\input amstex
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\def\gap{\vskip 0.1in\noindent}
\def\ref#1#2#3#4#5#6{#1, {\it #2,} #3 {\bf #4} (#5), #6.}
%References
\def\aron {1} %Aronszajn
\def\aubry {2} %Aubry
\def\auband {3} %Aubry-Andre
\def\avrsimids {4} %Avron-Simon IDS
\def\avrsimjm {5} %Avron-Simon JM
\def\blt {6} %Bell-Lima-Test
\def\bel {7} %Belokolos
\def\choj {8} %Chojnacki
\def\chudel {9} %Chulaevsky-Delyon
\def\deisim {10} %Deift-Simon
\def\delyon {11} %Delyon
\def\ds {12} %Dinaburg-Sinai
\def\dono {13} %Donoghue
% \def\el {x} %Eliasson
\def\fsw {14} %Frohlich, Spencer, Wittwer
\def\gessim {15} %Gesztesy-Simon (the xi function)
\def\gold {16} %Goldstein
\def\helsjo {17} %Helffer-Sjostrand
\def\jitocmp {18} %Jitomirskaya CMP
\def\jitoparis {19} %Jitomirskaya Paris
\def\js{20} %Jit-Simon
\def\lastac {21} %Last CMP93
\def\lastzero{22} %Last CMP94
\def\lastparis {23} %Last Paris
\def\lassim {24} %Last-Simon
\def\mz {25} %Mand-Zhit
\def\simon {26} %Simon
\def\sinai {27} %Sinai
\topmatter
\title Duality and Singular Continuous Spectrum in the
Almost Mathieu Equation
\endtitle
\rightheadtext{Duality and Singular Continuous Spectrum}
\author A.~Gordon$^{1,2}$, S.~Jitomirskaya$^{3,2}$, Y.~Last$^4$,
and B.~Simon$^{4,5}$
\endauthor
\leftheadtext{Gordon, Jitomirskaya, Last, and Simon}
\thanks $^1$ Department of Mathematics, University of North Carolina
at Charlotte, Charlotte, NC 28223.
\endthanks
\thanks $^2$International
Institute of Earthquake Prediction Theory and Mathematical Geophysics.
Moscow 113556, Russia.
\endthanks
\thanks$^3$ Department of Mathematics, University of California,
Irvine, CA 92717. This material is based upon work supported
by the National Science Foundation under Grant Nos.~DMS-9208029
and DMS-9501265. The Government has certain rights in this material.
\endthanks
\thanks$^4$ Division of Physics, Mathematics, and Astronomy,
California Institute of Technology, Pasadena, CA 91125
\endthanks
\thanks$^5$ This material is based upon work supported by the
National Science Foundation under Grant No.~DMS-9401491.
The Government has certain rights in this material.
\endthanks
\thanks To be submitted to {\it{Acta Math}}.
\endthanks
\date January 29, 1996
\enddate
\abstract We study the almost Mathieu operator
$(h_{\lambda,\alpha,\theta}u)(n)=u(n+1)+u(n-1)+
\lambda\cos (\pi\alpha n+\theta)u(n)$
on $\ell^2(\Bbb Z)$, and prove that the dual of point spectrum
is absolutely continuous spectrum. We use this to show that for
$\lambda = 2$ it has purely singular continuous spectrum for
a.e.~pairs $(\alpha, \theta)$. The $\alpha$'s for which we prove
this are explicit.
\endabstract
\endtopmatter
\document
\flushpar {\bf \S 1. Introduction}
\vskip 0.1in
Our main goal in this paper is to study the almost Mathieu operator
on $\ell^2(\Bbb Z)$ defined by
$$
(h_{\lambda,\alpha,\theta}u)(n)=u(n+1)+u(n-1)+\lambda
\cos(\pi\alpha n+\theta)u(n). \tag 1.1
$$
Our results in Section 2 on measurability of (normalizable)
eigenfunctions may be of broader applicability. For background on
(1.1), see [\jitoparis,\lastparis].
Our main result here concerns (1.1) at the self-dual point $\lambda
=2$.
\proclaim{Theorem 1} Let $\alpha$ be an irrational so there exist
$q_n \to\infty$ and $p_n$ in $\Bbb Z$ with
$$
q^2_n \left|\alpha-\frac{p_n}{q_n}\right| \to 0 \tag 1.2
$$
as $n\to\infty$. Then for a.e.~$\theta$, $h_{\lambda=2,\alpha,\theta}$
has purely singular continuous spectrum.
\endproclaim
\remark{Remarks} 1. (1.2) is used because for such $\alpha$,
Last [\lastzero] has proven that the spectrum,
$\sigma_{\lambda,\alpha}$, of $h_{\lambda,\alpha,\theta}$
(which is $\theta$ independent [\avrsimids]) has
$|\sigma_{2,\alpha}|=0$ (where $|\cdot|$ denotes Lebesgue
measure). Our proof is such that for any other $\alpha$ with
$|\sigma_{\lambda=2,\alpha}|=0$ (presumably all irrational $\alpha$),
one has that $h_{\lambda=2,\alpha,\theta}$ has purely singular
continuous spectrum for a.e.~$\theta$. Prior to [\lastzero],
Helffer-Sj\"ostrand [\helsjo] have shown $|\sigma_{2,\alpha}|=0$
for a set of $\alpha$'s having all quotients of their continued
fraction expansion sufficiently large. While this set is nowhere
dense and of zero Lebesgue measure, it includes some $\alpha$'s
for which (1.2) doesn't hold.
2. The set of $\alpha$'s for which (1.2) holds is a dense $G_\delta$
whose complement has Lebesgue measure $0$, that is, (1.2) is generic
in both Baire and Lebesgue sense. Although we do not describe the
a.e.~$\theta$ explicitly, the set with singular continuous spectrum
contains a dense $G_\delta$ [\js], so it is also generic in both Baire
and Lebesgue sense.
3. Delyon [\delyon] proved that $h_{\lambda=2,\alpha,\theta}$ has no
eigenfunctions belonging to $\ell^1$ (for any $\alpha,\theta$). More
recently, Chojnacki [\choj] has proven that
$h_{\lambda=2,\alpha,\theta}$ must have at least some continuous
spectrum (for all $\alpha$'s and a.e.~$\theta$). His result does
not contradict, however, the possibility of mixed (overlapping
continuous and p.p.) spectrum. We note that the absence of
absolutely continuous spectrum is obvious whenever the spectrum
has zero Lebesgue measure.
