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Jacobi matrices, generalized eigenvectors, WKB-type asymptotics, absolutely continuous spectrum
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\newtheorem{algorithm}{Algorithm}
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\newtheorem{case}{Case}
\newtheorem{claim}{Claim}
\newtheorem{conclusion}{Conclusion}
\newtheorem{condition}{Condition}
\newtheorem{conjecture}{Conjecture}
\newtheorem{corollary}{Corollary}
\newtheorem{criterion}{Criterion}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{exercise}{Exercise}
\newtheorem{lemma}{Lemma}
\newtheorem{notation}{Notation}
\newtheorem{problem}{Problem}
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\newtheorem{summary}{Summary}
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\begin{document}
\title[WKB asymptotics]{On the existence of WKB-type asymptotics for the generalized eigenvectors of
discrete string operators}
\date{Apr, 2002}
\author{Sergei Belov and Alexei Rybkin}
\address{Department of Mathematical Sciences\\
University of Alaska Fairbanks\\
PO Box 756660\\
Fairbanks, AK 99775}
\email{ftsmb1@uaf.edu}
\address{Department of Mathematical Sciences\\
University of Alaska Fairbanks\\
PO Box 756660\\
Fairbanks, AK 99775}
\email{ffavr@uaf.edu}
\date{\today}
\subjclass{47B36, 47B37, 47B39}
\keywords{Jacobi matrices, generalized eigenvectors, WKB-type asymptotics, absolutely
continuous spectrum}

\begin{abstract}
Let $J$ be a Jacobi real symmetric matrix on $l_{2}$ with zero diagonal and
non-diagonal entries of the form $\{1+p_{n}\}$. If $p_{n-1}\pm
p_{n}=O(n^{-\alpha })$ with some $\alpha >2/3$, then we prove the existance
of bounded solutions of $Ju=\lambda u$ for a.e. $\lambda \in (-2,2)$ with
the WKB-type asymptotic behavior.
\end{abstract}

\maketitle

\section{Introduction}

The main object of this note is a symmetric Jacobi matrix $J$ of the
form
\begin{equation}
J=\left(
\begin{array}{ccccc}
b_{1} & a_{1} & 0 & 0 & \cdots \\
a_{1} & b_{2} & a_{2} & 0 & \cdots \\
0 & a_{2} & b_{3} & a_{3} & \cdots \\
\vdots & \vdots & \vdots &  &
\end{array}
\right) \ \ \   \label{1.1}
\end{equation}
with $a_{n}>0$ and $b_{n}$ $\in \mathbb{R}$. Under certain conditions on $%
a_{n}$'s$,b_{n}$'s$\,\ $expression $\left( \ref{1.1}\right) $ represents a
self-adjoint operator on $l_{2}\left( \mathbb{N}\right) .$ We are especially
interested in $J$'s close to the free matrix $J_{0}$ defined by $\left( \ref
{1.1}\right) $ with $a_{n}=1$ and $b_{n}$ $=0$ (the discrete free
Schr\"{o}dinger operator). If $\lim a_{n}=1$ and $\lim b_{n}=0,n\rightarrow
\infty ,$ then $J$ is a compact perturbation of $J_{0}.$ Under the
additional condition
\begin{equation}
\sum_{n\geq 1}\left( a_{n}-1\right) ^{2}+\sum_{n\geq 1}b_{n}^{2}<\infty ,
\label{1.2}
\end{equation}
the difference $J-J_{0}$ becomes a Hilbert-Schmidt operator on $l_{2}.$ A
complete description of the spectra of all such Jacobi matrices was recently
found by Killip-Simon in $\left[ {6}\right] $\bigskip . In particular the
absolutely continuous spectrum fills $\left( -2,2\right) .$

