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\begin{document}
\begin{titlepage}
\Large
\title{Anderson Localization for the Almost Mathieu Equation: A
Nonperturbative Proof.}
\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 diophantine rotation angle $\omega$
and a.e. phase $\theta$
the almost Mathieu operator $(H(\theta)\Psi )_n = \Psi_{n-1}
+\Psi_{n+1} + \lambda
\cos(2\pi(\theta+n \omega))\Psi_n$ has pure point spectrum with exponentially
decaying eigenfunctions for $\lambda \ge 15.$ We also prove the existence of
some
pure point spectrum for any $\lambda \ge 5.4.$
\end{abstract}
\newpage\clearpage
\noindent{\bf 1. INTRODUCTION}
\vskip .25in
In this paper we study localization for the almost-Mathieu
operator on $\ell ^2({\bf Z}):$\\
$$
(H(\theta)\Psi )_n = \Psi_{n-1} +\Psi_{n+1} + \lambda
\cos(2\pi(\theta+n \omega))\Psi_n
$$
The almost-Mathieu operator attracted a lot of interest especially in
the last decade. For
references before 1985 see [1]. Some of the later
references are [2-7 ].
While it is very well understood and commonly believed that for
diophantine $\omega $ and $|\lambda|> 2$ the operator $H(\theta)$
should
have pure point spectrum
with exponentially decaying eigenfunctions for almost every $\theta $ this is
not yet rigorously
proved .
Localization
was proved by Sinai [2] and Fr\"{o}hlich, Spencer and Wittwer [3] in the
perturbative regime:
$|\lambda| $ "big enough." The methods developed in [2] and [3] are very
different but have to
overcome one common difficulty: the absence of a Wegner-type estimate
that would give
control of eigenvalue splitting . In both [2] and [3]
the gaps
in the spectrum were
estimated by special inductive multi-scale procedures and these were
the hardest parts of the
proofs. The estimation of the gaps is in fact the main difficulty in
the proof of
localization for any potential that is not random enough to be treated
by a Wegner-type argument.
In this paper we present a new approach to this difficulty: avoiding
rather than fighting it.
That makes the proof of localization shorter and more elementary.
The proof presented uses many basic ideas of the one in [3], but it
is neither a perturbative nor
a multiscale type of argument. Namely just one big enough scale
is employed. It allows
extension of the proof to the range of "not so big" values of
$\lambda $.(from now on we assume without loss of generality that
$\lambda \ge 0.$) The argument as it is works
for $\lambda \ge 15$, but we believe it should be extendible for smaller
$\lambda $, may be even up to
the critical value $\lambda = 2 $. The number 15 (which is actually
14.97...) looks
ugly of course and is an artifact of the "short proof". Surprisingly
the method also allows us to prove
localization in the center of the band for $\lambda \ge 5.4$ which somehow
contradicts the
intuition gained in
experience with random potentials that localization should be easier
to prove near the edges of
the spectrum.
There is of course a price to pay: the method looses the explicit
control of the appearance of the
gaps and, unlike the method of [2], cannot prove the Cantor character
of the spectrum.
Nor can it study the scaling properties of eigenfunctions.\\
We will denote the spectrum of $H(\theta)$ by $\sigma $ and the pure
point part of the spectrum
by $\sigma_{pp} $. It is well known that for irrational $\omega$
both $\sigma $ and
$\sigma_{pp} $(understood as the
closure of the set of eigenvalues) do not depend on the phase
$\theta$ for a.e $\theta$. But the dependence on the two other
parameters, $\lambda$ and $\omega,$
is very nontrivial. For $\lambda >2 $ and irrational $\omega$ the
Lyapunov exponent is positive;
that proves the absence of the absolutely continuous part of the
spectrum [8,9]. By Aubry duality
for $\lambda <2 $ there is no pure point spectrum [4], and there was
recently great progress [7] in
proving that in this case the spectrum is absolutely continuous for any
irrational $\omega$.
