree $k$ in $\cos(2\pi(\theta+{k-1 \over 2} \omega))$ :
$$
P_k(\theta,E) = \displaystyle \sum_{j=0}^k
b_j(E) \cos^j(2\pi(\theta+{k-1 \over 2} \omega))
$$
To simplify the notation we will sometimes omit the dependence on E.
It is easy to see that $b_k = 2\lambda ^k$.
We now fix $E \in R; \;1 < m_1 < {\lambda \over 2}.$
Given $k>0$ we set
$$
A_k = \{x: |P_k(\theta + x\omega )| > m_1^k \}
$$
For any $x_1,\; x_2=x_1+k-1,\; x_1 \le y \le x_2$ we have
\begin{eqnarray}
|G_{[x_1,x_2]}(x_1,y)| &=& \left| { P_{x_2-y}(\theta + (y+x_1)\omega )
\over P_k(\theta +x_1\omega )}\right| \nonumber\\[0.25 in]
|G_{[x_1,x_2]}(y,x_2)| &=& \left| { P_{y-x_1}(\theta +x_1\omega )
\over P_k(\theta +x_1\omega )}\right|
\end{eqnarray}
\\
Since $P_k(\theta ,E)$ is one of the entries of the matrix $M_k(\theta ,E)$ we
have an evident upper bound $P_k(\theta ,E)\le ||M_k(\theta ,E)||.$ It is
proved in [12] that for {\bf all} $\theta ,E,\lambda $ the inequality
$$\limsup |k|^{-1}\ln ||M_k(\theta ,E,\lambda)||\le \gamma(E,\lambda)$$ holds.
Thus for any $\epsilon > 1$ there exists $k(\epsilon,\theta, E )$
such that for
$k>k(\epsilon,\theta, E )$ we have
\begin{equation}
|P_k(\theta ,E)| < (\epsilon \gamma(E,\lambda ))^k.
\end{equation}
It follows from (3),(4) that for $x_1\;\in \; A_k, \;
x_2= x_1+k-1, \;k>k(\epsilon,\theta, E ),\;{1\over m_1}< m<1$ and
$y\in [x_1,x_2] $ such that
$$k\left( 1- {\ln(m_1m)\over \ln(\epsilon\gamma(E,\lambda ))}\right) < y -x_1<
k\left( {\ln(m_1m)\over \ln(\epsilon \gamma(E,\lambda ))}\right) $$
we have
\begin{equation}
|G_{[x_1,x_2]}(y,x_i)|k_1(\theta ,E )$ if the points $x_1,x_2$ satisfy
\\
1) $x_i, x_i+1,...,x_i+[{k+1\over 2}]\notin A_k$, $i=1,2$\\
2) $dist(x_1,x_2)>[{k+1\over 2}]$\\
then
$$dist(x_1,x_2) >\alpha^{ k}$$
with $\alpha=\alpha(m_1,m,\lambda ,E,s,r)>1.$}
\vskip .25in
Indeed, let us
suppose that ${\ln{\lambda \over 2} \over \gamma(E,\lambda)}
>{3\over 4}.$ Then there exist $1< m_1<{\lambda \over
2},\; m<1$ and $\epsilon >1$ such that
$ 2c_{\lambda,\epsilon }-1 > {1 \over 2}$.
Let $k$ be bigger than $\max[k(\epsilon ,\theta ,E),k_1(\theta ,E)]$.
Suppose $x'_1, x'_2$ are $(m,k)$-singular and $dist(x'_1, x'_2) >[{k+1\over 2}].$
Since $2c_{\lambda,\epsilon }-1 > {1 \over 2},\;$ we obtain
using Proposition 1 that the points $x_1=x'_1-c_{\lambda,\epsilon }k$ and
$x_2=x'_2-c_{\lambda,\epsilon }k$ satisfy the conditions of Lemma 2.
Applying Lemma 2 we get that $dist(x'_1, x'_2) >
\alpha^{ k}>k$ for large $k$ which proves the
statement of the Quasilemma.
\hfill $\Box$ \vskip .50in
\noindent{\bf 5. Acknowledgement}
\vskip .25in
It is a pleasure to thank A. Klein for many useful
discussions.
\newpage\clearpage
\begin{center}
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\end{center}
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\end{document}