In the present note we deal with discrete string operators of the form

\begin{equation}
J=J_{0}+\left(
\begin{array}{ccccc}
0 & p_{1} & 0 & 0 & \cdots \\
p_{1} & 0 & p_{2} & 0 & \cdots \\
0 & p_{2} & 0 & p_{3} & \cdots \\
\vdots & \vdots & \vdots &  &
\end{array}
\right) .\ \ \   \label{1.3}
\end{equation}
It was proven by Janas-Naboko in $\left[ 5\right] $ that under conditions
\begin{equation}
p_{n-1}\pm p_{n}=O\left( n^{-\alpha _{\pm }}\right) ,\alpha _{+}>3/4,\alpha
_{-}>2/3,\alpha _{+}+\alpha _{-}>3/2  \label{1.4}
\end{equation}
the equation ($u:=\{u_{n}\}_{n\geq 1}$)
\begin{equation}
Ju=\lambda u  \label{1.5}
\end{equation}
has two linearly independent bounded solutions $u$ and $\overline{u}$ with
the WKB-type asymptotic behavior ($\varepsilon _{n}:=\frac{1}{2}\frac{%
p_{n-1}-p_{n}}{1+p_{n}}$)

\begin{equation}
u_{n}\left( \theta \right) \sim e^{in\theta }\exp i\sum_{k=2}^{n}\left\{
p_{k}\cot \theta -\varepsilon _{k}\sum_{m\geq 0}\varepsilon _{m+k}\sin
2m\theta \right\} ,n\rightarrow \infty ,  \label{1.6}
\end{equation}
for almost all $\lambda :=2\cos \theta \in \left( -2,2\right) .$

By the Gilbert-Pearson subordinacy theory $\left[ 4\right] $, the existence
of asymptotics $\left( \ref{1.6}\right) $ implies that the absolutely
continuous spectrum of $J$ is $\left( -2,2\right) .\,$Note that $\left[ {5}%
\right] $ belongs to a recent sequence of papers on conservation of
absolutely continuous spectra of Schr\"{o}dinger operators and Jacobi
matrices under slowly decaying perturbations which began with Kiselev $\left[
7\right] ,\left[ 8\right] $, Christ-Kiselev $\left[ 1\right] ,$ and Remling $%
\left[ 11\right] .\,$The original approach was based upon a subtle analysis
of WKB-type asymptotics. It was discovered soon by Deift and Killip $\left[ 3%
\right] $ that the final solution to the question on the support of
absolutely continuous spectra of Schr\"{o}dinger operators and Jacobi
matrices with $L_{2}$ (respectively $l_{2}$) potentials follows from an
elegant idea based on certain trace formulas. This idea was immediately
employed by Molchanov-Novitskii-Vainberg $\left[ 10\right] $ (the
Schr\"{o}dinger operator case) and Laptev-Naboko-Safronov $\left[ 9\right] $
(the Jacobi matrix case) to come to a number of important conclusions on the
absolutely continuous spectrum. The culminating paper in this series is the
above mentioned preprint $\left[ 6\right] $ by Killip-Simon. The
Gilbert-Pearson theory and the WKB-analysis hence do not appear to be
necessary tools in this circle of questions. However the existence of
generalized eigenfunctions of absolutely continuous spectrum with the WKB
asymptotic behavior has become a separate issue interesting in its own right
(e.g. it is a principal ingredient in establishing existence of wave
operators $\left[ {2}\right] $). The main result of this paper is a further
step in improving conditions\ $\left( \ref{1.4}\right) $ and asymptotics $%
\left( \ref{1.6}\right) .$ Namely, we prove the following theorem.

\begin{theorem}
Let $J$ be a symmetric Jacobi matrix of form $\left( \ref{1.3}\right) $
subject to the condition $p_{n-1}\pm p_{n}=O(n^{-\alpha })$ with some $%
\alpha >2/3$. Then, for almost all $\lambda =2\cos \theta \in (-2,2)$, there
exist two linearly independent bounded solutions $u$ and $\overline{u}$ of
equation $\left( \ref{1.5}\right) $ with the WKB-type asymptotics
\begin{equation}
u_{n}\left( \theta \right) \sim e^{in\theta }\exp \left\{ i\cot \theta
\sum_{k=1}^{n}p_{k}\right\} ,\text{ }n\rightarrow \infty .\ \   \label{1.7}
\end{equation}
\end{theorem}

The authors thank Sergei Naboko for originating the interest in this area
and encouragements. \newline

\section{Notation and Preliminaries}

\bigskip

We will follow standard notation with a few exceptions. Given sequence of
matrices $\left\{ M_{n}\right\} _{n\geq 1},$ we write
\begin{equation*}
\prod_{k=1}^{n}M_{k}:=M_{n}M_{n-1}\cdots M_{1}.
\end{equation*}
We also agree to write $M_{n}=\tilde{O}(n^{-\alpha })$ if $\left\|
M_{n}\right\| =O(n^{-\alpha -\epsilon }),\epsilon >0,n\rightarrow \infty .$
Through the paper, we denote by $C$ ($M$) any scalar (matrix-valued )
quantities (with variables or without) which specific values are not of
importance to us.