But for $\lambda >2 $ the arithmetic nature of $\omega$ starts to
play a major role. Despite the
positivity of the Lyapunov exponents, for Liouville $\omega$
(abnormally well approximated by
rationals) the spectrum of $H(\theta)$ is purely singular continuous.
Anderson localization
(pure point spectrum with exponentially decaying eigenfunctions) is
expected (and proved for $\lambda $
large) only for "typical" (diophantine or may be satisfying a
slightly weaker property)
values of $\omega$.
We say that an irrational number $\omega$ is diophantine if there
exist $r > 1$ and
$C>0$ such that
\begin{equation}
|q\omega -p| > {C \over q^{r}}
\end{equation}
for any
$p,q\in {\bf N}.$
Throughout this paper $\omega$ will be assumed to be diophantine.
We will prove\\
\vskip .25in
\noindent{\bf Theorem 1.}{\it
\begin{enumerate}\item For $\lambda > 5.4 $ there exists an interval
$[-\epsilon(\lambda) ,\epsilon(\lambda)]$ such that $\sigma_{pp}\cap
[-\epsilon(\lambda) ,\epsilon(\lambda)] \not= \emptyset$ and the spectrum of
$H(\theta)$
in this interval is (for a.e. $\theta$) pure point with exponentially
decaying eigenfunctions.
\item For $\lambda \ge 15 $ we have $\epsilon(\lambda)\ge \lambda+2$ which
means Anderson localization for a.e. $\theta.$
\end{enumerate}
}
\vskip .25in
\noindent{\bf Remarks.}
\begin{enumerate}\item The function $\epsilon(\lambda)$ is monotone increasing
in $\lambda $ and so is the function $\epsilon(\lambda)-\lambda.$ For example
$\epsilon(6)\ge 4,\;\epsilon(9)\ge 9$ and $\epsilon(14)\ge 15.$
\item As can be seen from the proof, the set of $\theta$'s for which we prove
localization, is given by an explicit condition (e.g., $\theta=0$ belongs to
this set). Thus Theorem 1 provides a lot of "concrete"
examples of operators with pure point spectrum.
\item With only minor changes in the argument (mainly in the proof of the
Proposition 5)
the same result can be proved for
$\omega$ satisfying a weaker condition than (1). Namely, let $g(x)$ be any
function such that for any $c>0$ we have $e^{-cx}=o(g(x)).$ Then the
same result holds for all $\omega$ such that for any
$p,q\in {\bf N}$ we have $|q\omega -p| > g(q).$
\end{enumerate}
\vskip .50in
\noindent{\bf 2. Proof of Theorem 1}
\vskip .25in
Let
$$M(E,\lambda ) = {1\over \sqrt{3}}|E+i+\sqrt {(E+i)^2-\lambda ^2}|$$
where by $\sqrt {(E+i)^2-\lambda ^2}$ we understand the value with
positive imaginary part.
We set
$$C(E,\lambda ) = {\ln{\lambda \over 2}\over
\ln M(E,\lambda )}-{3\over 4}
$$
Theorem 1 follows immediately from\\
\noindent{\bf Theorem 2.} {\it
If an interval $I$ is such that $C(E,\lambda )>0$ for all $E\in I$ then
the spectrum of $H(\theta)$ in $I$ is pure point and the corresponding
eigenfunctions are exponentially decaying.} \\
\noindent{\bf Proof of Theorem 1.} It is easy to see that $C(E,\lambda )$ is
monotone decreasing in $|E|$
and increasing in $\lambda $. Since $0\in \sigma ,$ (see, e.g., [6])
the first statement now follows from $C(0, 5.4)>0.$
For all $\lambda $ we have $\sigma \subset [- \lambda-2, \lambda+2].$
In order to finish the proof
of the second statement it remains to notice that
$C(\lambda+2,\lambda )$ is monotone increasing in $\lambda $ and
that $C(16.97, 14.97)>0.$ \hfill $\Box$ \vskip .50in
\noindent{\bf 3. Proof of Theorem 2}
\vskip .25in
We will use the notation $G_{[x_1,x_2]}(E)$ for 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 $x_2+1$.