\endremark
While Theorem 1 is our main result, we also prove
\proclaim{Theorem 2} If $\alpha$ is irrational and $\lambda$ is
such that $h_{4/\lambda,\alpha,\theta}$ has only p.p.\ spectrum for
a.e.~$\theta$, then $h_{\lambda,\alpha,\theta}$
has purely a.c.~spectrum for a.e.~$\theta$.
\endproclaim
\remark{Remarks} 1. It is known that if $\alpha$ has good
Diophantine properties, then for $\lambda< \frac{4}{15}$,
$h_{4/\lambda, \alpha,\theta}$ has only p.p.~spectrum [\jitocmp]
(see [\sinai,\fsw,\gold] for earlier results). For such
$\alpha,\lambda$, we conclude purely a.c.~spectrum of
$h_{\lambda,\alpha,\theta}$ for a.e.~$\theta$.
2. Existence of some a.c.~spectrum for small $\lambda$ and
Diophantine $\alpha$ has been proven by Bellissard, Lima, and Testard
[\blt], who applied ideas earlier developed by Dinaburg-Sinai [\ds]
and Belokolos [\bel]. Such existence (but not necessarily purely
a.c.~spectrum) is now known for all $\lambda < 2$ and all
$\alpha, \theta$ [\gessim,\lastac]. Purely a.c.~spectrum for
(unspecified) small $\lambda$ and Diophantine $\alpha$ has been
proven by Chulaevsky and Delyon [\chudel] using duality. Their
proof uses detailed information from Sinai's proof of
localization [\sinai].
\endremark
We also provide a new proof of
\proclaim{Theorem 3} If $\alpha$ is irrational and $\lambda<2$,
then for a.e.~$\theta$, $h_{\lambda,\alpha,\theta}$ has no point
spectrum.
\endproclaim
\remark{Remark} Delyon [\delyon]
has proven that there is no point spectrum for all $\theta$,
which is strictly stronger. Moreover, his proof is much simpler.
Our proof has a certain methodological advantage in that we don't
use the positivity of the Lyapunov exponent for the dual model.
\endremark
Our proof of Theorems 1--3 depends on a precise version of Aubry
duality [\aubry,\auband]. Recall that one way to understand duality
is to note the following: Suppose $a_n$ solves
$$
a_{n+1}+a_{n-1}+\lambda\cos(\pi\alpha n+\theta)a_n =Ea_n \tag 1.3
$$
with $a_n \in \ell^1$. Define
$$
\varphi(x)=\sum a_n e^{i(\pi\alpha n+\theta)x} \tag 1.4
$$
which is continuous on $\Bbb R$ with
$$
\varphi\biggl(x+\frac{2}{\alpha}\biggr) =e^{2i\theta/{\alpha}}
\varphi(x).
\tag 1.5
$$
For any $\eta$, the sequence
$$
u(n)=\varphi\biggl(n+\frac{\eta}{\pi\alpha}\biggr) \tag 1.6
$$
is seen to obey
$$
u(n+1)+u(n-1)+\frac{4}{\lambda}\,\cos(\pi\alpha n+\eta)u(n) =
\frac{2E}{\lambda}\,u(n) \tag 1.7
$$
by manipulating (1.4). Thus, nice enough normalizable eigenfunctions
at $(\lambda, E)$ yield Bloch waves at $(\frac{4}{\lambda}, \frac{2E}
{\lambda})$ for $h$ and, conversely, nice enough Bloch waves
(regularity of $\varphi$ implies decay of the Fourier coefficients
$a_n$ in (1.4)) yield normalizable eigenfunctions.
If we slough over what ``nice enough'' means, we have naive duality:
\roster
\item"{(D1)}" point spectrum at $\lambda\Rightarrow$ a.c.~spectrum
at $\frac{4}{\lambda}$
\item"{(D2)}" a.c.~spectrum at $\lambda\Rightarrow$ point spectrum
at $\frac{4}{\lambda}$
\item"{(D3)}" s.c.~spectrum at $\lambda\Rightarrow$ s.c.~spectrum
at $\frac{4}{\lambda}$
\endroster
where the last statement follows from the first two.
The surprise in Last [\lastac] is that this naive expectation is
false. There exist $\alpha$ (Liouville numbers) for which the
spectrum for $\lambda>2$ is purely singular continuous, but the
spectrum for $\lambda<2$ has an a.c.~component. Thus, (D2) need not
be true. In a sense, the main result of the paper is that (D1) is
still true. More explicitly, we show that the dual of point spectrum
is a.c.~spectrum, in the sense that some p.p.~spectrum for $\lambda$
implies some a.c.~spectrum for $\frac{4}{\lambda}$, and only
p.p.~spectrum for $\lambda$ implies only a.c.~spectrum for
$\frac{4}{\lambda}$. This strengthens Chojnacki's result [\choj],
which shows (in a more general context, though) that the dual of
point spectrum is continuous spectrum (but not necessarily
a.c.~spectrum).
Thus, there is a kind of more precise duality for the almost Mathieu
operator:
\roster
\item"{(D1$^\prime$)}" point spectrum at $\lambda\Rightarrow$
a.c.~spectrum at $\frac{4}{\lambda}$
\item"{(D2$^\prime$)}" a.c.~spectrum at $\lambda\Rightarrow$ point
or s.c.~spectrum at $\frac{4}{\lambda}$
\item"{(D3$^\prime$)}" s.c.~spectrum at $\lambda\Rightarrow$
a.c.~spectrum (or s.c.~spectrum) at $\frac{4}{\lambda}$.
\endroster
It is unclear if there is any s.c.~spectrum for $\lambda<2$.
Note that while we prove (D1$^\prime$) (and thus (D3$^\prime$))
we do not prove (D2$^\prime$). It follows, of course, from the
known results for the almost Mathieu operator, but we want to point out
that this implication fails in certain more general contexts [\mz].
In Section 2, we prove that if there is point spectrum,
one can always choose
the eigenvalues and eigenvectors to be measurable in
$\theta.$
That allows us to represent the set of all
eigenvalues of $h_{\lambda,\alpha,\theta}$
as a union of values of a measurable
multivalued function along the trajectory of the rotation by
$\alpha$ --- an object first introduced by Sinai [\sinai].