We reserve $A$ and $B$ for the following two matrices
\begin{equation}
A(a)=\left(
\begin{array}{cc}
a & 0 \\
0 & \overline{a}
\end{array}
\right) ,\ B(a)=\left(
\begin{array}{cc}
0 & \overline{a} \\
a & 0
\end{array}
\right) ,\ a\in \mathbf{C},  \label{2.1}
\end{equation}
which will be basic blocks in our construction. We are going to
systematically  use their elementary properties:\newline
\begin{eqnarray*}
A(a)A(b) &=&A(ab)\newline
;\,\,\,A^{n}(a)=A(a^{n}),\newline
n\in \mathbf{Z;} \\
B(a)B(b) &=&A(\overline{a}b);\,\,\,(I+B(a))^{-1}=(1-|a|^{2})^{-1}(I-B(a)); \\
A(a)B(b) &=&B(\overline{a}b)\newline
;\,\,\ B(a)A(b)=B(ab)\newline
.
\end{eqnarray*}
The following lemma will play a crucial role in our arguments.

\begin{lemma}
Let $A_{n}=A\left( a_{n}\right) ,$ where $\{a_{n}\}$ is a complex sequence
and $\inf \{\left| a_{n}\right| \}>0.$ If $\left\{ M_{n}\right\} $ is a
sequences of $2\times 2$ matrices and $M_{n}=\tilde{O}(n^{-1})$ then

\begin{equation}
\prod_{k=1}^{n}(A_{k}+M_{k})=A\left( \prod_{k=1}^{n}a_{k}\right) \cdot
\left\{ M+\tilde{O}(1)\right\} ,n\rightarrow \infty ,  \label{2.2}
\end{equation}
where $M$ is some $2\times 2$ matrix. If $\sup\limits_{n\geq 1}\left|
a_{n}\right| ^{-1}\left\| M_{n}\right\| <1/4$ then $M$ is invertible.
\end{lemma}

\begin{proof}
All matrices $A_{k}$ are obviously invertible and
\begin{equation*}
A_{k}+M_{k}=A_{k}(I+A_{k}{}^{-1}M_{k}).
\end{equation*}
\newline
It immediately follows by induction from
\begin{equation*}
(I+A_{k}{}^{-1}M_{k})A_{k-1}=A_{k-1}\left(
I+A_{k-1}{}^{-1}A_{k}{}^{-1}M_{k}A_{k-1}\right)
\end{equation*}
that
\begin{equation}
\prod_{k=1}^{n}(A_{k}+M_{k})=\prod_{k=1}^{n}A_{k}\cdot
\prod_{k=1}^{n}\left\{ I+\left( \prod_{m=1}^{k}A_{m}\right) ^{-1}B_{k}\left(
\prod_{m=1}^{k-1}A_{m}\right) \right\} .  \label{2.3}
\end{equation}

Due to the specific form $\left( \ref{2.1}\right) $ of our matrices $A_{k}$
we have
\begin{equation*}
\prod_{k=1}^{n}A_{k}=A(\alpha _{n}),\text{ where }\alpha
_{n}:=\prod_{k=1}^{n}a_{k},
\end{equation*}
and $\left( \ref{2.3}\right) $ then reads

\begin{equation}
\prod_{k=1}^{n}(A_{k}+M_{k})=A(\alpha _{n})\cdot \prod_{k=1}^{n}\left\{
I+A(\alpha _{k}^{-1})M_{k}A(\alpha _{k-1})\right\} .  \label{2.4}
\end{equation}