Let us denote
$$P_k(\theta,E)=\det \left[ (H(\theta)-E)\bigg|_{[0,k-1]}\right] .$$
Notice that $P_k(\theta,E)$ 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 $k>0$ let us denote
$$
A_k = \{x: |P_k(\theta + x\omega )| > m_1^k \}
$$
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}
\\
{\bf Proposition 1.}
{\it For any $\epsilon > 1$ there exists $k(\epsilon, E )$
such that for
$k>k(\epsilon, E )$ and all $\theta$ we have
$$|P_k(\theta ,E)| < (\epsilon M(E,\lambda ))^k.$$ }
{\bf Proof.}
We define
$$B(\theta ,E) =\left( \matrix{ E-\lambda \cos2\pi\theta &1\cr
-1&0\cr}\right) ,
B_k(\theta ,E)= B(\theta +k\omega ,E),$$
$$M_k(\theta ,E)= B_k(\theta ,E)... B_0(\theta ,E).$$
For $A= \left( \matrix{a&b\cr c&d \cr}\right) $ we define the norm of $A$
as $||A||=\max(\sqrt{a^2+c^2},\sqrt{b^2+d^2}).$\\
Let $||A||_{op}$ be the operator norm.
It is well known that $| P_k(\theta ,E)|\le || M_k(\theta ,E)||_{op}\le
\displaystyle \prod_{j=0}^k||B_j(\theta ,E)||_{op}. $\\
For any matrix $A$ of the form $\left( \matrix{ c &1\cr
-1&0\cr}\right)$
we have $||A||_{op}\le{2\over \sqrt{3}}||A||$ thus
$| P_k(\theta ,E)|\le\left({2\over \sqrt{3}}\right)^{k+1}\displaystyle
\prod_{j=0}^k||B_j(\theta ,E)||. $
Since $||B(\theta ,E)||$ is a continuous function of $\theta$ we can
use strict ergodicity of the rotation by the irrational angle $\omega
$ to show that for any $\epsilon > 1$ and $k$ large enough we have
$$| P_k(\theta ,E)|\le \epsilon^k\left({2\over \sqrt{3}}\right)^{k}
e^{{k \over 2\pi} \displaystyle
\int_0^{2\pi}\ln||B(\theta ,E)||\,d\theta} .$$
The last integral can be computed directly and is equal to $
\pi\sqrt{3}\ln M(E,\lambda )$.
\hfill $\Box$ \vskip .25in
It follows from Proposition 1 and (2) that for $x_1\;\in \; A_k, \;
x_2= x_1+k-1, \;k>k(\epsilon ),\;{1\over m_1}< m_2<1$ and $y\in [x_1,x_2] $
such that
$$k\left( 1- {\ln(m_1m_2)\over \ln(\epsilon M(E,\lambda ))}\right) < y -x_1<
k\left( {\ln(m_1m_2)\over \ln(\epsilon M(E,\lambda ))}\right) \nonumber $$
we have
\begin{equation}
|G_{[x_1,x_2]}(y,x_i)|k_1(\theta ,E )$ if the points $x_1,x_2$ are such
that \\
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 $\alpha =\alpha (m_1,\lambda , E)>1.$}\vskip .25in
The proof of Lemma 3 will be given in the section 4.\vskip .25in
\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 $.\vskip .25in
It is well known that to prove pure point spectrum one only needs to prove
that generalized eigenfunctions belong to $\ell ^2$ ([10]; see also
[3],[11]).
Let $E(\theta )$ be a generalized eigenvalue of $H_{\theta }$, $\Psi (x)$
the corresponding generalized eigenfunction.