In Section 3, we use this representation and analyze duality to
show point spectrum implies there are spectral measures for the dual
problem (at coupling $\frac{4}{\lambda}$) so that $d\mu_\theta$ is
independent of $\theta$. Here we use some important ingredients from
[\chudel] and [\mz].
In Section 4, we show that results of Deift-Simon [\deisim] imply
that the singular components of the spectral measures $d\mu_\theta$
and $d\mu_{\theta'}$ are a.e.~disjoint. Thus, the $\theta$
independence of Section 3 implies that $d\mu_\theta$ is purely
absolutely continuous. We make this precise and prove Theorems 1--3
in Section 5.
A.G. is grateful for the hospitality of the Division of Physics,
Mathematics and Astronomy of Caltech and the Department of Mathematics
at UC Irvine.
S.J.~and Y.L.~would like to thank J.~Avron for the hospitality of the
Institute for Theoretical Physics at the Technion where parts of this
work were done.
\vskip 0.3in
\flushpar {\bf \S 2. Measurability of Eigenfunctions}
\vskip 0.1in
To emphasize that ``measurability'' here means in Borel sense rather
than up to sets of measure zero in the completed measure, we'll
initially discuss a setup with no measure! At the end of this section,
we'll link this to the almost periodic situation that is the main focus
of this paper.
Let $A<\infty$ be positive and fixed. Let $\Omega=[-A,A]^{\Bbb Z}$,
that is, $\omega\in\Omega$ is a sequence $\{\omega_n\}^\infty_
{n=-\infty}$ with $|\omega_n|\leq A$. $\Omega$ is a separable compact
metric space with Baire = Borel sets. We call these sets ``measurable.''
For each $\omega\in\Omega$, define a self-adjoint operator on
$\ell^2(\Bbb Z)$ by
$$
(h_\omega u)(n)=u(n+1)+u(n-1)+\omega_n u(n).
$$
A critical fact we'll use below is that (normalizable) eigenvalues
are always simple.
Given any normalized eigenvector $u$ for $h_\omega$, we define $j(u)$
to be the leftmost maximum for $|u|$, that is, that $j$ with
$$\alignat2
|u(j)|&\geq |u(k)| \qquad && \text{all } k \\
& > |u(k)| \qquad && \text{all } k0$. Since $\sum|u(k)|^2 =1$,
we have $u(k)\to 0$ as $|k|\to\infty$, so it has a leftmost maximum.
If $j$ is the leftmost maximum for $u$, we say that $u$ is attached
to $j$.
Let $\{u_n\}$ be the collection of eigenvectors of $h_\omega.$ Define
$N_j(\omega)$ to be the number of eigenvectors of $h_\omega$ attached
to $j$. One of our goals below will be to prove
\proclaim{Theorem 2.1} $N_j(\omega)$ is a measurable function on
$\Omega$.
\endproclaim
Let $\Omega_{j,k}=\{\omega\mid N_j(\omega)\geq k\}$ for $k=1,2,\dots$.
On $\Omega_{j,1}$, define $u_1(n;\omega,j)$ to be the eigenfunction
attached to $j$ with maximal value of $u_1(j)$. Let $e_1(\omega,j)$
be its eigenvalue. If there are multiple eigenfunctions attached to
$j$ with the same value of $u_1(j)$, pick the one with the largest
energy. Since $\sum\limits_n |u_n(j)|^2 \leq 1$ (by Parseval's
inequality for $\delta_j$), there are only finitely many $u$'s with
this maximum value of $u(j)$ and so we can pick the one with largest
$e$.
Similarly, on $\Omega_{j,2}$ we can define $u_2(n;\omega,j)$ by
picking the attached eigenvector with second largest $u(j; \omega,j)$
again breaking ties by choosing the largest energy. In this way, we
define $u_l (n;\omega,j)$ and $e_l (\omega,j)$ on $\Omega_{j,l}$ so
that
\roster
\item"{(1)}" $\{u_l (\,\cdot\,; \omega,j)\}^{N_j(\omega)}_{l=1}$ is
the set of eigenvectors attached to $j$,
\item"{(2)}" $u_l(j; \omega,j)\geq u_{l+1}(j;\omega,j)$, and if
equality holds, then $e_l (\omega,j) >e_{l+1}(\omega,j)$.
\endroster
Extend $u_l$ and $e_l$ to all of $\Omega$ by setting to $0$ on
$\Omega\backslash\Omega_{j,l}$. Then we'll prove that
\proclaim{Theorem 2.2} $e_l (\omega,j)$ and $u_l (n;\omega,j)$ are
measurable functions on $\Omega$ for each fixed $l,j$ \rom(and $n$\rom).
\endproclaim
Notice that
\proclaim{Proposition 2.3} For each $\omega$, $l,l'$ and $j\neq k$,
$$
\sum^\infty_{n=-\infty} \overline{u_l (n; \omega,j)}\,
u_{l'}(n; \omega,k)=0. \tag 2.1
$$
\endproclaim
This is true because the $u$'s are distinct eigenfunctions (since
they are attached to distinct points) and so orthogonal.
As a first preliminary to the proofs of Theorems 2.1, 2.2, we make
several simplifying remarks:
\roster
\item"{(i)}" Without loss, we can take $j=0$.
\item"{(ii)}" Instead of looking only at eigenfunctions attached
to $j=0$, we can look at eigenfunctions with $u(0)\neq 0$
normalizing by $u(0)>0$. If we define $\tilde N_0(\omega)$ and
$\tilde u_l (n;\omega,0)$ analogously by requiring the analog of
(1) and (2) and prove measurability, we recover Theorems 2.1, 2.2
by noting that since $\tilde u_1 (n;\omega,0)$ is measurable,
$\{\omega\mid\tilde u_1 \text{ is attached to } 0\}$ is measurable,
and similarly for $\tilde u_2, \dots$. Then define $u_1$ in the
obvious way and see that it is measurable.
\item"{(iii)}" It will be convenient to deal with that multiple
$\eta$ of $u$ with $\eta(0)=1$. $u$ and $\eta$ are related, of
course, by $\eta(n) =u(n)/u(0)$ and $u(n)=\eta(n)/\bigl(\sum
\limits^\infty_{j=-\infty}|\eta(j)|^2\bigr)^{1/2}$. By these
relations, if we show $\eta$ is measurable, so is $u$. Notice that
since $u(0)=1/\|\eta\|$, ordering by maximal $u(0)$ is the same
as ordering by minimal $\|\eta\|$.