Since $||A\left( a\right) ||\leq 2\left| a\right| $ one has

\begin{eqnarray*}
\left\| A(\alpha _{k}^{-1})M_{k}A(\alpha _{k-1})\right\|  &\leq &4\left|
\alpha _{k}^{-1}\right| \left| \alpha _{k-1}\right| \left\| M_{k}\right\|
=4\left| a_{k}\right| ^{-1}\left| \alpha _{k-1}\right| ^{-1}\left| \alpha
_{k-1}\right| \left\| M_{k}\right\|  \\
&=&4\left| a_{k}\right| ^{-1}\left\| M_{k}\right\| \leq C\left\|
M_{k}\right\| =\tilde{O}(k^{-1}),
\end{eqnarray*}

and hence the product on the right hand side of $\left( \ref{2.4}\right) $
converges to some invertible $M$.
\end{proof}

The following lemma can be extracted from some more general results of $%
\left[ 1\right] .$

\begin{lemma}
Let $K$ be a linear operation defined on finite vectors $x=\{x_{n}\}$ by
\begin{equation}
(Kx)(\theta )=\varphi (\theta )\sum_{n\geq 1}\xi _{n}(\theta )x_{n},
\label{2.5}
\end{equation}
where $\varphi $ is a smooth indicator of a fixed compact interval $I$, and $%
|\xi _{n}(\theta )|=1$. If
\begin{equation}
||Kx||_{L_{2}(I)}\leq C||x||_{l_{2}},\forall x\in l_{2},  \label{2.6}
\end{equation}
then the series $\sum\limits_{n\geq 1}\xi _{n}(\theta )x_{n}$ and $%
\sum\limits_{n\geq m}\xi _{n}(\theta )\overline{\xi }_{m}(\theta
)x_{n}y_{m}\,$converge for a.e. $\theta \in I$ and every $x,y\in l_{2}$.
\end{lemma}

The proposition below can easily be assembled from the relevant statements
of $\left[ 1\right] .$ For the reader's convenience, we present here its
streamlined proof.

\begin{proposition}
Let $x_{n},y_{n},z_{n}=\tilde{O}(n^{-2/3})$ and real. Define $\xi
_{n}(\theta ):=\exp i(2n\theta +\cot \theta \sum_{k=1}^{n}z_{k})$, $0<\theta
<\pi $. Then the series
\begin{equation}
\sum_{n\geq 1}\xi _{n}(\theta )x_{n},\text{ }\sum\limits_{n\geq m}\xi
_{n}(\theta )\overline{\xi }_{m}(\theta )x_{n}y_{m}  \label{2.7}
\end{equation}
converge for a.e. $\theta \in (0,\pi )$ and \ for those $\theta :$%
\begin{equation}
\sum_{k\geq n}\xi _{k}(\theta )x_{k}=O(n^{-1/6}).  \label{2.8}
\end{equation}
\end{proposition}