\\
\noindent{\bf Lemma 4.} {\it For every $x\;\in \; {\bf Z}$ such that $\Psi (x)
\neq 0$ there exists $k_0=k_0(x,m_2,\theta ,E)$ such that for $k>k_0$ the
point $x$ is $(m_2,k)$-singular.}\vskip .25in
Lemma 4 is the same kind of statement as Lemma 3.1 in [3]
and so is the proof.\\
We now can finish the proof of the theorem.
Suppose $C(\lambda ,E)>0$. Then there exist $1< m_1<{\lambda \over
2},\; m_2<1$ and $\epsilon >1$ such that
$ 2c_{\lambda,\epsilon }-1 > {1 \over 2}$.
Assume without loss of generality that $\Psi (0) \not= 0$.
Let $|x|$ be bigger than $\max[k(\epsilon ,E),k_1(\theta ,E)]$.
Suppose $x$ is $(m_2,|x|)$-singular. Since 0 is $(m_2,|x|)$-singular, and
$2c_{\lambda,\epsilon }-1 > {1 \over 2},\;$ we obtain
using Proposition 2 that the points $x_1=x-c_{\lambda,\epsilon }|x|$ and
$x_2=-c_{\lambda,\epsilon }|x|$ satisfy the conditions of Lemma 3 with $k=|x|$
for $|x|$ large enough. Applying Lemma 3 we get that $dist(x_1,x_2)=|x| >
\alpha ^{|x|}$ which gives a contradiction for $|x|$ large and implies that
$x$ is $(m_2,|x|)$-regular. Thus we have that there exists an interval
$[x_1,x_2]$
containing $x$ such that
$$|x_i-x| \le |x|,\;\; |G_{[x_1,x_2]}(x,x_i)|\le m_2^{|x|},\;i=1,2.$$
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:
$$|\Psi (x)| \le 2C(1+2|x|)m_2^{|x|}.$$
\hfill $\Box$ \vskip .50in
\noindent{\bf 4. Proof of Lemma 3}
\vskip .25in
Let $x_1,x_2$ be as in Lemma 3.
We write:
$$S_k=\{ \theta : |P_k(\theta)|2$ there exists $k_0$ such
that for $k>k_0$ and $j\in [0,k]$
we have that $|\tilde Q_k^{(j)}(z)|0,$ implies there exists
$s\in [0,k], s\neq j$
such that}
$$|z-z_s|<(ba)^{k\over 4}$$
{\bf Proof.}
Suppose $|z-z_s| > (ba)^{k\over 4}$
for all $s\in [0,k], s\neq j$.We can assume without loss of generality
that $j\le [{k+1\over 2}].$ We can write
\begin{eqnarray}
\ln|\tilde Q _k^{(j)}(z)|=\displaystyle \sum _{s=0,...,[{k+1\over 2}];s\neq j}
\ln|(z-\cos(2\pi(\theta_1^\prime
+ s\omega)))| +\displaystyle \sum _{s=0}^{k-1-[{k+1\over 2}]}
\ln|(z-\cos(2\pi(\theta_2^\prime
+ s\omega)))| \ge \nonumber \\[0.25 in]
4\ln (ba)^{k\over 4}+ \displaystyle
\sum _{s=0,...,[{k+1\over 2}],s\neq j}
f_k(z-\cos(2\pi(\theta_1^\prime + s\omega)))
+\sum _{s=0,...,k-1-[{k+1\over 2}]}f_k(z-\cos(2\pi(\theta_2^\prime + s\omega)))
\end{eqnarray}
where $$f_k(z)=
\left\{\;
\begin{array}{cc}
& ln|z |, \; |z |> {C_1\over k^{2r}} \\[0.25 in]
& ln{C_1\over k^{2r}}, \; |z |\le {C_1\over k^{2r}},
\end{array} \right.$$
$C_1={2^{2r}C^2 \over 18},$ $C$ and $r$ are as in (1),
since the diophantine property (1) implies that there exist not more than 4
points $s\in [0,k]$ such that
$|z-z_s| \le C_1 k^{-2r}.$
Let $\omega =[k_1,...,k_n,...]$ be the continuous fractions expansion of
$\omega $; ${p_n\over q_n }= [k_1,...,k_n]$ - the $n^{th}$ approximant.