\endroster
\remark{Note} Since the passage from $\eta$ to $u$ involves an
infinite sum, weak continuity of $\eta$ does not imply weak
continuity of $u$, only weak measurability. The use of $\eta$ below
is critical because it, not $u$, is weakly continuous.
\endremark
As a second preliminary, we note a few standard facts about Borel
functions.
\proclaim{Lemma 2.4} Let $X$ be a complete metric space, $Y\subset X$
an arbitrary subspace, and $B_X,B_Y$ their Borel subsets. Then
$$
B_Y = \{Y\cap A\mid A\in B_X\}. \tag 2.2
$$
\endproclaim
\demo{Proof} Let $\tilde B_Y$ be the right side of (2.2). Then $B_Y
\subset \tilde B_Y$ since $\tilde B_Y$ is clearly a sigma algebra
containing the closed sets in $Y$ since if $C$ is closed in $Y$ and
$D$ is its closure in $X$\!, then $D\cap Y=C$. Conversely, let
$\tilde B_X =\{A\subset X\mid A\cap Y\in B_Y\}$. $\tilde B_X \supset
B_X$ since $\tilde B_X$ is a sigma algebra containing the open sets.
Thus, $\tilde B_Y \subset B_Y$. \qed
\enddemo
\proclaim{Lemma 2.5} Let $X$ be a complete metric space, $Y\subset X$
with $Y\in B_X$, the Borel subsets of $X$. Then any $C\in B_Y$, the
Borel subset of $Y$\!, is Borel as a subset of $X$.
\endproclaim
\demo{Proof} By Lemma 2.4, $C=Y\cap A$ with $A\in B_X$. Since $Y\in
B_X$, so is $C$. \qed
\enddemo
\proclaim{Proposition 2.6} Let $X$ be a complete metric and
$B_X$ its Borel subsets. Let $X=\operatornamewithlimits{\cup}
\limits^{\infty}_{n=1} X_n$ with $X_n \in B_X$. Let $f: X\to\Bbb R$
be such that for each $n$, $f_n\equiv f\restriction X_n$ is a
Borel function from $X_n$ to $X$. Then $f$ is Borel.
\endproclaim
\demo{Proof} Let $(a,b)\subset\Bbb R$. Then $f^{-1}(a,b)=
\operatornamewithlimits{\cup}\limits^{\infty}_{n=1} f^{-1}_n (a,b)$
is Borel by Lemma 2.5. \qed
\enddemo
\example{Example} Let $X=[0,1]$ and let $f=\chi_{[1/2,1]}$ the
characteristic function of $[\frac{1}{2}, 1]$. Let $X_n =[\frac{1}
{2}, 1]\cup [0, \frac{1}{2} -\frac{1}{n}]$. Then $f\restriction
X_n : X_n \to\Bbb R$ is continuous, so $f$ is Borel. This example
shows that Proposition 2.6 is false if a Borel function is replaced
by a continuous function. It is useful to keep in mind, given the
continuous function argument we use below.
\endexample
As a final preliminary, we note the following elementary fact:
\proclaim{Proposition 2.7} Suppose $\omega^{(m)}\in\Omega$ and
$\omega^{(m)} \to \omega^{(\infty)}$ and let $h_m \equiv
h_{\omega^{(m)}}$. Suppose for each finite $m$, there is
$\eta^{(m)}$ with
\roster
\item"\rom{(i)}" $h_m \eta^{(m)} =e_m \eta^{(m)}$
\item"\rom{(ii)}" $\eta^{(m)}(0)=1$
\item"\rom{(iii)}" $\operatornamewithlimits{\sup}\limits_{m}
\|\eta^{(m)}\|\equiv C <\infty$.
\endroster
Then there exists a subsequence $m_i$, $\eta^{(\infty)}$ and
$e_\infty$ so that
\roster
\item"\rom{(1)}" $h_\infty \eta^{(\infty)} = e_{\infty}
\eta^{(\infty)}$
\item"\rom{(2)}" $\eta^{(\infty)}(0)=1, \qquad \|\eta^{(\infty)}\|
\leq C$
\item"\rom{(3)}" $e_{m_i}\to e_\infty$, $\eta^{(m_i)}(n) \to
\eta^{(\infty)}(n)$ for each $n$.
\endroster
\endproclaim
\demo{Proof} $|e_m|\leq 2+A$, so by compactness of the unit ball of
$\ell^2 (\Bbb Z)$ in the weak topology, we can pick a subsequence so
that (3) holds. (2) is obvious and (1) holds by taking pointwise
limits in the equation (i). \qed
\enddemo
In the proof below, we have to worry about three possibilities that
destroy continuity of a function like $\tilde N_0$. First,
$\operatornamewithlimits{\sup}\limits_{m} \|\eta^{(m)}\|=\infty$.
We'll avoid this by looking at subsets with $\|\eta\|\leq k$, get
measurability, and take $k$ to infinity. Second, as $\omega^{(m)}
\to \omega^{(\infty)}$, two distinct eigenvalues of $h_m$ can
converge to a single $e$ so that $h_\infty$ has fewer eigenvalues.
We'll avoid this by looking at subsets where eigenvalues stay at
least a distance $2^{-\ell}$ from each other. Then we'll take
$\ell$ to infinity. Third, after we restrict to $\eta$'s with
$\|\eta\|\leq k$, in a limit a bunch of $\eta$'s with $\|\eta\|\geq
k$ can approach one with $\|\eta\|=k$ increasing $N$. We'll handle
this by proving semicontinuity rather than continuity as the
starting point of a proof of measurability.
\demo{Proof of Theorems {\rom{2.1, 2.2}}} For each pair of positive
integers $k,p$, define $M_{k,p}(\omega)$ to be the maximum number
of $m$'s so that $\|\eta_m\|\leq k$ and $|e_m -e_{m'}|\geq 2^{-p},$
for all $m'\not= m.$ Since Parseval's inequality implies that
$\sum\limits_{m} 1/\|\eta_m\|^2 \leq 1$, there are at most $k^2$
$m$'s with $\|\eta_m\|\leq k$ and so we can determine the maximum
number with $|e_m -e_{m'}|\geq 2^{-p}$.