\begin{proof}
Consider a linear operator $K$ defined on $x=\{x_{n}\}\in l_{2}$ by (\ref
{2.5}) with an interval $I\subset (0,\pi )$ and $\varphi (0)=0=\varphi (\pi )
$. We show that estimate (\ref{2.6}) holds.\newline
To this end, consider
\begin{equation}
||Kx||_{L_{2}(0,\pi )}^{2}=\int_{0}^{\pi }\left( \sum_{n,m\geq 1}\xi
_{n}(\theta )\overline{\xi _{m}}(\theta )x_{n}x_{m}\right) \varphi
^{2}(\theta )d\theta =\sum_{n,m\geq 1}K(n,m)x_{n}x_{m},  \label{2.9}
\end{equation}
where
\begin{equation*}
K(n,m):=\int_{0}^{\pi }e^{2i(n-m)\theta }F_{nm}(\theta )d\theta ,
\end{equation*}
\begin{equation*}
F_{nm}(\theta ):=e^{iZ_{nm}\cot \theta }\varphi ^{2}(\theta
),Z_{nm}:=\sum_{k=1}^{n}z_{k}-\sum_{k=1}^{m}z_{k}.
\end{equation*}
Let us estimate $K(n,m)$. Assume $n\neq m$. Twice by parts yields
\begin{equation}
K(n,m)=\left( \frac{1}{2i(n-m)}\right) ^{2}\int_{0}^{\pi }e^{2i(n-m)\theta
}F_{nm}^{\prime \prime }(\theta )d\theta .  \label{2.10}
\end{equation}
Note, that the integrated terms vanish in virtue of $\varphi (0)=0=\varphi
(\pi )$. By a direct computation,
\begin{equation*}
F_{nm}^{\prime \prime }(\theta )=e^{iZ_{nm}\cot \theta }\left\{ 2\left(
\left( \varphi ^{\prime }\right) ^{2}+\varphi \varphi ^{\prime \prime
}\right) +2iZ_{nm}\frac{\varphi ^{2}\cot \theta -2\varphi \varphi ^{\prime }%
}{\sin ^{2}\theta }-Z_{nm}^{2}\frac{\varphi ^{2}}{sin^{4}\theta }\right\} ,
\end{equation*}
and one easily derives
\begin{equation}
|F_{nm}^{\prime \prime }(\theta )|\leq C\left( 1+\left| Z_{nm}\right|
+\left| Z_{nm}^{2}\right| \right) ,\ \ C=C(\varphi ,I).  \label{2.11}
\end{equation}
By condition, $z_{n}=\tilde{O}(k^{-2/3})$ and hence $\{z_{n}\}\in l_{3/2}$.
By H\"{o}lder's inequality it follows from (\ref{2.11}) that $\left(
z=\{z_{n}\}\right) $
\begin{equation*}
|F_{nm}^{\prime \prime }(\theta )|\leq C\left( 1+||z||_{l_{3/2}}\cdot |n-m|^{%
\frac{1}{3}}+||z||_{l_{3/2}}^{2}\cdot |n-m|^{\frac{2}{3}}\right) \leq
C\left( 1+|n-m|^{\frac{2}{3}}\right) ,
\end{equation*}
and hence (\ref{2.10}) implies
\begin{equation}
|K(n,m)|\leq C|n-m|^{-2}|n-m|^{\frac{2}{3}}=C|n-m|^{-\frac{4}{3}},\ n\neq m
\label{2.12}
\end{equation}
If $n=m$, then
\begin{equation}
K(n,m)=\int_{0}^{\pi }\varphi ^{2}(\theta )d\theta <\pi   \label{2.13}
\end{equation}
and it follows from (\ref{2.9}), (\ref{2.12}), and (\ref{2.13}) that
\begin{equation}
||Kx||_{L_{2}(0,\pi )}^{2}\leq C\left( ||x||_{l_{2}}+\sum_{n\neq m\geq
1}|n-m|^{-\frac{4}{3}}|x_{n}||x_{m}|\right) .  \label{2.14}
\end{equation}
For the sum in (\ref{2.14}), by H\"{o}lder's inequality, one has
\begin{equation*}
\sum_{n\neq m}|n-m|^{-\frac{4}{3}}|x_{n}||x_{m}|=\sum_{n\neq m}|n-m|^{-\frac{%
2}{3}}|x_{n}||n-m|^{-\frac{2}{3}}|x_{m}|
\end{equation*}
\begin{equation*}
\leq \sum_{n\geq 1}\left( \sum_{m\neq n}|n-m|^{-\frac{4}{3}}\right)
x_{n}^{2}\leq C\sum_{n\neq m}|x_{n}|^{2}=C||x||_{l_{2}}^{2}
\end{equation*}
and estimate (\ref{2.6}) is established. So, our operator $K$ is subject to
the conditions of Lemma 2 and both series (\ref{2.7}) converge for a.e. $%
\theta \in I$ and hence for a.e. $\theta \in (0,\pi )$ since $I$ is an
arbitrary subinterval of $(0,\pi )$.\newline
Now we prove (\ref{2.8}). Introduce a new vector $\widetilde{x}$ by
\begin{equation*}
\tilde{x}_{n}:=n^{1/6}x_{n}=n^{1/6}\tilde{O}(n^{-2/3})=\tilde{O}(n^{-1/2}).
\end{equation*}
Since $\tilde{x}\in l_{2}$, by Lemma 2, the sequence
\begin{equation*}
G_{n}(\theta ):=\sum_{k=1}^{n}\xi _{k}(\theta )\tilde{x}_{k}
\end{equation*}
converges a.e. on $(0,\pi ).$ Performing Abel's transformation:
\begin{equation*}
\sum_{k\geq n}\xi _{k}(\theta )x_{k}=\sum_{k\geq n}k^{-1/6}\xi _{k}(\theta )%
\tilde{x}_{k}
\end{equation*}
\begin{equation*}
=\sum_{k\geq n}k^{-1/6}G_{k}(\theta )=-n^{-1/6}G_{n-1}(\theta )+\sum_{k\geq
n}\left( k^{-1/6}-(k+1)^{-1/6}\right) G_{k}(\theta )
\end{equation*}
\begin{equation*}
=O(n^{-1/6}),\ \text{a.e.}\ \theta \in (0,\pi ),
\end{equation*}
we arrive at (\ref{2.8}).
\end{proof}