We will use the following properties of the continuous fractions expansions
(see, e.g.,
[12]):
\begin{equation}
|q_i\omega -p_i|<{1\over q_{i+1}}
\end{equation}
and
\begin{equation}
q_i\ge \sqrt{2}^{\,i-1}
\end{equation}
Pick $n$ such that $q_n\le[{k+1\over 2}]< q_{n+1}.$ Inequality (9) implies that
$n+1<3\ln k$ for $k>64.$
We can represent $[{k+1\over 2}]$ as $[{k+1\over 2}]=b_nq_n+...+b_1q_1 +b_0$
where for each $0\le j\le n$
$$ b_j= \left[{[{k+1\over 2}]-b_nq_n-...-b_{j+1}q_{j+1}\over q_j }\right]$$
$$b_n=\left[{[{k+1\over 2}]\over q_n }\right].$$
Property (8) implies that
$$|b_iq_i\omega -b_ip_i|\le b_i|q_i\omega -p_i|\le k_{i+1}|q_i\omega -p_i|
<{q_{i+1}\over q_i}{1\over q_{i+1}}={1\over q_{i}},
\;i=0,...,
n$$
The right hand side of (7) can be estimated from below as:
$$
4\ln (ba)^{k\over 4}+(k+1)\displaystyle \int _0^{1}f_k(z-\cos2\pi\theta )d\theta
-2(n+1)Var f_k \ge $$
$$k\ln (ba)+(k+1) \displaystyle \int _0^{1 } \ln |z-\cos 2\pi\theta |d\theta
+6
\ln k\ln
{C_1\over k^{2r}}
\ge k\ln (ba/2)+6
\ln k\ln
{C_1\over k^{2r}}-\ln 2 > k\ln a
$$
Here we estimated the error in the ergodic theorem using a standard technique
(see, e.g., Lemma 4.1,
Ch.3,
[12]).
The contradiction proves the statement of the proposition.
\hfill $\Box$ \vskip .25in
Let us fix ${m_1\over \lambda} |j_{s_1}-j_{s_2}|-k-1> \left( {C\over
(2(ba) ^{k\over 4})
^{1\over 2 }}\right) ^{{1\over r} }-k-1 >m_3^k$$
with $m_3=(ba)^{-{1\over 9r}}$, for $k$ large enough.\\
The event $ii)$ can occur for specific set of $\theta$ only. Namely for
every $j_{s_1},j_{s_2}$ we define
$$ \Theta _{j_{s_1},j_{s_2}}=\{\ \theta : (12)\; holds\}\ .$$
For every pair $s_1,s_2$ the set $\Theta _{j_{s_1},j_{s_2}}$ is
independent of $E$ and has measure
$\left( 2(ba) ^{k\over 4}\right) ^{1\over 2 }.$
The total number of pairs $s_1,s_2$ such that $ j_{s_i}\in [0,[{k+1\over 2}]]
\cup [x_2-x_1,x_2-x_1+k-1-[{k+1\over 2}]],\;i=1,2,\;$ does not exceed
${1\over 4}(k+1)^2$.
Thus the set
$$\Theta _k = \displaystyle \cup_{s_1,s_2: j_{s_i}\in [0,[{k+1\over 2}]]
\cup [x_2-x_1,x_2-x_1+k+1-[{k+1\over 2}]],\;i=1,2} \Theta _{j_{s_1},j_{s_2}}$$ has measure
not exceeding ${1\over 4}(k+1)^2\left( 2(ba) ^{k\over 4}\right) ^{1\over 2 }$
and using Borel-Cantelli
lemma we obtain the statement of the lemma.
\hfill $\Box$ \vskip .50in
\noindent{\bf 5. Acknowledgement}
\vskip .25in
I am deeply grateful to A. Klein for his support and many fruitful
discussions.
\newpage\clearpage
\begin{center}
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\end{center}
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\end{document}