We claim that $S\equiv\{\omega\mid M_{k,p}(\omega)\geq l\}$ is
closed so $M_{k,p}(\,\cdot\,)$ is measurable. For if $\omega^{(m)}
\in S$ and $\omega^{(m)}\to \omega^{(\infty)}$, we can use
Proposition 2.7 and find $\eta$'s and $e$'s for $h_\infty$ by
taking limits. Clearly, the limiting $e$'s still obey the
$|e_m -e_{m'}|\geq 2^{-p}$ condition. Thus, $S$ is closed and
$M_{k,p}(\,\cdot\,)$ is measurable.
Now define
$$
M_{k,\infty}(\omega)=\#\text{ of $m$'s with }\|\eta_m\|\leq k.
$$
Because of simplicity of the spectrum,
$$
M_{k,\infty}(\omega)=\max\limits_{p}\, M_{k,p}(\omega)
$$
so
$$
\{\omega\mid M_{k,\infty}(\omega)\geq l\} =
\bigcup\limits^{\infty}_{p=1}\{\omega\mid M_{k,p}(\omega)\geq l\}
$$
is an $F_\sigma$, and so $M_{k,\infty}$ is measurable.
Next define
$$
\Sigma_{k,l,p}=\{\omega\mid M_{k,\infty}(\omega)=
M_{k,p}(\omega)=l\},
$$
the set of $\omega$'s with exactly $l$ eigenvalues with $\|\eta_m\|
\leq k$ so that $|e_m -e_{m'}|\geq 2^{-p}$. $\Sigma_{k,l,p}$ is a
Borel set. Let $e_1 >\cdots> e_l$ be the eigenvalues and $\eta_1,
\dots,\eta_l$ the eigenvectors (normalized by $\eta_i (0)=1$).
We claim that $e_i$ and $\eta_i$ are continuous on $\Sigma_{k,l,p}$.
For if $\omega^{(m)}\to \omega^{(\infty)}$, $e^{(m)}_i$ has limit
points which are distinct (as $i$ varies) since $|e^{(m)}_i -
e^{(m)}_j|\geq 2^{-p}$. By Proposition 2.7, these limit points must
be eigenvalues of $h_\infty$ with eigenvectors $\eta^{(\infty)}_i$
obeying $\|\eta^{(\infty)}_i\|\leq k$ and so these limit points
must be the $e^{(\infty)}_i$ since $h_\infty$ has only $l$ such
eigenvalues. That is, $e^{(\infty)}_i$ are the unique limits
points of $e^{(m)}_i$, so $e^{(m)}_i \to e^{(\infty)}_i$.
Similarly, the $\eta$'s converge by Proposition 2.7 and the
uniqueness of eigenvectors.
Now let
$$
\Sigma_{k,l}=\{\omega\mid M_{k,\infty}(w)=l\}=
\bigcup\limits_p \Sigma_{k,p,l}.
$$
By Proposition 2.6, $e_i$ and $\eta_i$ are measurable on
$\Sigma_{k,l}$.
Now change the labeling so that instead of $e_1 > e_2 >\cdots > e_l$,
we have $\|\eta_i\|\leq\|\eta_{i+1}\|$ with $e_i > e_{i+1}$ if
$\|\eta_i\|=\|\eta_{i+1}\|$. This involves a permutation $\pi$ so
that
$$\align
e^{(\text{new})}_i &= e^{\text{old}}_{\pi(i)} \\
\eta^{(\text{new})}_i &= \eta^{\text{old}}_{\pi(i)}.
\endalign
$$
Because $\eta^{(\text{old})}_i$ and $e^{(\text{old})}_i$ are Borel
functions, the set $\Sigma^{(\pi)}_{k,l}$ on which a given permutation
$\pi$ is used is a Borel set. $e^{(\text{new})}_i$ is built out of
Borel functions on each $\Sigma^{(\pi)}$ and so we have measurability
with the changed labeling.
Once we change labeling, $e_i$'s and $\eta_i$'s are defined
consistently on $\Sigma_{k,l}$ as $k,l$ vary, and so, using
Proposition 2.6, on all $\Omega$. \qed
\enddemo
Note that although the set of
all eigenvalues can be naively considered a nonmeasurable ``function'' of
$\omega,$ since it is invariant but nonconstant, we have shown it admits a
measurable selection.
As a final remark on the issue of this section, we want to rewrite
(2.1) in a useful way. Taking $\omega_n (\theta)=\lambda\cos(\pi\alpha
n+\theta)$ embeds the circle $\theta\in [0,\pi)$ into $\Omega$, so
we define $u_l (n;\theta)\equiv u_l (n;\theta,0)$, the eigenvectors
with leftmost maximum at $0$. Note that
$$
u_l (\,\cdot\,-j; \theta+j\alpha\pi)= u_{l} (\,\cdot\,;\theta, j).
$$
In particular, (2.1) becomes
$$
\sum_n \overline{u_{l'}(n-j;\theta+j\alpha\pi)}\, u_l(n;\theta)=0
\tag 2.3
$$
for all $l,l'$ (even with $l=l'$) and $j\neq 0$. If we extend $u_l
\equiv 0$ on the set where there aren't $l$ eigenvectors with
leftmost maximum at $0$, then (2.3) holds for all $\theta$.
\vskip 0.3in
\flushpar {\bf \S 3. Duality}
\vskip 0.1in
Fix $\alpha$ irrational throughout this section.
Let $\Cal H =L^2 (([0,2\pi),\frac{d\theta}{2\pi})\times\Bbb Z)$,
that is, functions $\varphi :[0,2\pi)\times\Bbb Z\to\Bbb C$ with
\linebreak $\sum\limits_n \int\frac{d\theta}{2\pi}|\varphi(\theta,n)|^2
<\infty$. Define $Q_\lambda$ on $\Cal H$ by
$$
(Q_\lambda\varphi)(\theta,n)=\varphi(\theta, n+1)+\varphi(\theta, n-1)
+\lambda\cos(\pi\alpha n+\theta)\varphi(\theta, n),
$$
that is, $Q_\lambda$ is the direct integral in $\theta$ of
$h_{\alpha,\lambda,\theta}$.
Define $U:\Cal H\to\Cal H$ by the formal expression:
$$
(U\varphi)(\eta,m)=\sum_n \int \frac{d\theta}{2\pi}\,
e^{-i(\eta+\pi \alpha m)n} e^{-im\theta}\varphi(\theta, n). \tag 3.1
$$
In terms of the Fourier transform $\widehat{\varphi}(m,\eta)$ we
have
$$
(U\varphi)(\eta,m)=\widehat{\varphi}(m,\eta+\pi \alpha m), \tag 3.2
$$
which gives a precise definition even for cases where the sum in $n$
may not converge absolutely and shows that $U$ is unitary. Here is a
precise version of Aubry duality [2,3]:
\proclaim{Theorem 3.1}
$$
Q_\lambda U = \frac{\lambda}{2}\,UQ_{4/\lambda}
$$
\endproclaim
\demo{Proof} A straightforward calculation. For example, if
$(T\varphi)(\theta,n)=\varphi(\theta,n+1)$, then $(U^{-1}TU\varphi)
(\theta, n) = e^{-i(\pi\alpha n+\theta)}\varphi(\theta, n)$ so
$(U^{-1}Q_\lambda U\varphi)(\theta, n)=2\cos(\pi\alpha n+\theta)
\varphi(\theta, n)+
%\mathbreak
\frac{\lambda}{2}[\varphi(\theta, n+1)+\varphi(\theta, n-1)]
= \frac{\lambda}{2}(Q_{4/\lambda}\varphi)(\theta, n)$. \qed
\enddemo
\remark{Remark} Theorem 3.1 also provides a proof of duality for the
integrated density of states first rigorously proven in [\avrsimjm].
For let
$$
g(\theta, n)=\delta_{n0}.
$$
Then, $Ug=g$. Moreover, if $k_\lambda(E)$ is the integrated density
of states, then for any continuous function $f$,
$$
\langle g, f(Q_\lambda)g\rangle = \int f(E)\,dk_\lambda(E).
$$
Thus, Theorem 3.1 which implies
$$\align
\langle g, f(Q_\lambda) g\rangle &= \langle Ug, Uf(Q_\lambda)
U^{-1} Ug\rangle \\
&= \langle g, f(\tfrac{\lambda}{2}Q_{4/\lambda})g\rangle
\endalign
$$
yields the duality of $k$.
\endremark
We need one more simple calculation:
\proclaim{Proposition 3.2} For any $\varphi\in\Cal H,\; l\in \Bbb Z$,
define a unitary operator $S_l$ by
$$
(S_l\varphi)(\theta, n)=\varphi(\theta +\pi\alpha l, n-l). \tag 3.3
$$
Then
$$
(US_l\varphi)(\eta,m)=e^{-il\eta}(U\varphi)(\eta,m). \tag 3.4
$$
\endproclaim
\demo{Proof} Let $\varphi$ be such that there exists $N_0$ with
$\varphi(n,\theta)=0$ if $|n|>N_0$. Then (3.4) is a simple change
of variables in the integral (3.1). Since such $\varphi$'s are
dense and $S_l$ is bounded, (3.4) holds for all $\varphi$. \qed
\enddemo
\proclaim{Proposition 3.3} Let $\varphi\in\Cal H$ so that for
all $l\neq 0$, $\langle S_l \varphi,\varphi\rangle=0$. Then
$\sum\limits_m |(U\varphi)(\eta,m)|^2=g(\eta)$ is a.e.~independent
of $\eta$.
\endproclaim
\demo{Proof} Note that since $U\varphi\in\Cal H$, $g\in L^1 ([0,
2\pi),\frac{d\theta}{2\pi})$. We compute $\int e^{il\eta}g(\eta)
\frac{d \eta}{2\pi}=\mathbreak \langle U\varphi, e^{il\eta}
U\varphi\rangle =\langle U\varphi, US_{-l}\varphi\rangle =\langle
\varphi, S_{-l}\varphi\rangle=0, \; l \not= 0,$ by hypothesis.
By the weak-$*$ density of finite linear combinations of
$\{e^{il\eta}\}^\infty_{l=-\infty}$ in $L^\infty$, we conclude
that $g(\eta)$ is constant. \qed
\enddemo
We come now to the main result of this section.
\proclaim{Theorem 3.4} Fix $\lambda$ and fix $\alpha$ irrational.
Let $u_l(\,\cdot\,;\theta, j)$ be the measurable function described
in Theorem {\rom{2.2}} for the Hamiltonian $h_{4/\lambda, \alpha,
\theta}$. Let $f(\theta)$ be an arbitrary function in $L^2
([0,2\pi), \frac{d\theta}{2\pi})$. Let $\varphi(\theta, n)=f(\theta)
u_l (n;\theta, j)$ for some fixed $l,j$. For each $\eta$, let
$$
\psi_\eta (n)=(U\varphi)(\eta, n)
$$
and let $d\mu_\eta (E)$ be the spectral measure for Hamiltonian
$h_{\lambda, \alpha, \eta}$ and vector $\psi_\eta$. Then $d\mu_\eta$
is a.e.~$\eta$ independent.
\endproclaim
\demo{Proof} By (2.3), $S_k \varphi$ is orthogonal (in $\Cal H$) to
$\varphi$ for any $k\neq 0$. Moreover, since
$$
\biggl( F\biggl(\frac{\lambda}{2}\, Q_{4/\lambda}\biggr)\varphi\biggr)
(\theta, n) = F\biggl(\frac{\lambda}{2}\,e_l (\theta, j)\biggr)
\varphi(\theta, n),
$$
we have that $S_k\varphi,\, k \not= 0,$ is orthogonal to
$F(Q_{4/\lambda})\varphi$ for any continuous function $F$. As in
Proposition 3.3, we conclude that
$$
\sum_m \overline{(U\varphi)(\eta,m)}\, \biggl(UF\biggl(\frac{\lambda}
{2}\,Q_{4/\lambda}\biggr)\varphi\biggr)(\eta,m) \tag 3.5
$$
is independent of $\eta$. But by Theorem 3.1, $UF(\frac{\lambda}{2}
Q_{4/\lambda})=F(Q_\lambda)U$. So (3.5) is just $\int F(E)\,d\mu_\eta
(E)$. Since this is a.e.~$\eta$ independent for each continuous $F$
(and the set of continuous $F$'s is separable), we conclude that
$d\mu_\eta (E)$ is a.e.~$\eta$ constant. \qed
\enddemo
\vskip 0.3in
\flushpar {\bf \S 4. A.E.~Mutual Singularity of the Singular Parts}
\vskip 0.1in
In this section we want to note a simple consequence of Deift-Simon
[\deisim]:
\proclaim{Theorem 4.1} Let $h_\omega$ be an ergodic family of
Jacobi matrices and let $d\mu^s_\omega$ be the singular part of a
spectral measure for $h_\omega$. Then for a.e.~$\omega,\omega'$,
$d\mu^s_\omega$ and $d\mu^s_{\omega'}$ are mutually singular.
\endproclaim
\remark{Remark} This is an analog of the celebrated result of
Aronszajn [\aron]-Donoghue [\dono] of mutual singularity under rank
one perturbations; see [\simon].
\endremark
\demo{Proof} Let $G_\omega (n,m;z)$ be the Green's function for
$h_\omega$ (matrix elements of $(h_\omega -z)^{-1}$) and let
$\frak S_\omega$ be the set of $E_0$ in $\Bbb R$ so that
$$
\varlimsup\limits_{\epsilon\downarrow 0}\, \{\text{Im}[G_\omega
(0,0;E_0 +i\epsilon)]+\text{Im}[G_\omega (1,1,E_0 +i\epsilon)]\}
=\infty.
$$
By the theorem of de la Vall\'ee-Poussin, $d\mu^s_\omega$ is
supported by $\frak S_\omega$. Deift-Simon [\deisim] prove that
for every $E_0\in\Bbb R$, $\{\omega'\mid E_0\in\frak S_{\omega'}\}$
has measure $0$. Thus, integrating on $E$ with respect to
$d\mu_\omega$, we see that
$$
\mu_\omega (\frak S_{\omega'})=0
$$
for a.e.~$\omega'$. Thus for each fixed $\omega$, $d\mu^s_{\omega'}$
is mutually singular to $d\mu_\omega$ for a.e.~$\omega'$. \qed
\enddemo
\vskip 0.3in
\flushpar {\bf \S 5. Putting It All Together}
\vskip 0.1in
\proclaim{Theorem 5.1} Fix $\lambda$ and fix $\alpha$ irrational.
Let $u_l(\,\cdot\,;\theta, j)$ be the measurable function described
in Theorem {\rom{2.2}} for the Hamiltonian $h_{4/\lambda, \alpha,
\theta}$. Let $f(\theta)$ be an arbitrary function in $L^2 ([0,2\pi),
\frac{d\theta}{2\pi})$. Let $\varphi(\theta, n)=f(\theta) u_l
(n;\theta, j)$ for some fixed $l,j$. For each $\eta$, let
$\psi_\eta (n)=(U\varphi)(\eta, n)$ and let $d\mu_\eta (E)$ be
the spectral measure for Hamiltonian $h_{\lambda, \alpha, \eta}$
and vector $\psi_\eta$. Then $d\mu_\eta$ is purely a.c.~for
a.e.~$\eta$.
\endproclaim
\demo{Proof} By Theorem 3.4, $d\mu_\eta$ is a.e.~constant. By Theorem
4.1, this means that $d\mu^s_\eta$ is a.e.~zero. \qed
\enddemo
\proclaim{Theorem 5.2} Fix $\lambda$ and fix $\alpha$ irrational.
If $h_{4/\lambda, \alpha, \theta}$ has point spectrum for a set of
$\theta$'s of positive measure, then $h_{\lambda,\alpha,\theta}$
has some a.c.~spectrum for a.e.~$\theta$.
\endproclaim
\remark{Remark} It then follows by [\lassim] that $h_{\lambda,\alpha,
\theta}$ has some a.c.~spectrum for {\it{all}} $\theta$.
\endremark
\demo{Proof} $\{u_l (\,\cdot\, ;\theta,j)\}_{l,j}$ span the point
spectrum for $h_{4/\lambda,\alpha,\theta}$, so if there is point
spectrum, some $u_1 (\,\cdot\,;\,\cdot\, ,j)$ is a non-zero
function in $\Cal H$. Thus by Theorem 5.1, $d\mu_\eta$ is
a.e.~purely absolutely continuous. Since $\int d\mu_\eta (E)\,
d\eta=\sum\limits_n \int |u_1 (n;\theta, j)|^2 \frac{d\theta}{2\pi}
>0$, we conclude that $\int d\mu_\eta (E)\neq 0$ for a set of
$\eta$'s of positive measure and so for a.e.~$\eta$ since
$d\mu_\eta (E)$ is a.e. $\eta$ independent. \qed
\enddemo
\proclaim{Theorem 5.3} Fix $\lambda$ and fix $\alpha$ irrational.
If $h_{4/\lambda,\alpha,\theta}$ has only point spectrum for
a.e.~$\theta$, then $h_{\lambda,\alpha,\theta}$ has only
a.c.~spectrum for a.e.~$\theta$.
\endproclaim
\demo{Proof} Let $f_m (\theta)$ be an orthonormal basis for $L^2
([0,2\pi), \frac{d\theta}{2\pi})$. By hypothesis for a.e.~$\theta$,
$\{u_l (\,\cdot\, ;\theta,j)\}_{l,j}$ is an orthonormal basis for
$\ell^2 (\Bbb Z)$ where we run over those $l,j$ for which $u_l
(\,\cdot\, ;\theta,j)\neq 0$. It follows that if $\varphi_{m,l,j}
(\theta,n)\equiv f_m (\theta) u_l (n;\theta, j)$, then
$\{\varphi_{m,l,j}\}_{m,l,j}$ is a complete orthogonal set (but
not necessarily normalized). By the unitarity of $U$\!,
$\{U\varphi_{m,l,j}\}_{m,l,j}$ is also a complete orthogonal set.
Thus for a.e.~$\eta$, $\{U\varphi_{m,l,j}(\eta,\,\cdot)\}$ is a
complete set. But these vectors lie in the a.c.~spectral subspace
by Theorem 5.1. \qed
\enddemo
\demo{Proof of Theorem {\rom{1}}} Last [\lastzero]
has shown for such $\alpha$,
the Lebesgue measure of the spectrum $\sigma_{\lambda=2,\alpha}$
is zero. It follows there is no a.c.~spectrum for such $\alpha$
and $\lambda=2$. By Theorem 5.2, there can't be any point spectrum
a.e.~since such point spectrum would imply a.c.~spectrum. Thus for
a.e.~$\theta$, the spectrum is purely singular continuous. \qed
\enddemo
\demo{Proof of Theorem {\rom{2}}} This is just a restatement of
Theorem 5.3. \qed
\enddemo
\demo{Proof of Theorem {\rom{3}}} Let $\lambda < 2$. Then
$h_{4/\lambda, \alpha,\theta}$ has no a.c.~spectrum since
$\frac{4}{\lambda} >2$ [\auband,\avrsimids]. Thus by Theorem 5.2,
$h_{\lambda,\alpha,\theta}$ can have no point spectrum.
\qed
\enddemo
\vskip 0.3in
\flushpar{\bf References}
\vskip 0.1in\noindent
\item{\aron.}\ref{N.~Aronszajn}{On a problem of Weyl in the theory
of Sturm-Liouville equations}{Am. J.~Math.}{79}{1957}{597--610}
\gap
\item{\aubry.}\ref{S.~Aubry}{The new concept of transition by
breaking of analyticity}{Solid State Sci.}{8}{1978}{264}
\gap
\item{\auband.}\ref{S.~Aubry and G.~Andre}{Analyticity breaking
and Anderson localization in incommensurate lattices}{Ann.~Israel
Phys.~Soc.}{3}{1980}{133--140}
\gap
\item{\avrsimids.}\ref{J.~Avron and B.~Simon}{Almost periodic
Schr\"odinger operators, II. The integrated density of states}
{Duke Math.~J.}{50}{1983}{369--391}
\gap
\item{\avrsimjm.}\ref{J.~Avron and B.~Simon}{Singular continuous
spectrum for a class of almost periodic Jacobi matrices}
{Bull.~Amer.~Math.~Soc.}{6}{1982}{81--85}
\gap
\item{\blt.}\ref{J.~Bellissard, R.~Lima, and D.~Testard}
{A metal-insulator transition for the almost Mathieu model}
{Commun.~Math.~ Phys.}{88}{1983}{207--234}
\gap
\item{\bel.} E.D.~Belokolos, {\it{A quantum particle in a one-dimensional
deformed lattice. Estimates of lacunae dimension in the spectrum}}
(in Russian), Teoret.~Mat.~Fiz. {\bf{25}} (1975), 344--57.
\gap
\item{\choj.}\ref{W.~Chojnacki}{A generalized spectral duality
theorem}{Commun.~Math.~Phys.}{143}{1992}{527--544}
\gap
\item{\chudel.}\ref{V.~Chulaevsky and F.~Delyon}{Purely absolutely
continuous spectrum for almost Mathieu operators}{J.~Statist.~Phys.}
{55}{1989}{1279--1284}
\gap
\item{\deisim.}\ref{P.~Deift and B.~Simon}{Almost periodic
Schro\"odinger operators, III. The absolutely continuous spectrum
in one dimension}{Commun.~Math.~Phys.}{90}{1983}{389--411}
\gap
\item{\delyon.}\ref{F.~Delyon}{Absence of localization for
the almost Mathieu equation}{J.~Phys.}{A 20}{1987}{L21--L23}
\gap
\item{\ds.}\ref{E.~Dinaburg and Ya.~Sinai}{The one-dimensional
Schr\"{o}dinger equation with a quasi-periodic potential}
{Funct.~Anal.~ Appl.}{9}{1975}{279--289}
\gap
\item{\dono.}\ref{W.~Donoghue}{On the perturbation of the spectra}
{Commun.~Pure Appl.~Math.}{18}{1965}{559--579}
\gap
%\item{\el.}\ref{L.H.~Eliasson}{Floquet solutions for the
%one-dimensional quasi-periodic Schr\"{o}dinger equation}
%{Commun.~Math.~Phys.}{146}{1992}{447--482}
%\gap
\item{\fsw.}\ref{J.~Fr\"ohlich, T.~Spencer, and P.~Wittwer}
{Localization for a class of one-dimensional quasi-periodic
Schr\"odinger operators}{Commun.~Math.~Phys.}{132}{1990}{5--25}
\gap
\item{\gessim.} F.~Gesztesy and B.~Simon, {\it{The xi function}},
to appear in Acta Math.
\gap
\item{\gold.} M.~Goldstein, {\it{Laplace transform method in
perturbation theory of the spectrum of Schr\"o-dinger operators,
II. One-dimensional quasi-periodic potentials}}, preprint (1992).
\gap
\item{\helsjo.}\ref{B.~Helffer and J.~Sj\"ostrand}
{Semi-classical analysis for Harper's equation, III. Cantor
structure of the spectrum}{M\'em.~Soc.~Math.~France (N.S.)}{39}
{1989}{1--139}
\gap
\item{\jitocmp.} S.~Jitomirskaya, {\it{Anderson localization for
the almost Mathieu equation; A nonperturbative proof}}, to appear
in Commun.~Math.~Phys.
\gap
\item{\jitoparis.} S.~Jitomirskaya, {\it{Almost everything about
the almost Mathieu operator, II}}, Proc.~XI Intl.~Congress of
Mathematical Physics, Paris (D.~Iagolnitzer, ed.), pp.~373--382,
International Press, 1995.
\gap
\item{\js.}\ref{S.~Jitomirskaya and B.~Simon}
{Operators with singular continuous spectrum, III. Almost periodic
Schr\"odinger operators}{Commun.~Math.~Phys.}{165}{1994}{201--205}
\gap
\item{\lastac.}\ref{Y.~Last}{A relation between a.c.~spectrum of
ergodic Jacobi matrices and the spectra of periodic approximants}
{Commun.~Math.~Phys.}{151}{1993}{183--192}
\gap
\item{\lastzero.}\ref{Y.~Last}{Zero measure for the almost Mathieu
operator}{Commun.~Math.~Phys.}{164}{1994}{421--432}
\gap
\item{\lastparis.} Y.~Last, {\it{Almost everything about the almost
Mathieu operator, I}}, Proc.~XI Intl.~Congress of Mathematical
Physics, Paris (D.~Iagolnitzer, ed.), pp.~366--372, International Press,
1995.
\gap
\item{\lassim.} Y.~Last and B.~Simon,
{\it{Eigenfunctions, transfer matrices, and
a.c.~spectrum of one-dimensional Schr\"odinger operators}},
in preparation.
\gap
\item{\mz.}\ref{V.~Mandelshtam and S.~Zhitomirskaya}
{1D-Quasiperiodic operators. Latent symmetries}{Commun.~Math.~Phys.}
{139}{1991}{589--604}
\gap
\item{\simon.}\ref{B.~Simon}{Spectral analysis of rank one
perturbations and applications}{CRM Proc.~Lecture Notes}{8}{1995}
{109--149}
\gap
\item{\sinai.}\ref{Ya.~Sinai}{Anderson localization for
one-dimensional Schr\"odinger operator with quasi-periodic
potential}{J.~Statist.~Phys.}{46}{1987}{861--909}
\gap
\enddocument