\section{Proof of Theorem 1}

Observe first that due to the asymptotic nature of the theorem we are free
to drop in matrix $\left( \ref{1.3}\right) $ a finite number of rows and
columns. Rewriting equation $\left( \ref{1.5}\right) \,\,$in the form

\begin{equation*}
(1+p_{n-1})u_{n-1}+(1+p_{n})u_{n+1}=\lambda u_{n},\,n=2,3,...,
\end{equation*}
or in the matrix form
\begin{equation*}
\left(
\begin{array}{c}
u_{n} \\
u_{n+1}
\end{array}
\right) =\left(
\begin{array}{cc}
0, & 1 \\
-\frac{1+p_{n-1}}{1+p_{n}}, & \frac{\lambda }{1+p_{n}}
\end{array}
\right) \left(
\begin{array}{c}
u_{n-1} \\
u_{n}
\end{array}
\right) ,\ \ \ \ n=2,3...,
\end{equation*}
immediately implies
\begin{equation}
\left(
\begin{array}{c}
u_{n} \\
u_{n+1}
\end{array}
\right) =\prod_{k=2}^{n}\left(
\begin{array}{cc}
0, & 1 \\
-\frac{1+p_{k-1}}{1+p_{k}}, & \frac{\lambda }{1+p_{k}}
\end{array}
\right) \left(
\begin{array}{c}
u_{1} \\
u_{2}
\end{array}
\right) .\ \ \   \label{3.1}
\end{equation}
Set $\lambda :=2cos\theta =\xi +\overline{\xi }$, where $\xi :=e^{i\theta
},\theta \in (0,\pi ),$ and
\begin{equation*}
U:=\left(
\begin{array}{cc}
1 & 1 \\
\xi & \overline{\xi }
\end{array}
\right) .
\end{equation*}
Introducing the sequence of matrices
\begin{equation}
T_{n}:=U^{-1}\left(
\begin{array}{cc}
0, & 1 \\
-\frac{1+p_{n-1}}{1+p_{n}}, & \frac{\lambda }{1+p_{n}}
\end{array}
\right) U  \label{3.2}
\end{equation}
transforms $(\ref{3.1})$ into
\begin{equation}
\left(
\begin{array}{c}
u_{n} \\
u_{n+1}
\end{array}
\right) =U\left( \prod_{k=2}^{n}T_{k}\right) U^{-1}\left(
\begin{array}{c}
u_{1} \\
u_{2}
\end{array}
\right) =:T(n)\left(
\begin{array}{c}
u_{1} \\
u_{2}
\end{array}
\right) .  \label{3.3}
\end{equation}
By involved but a straightforward computation one verifies that
\begin{equation}
T_{n}=\left\{ I+A(\tilde{z_{n}})-B(\tilde{z_{n}})\right\} \cdot A(\xi ),
\label{3.4}
\end{equation}
where $A$ and $B$ are defined by $(\ref{2.1})$ and
\begin{equation}
\text{ }\tilde{z}_{n}:=\frac{z_{n}}{1+p_{n}},\,\,\,\ z_{n}:=\frac{%
p_{n-1}-p_{n}}{2}+i\frac{p_{n-1}+p_{n}}{2}\cot \theta .  \label{3.5}
\end{equation}
To derive the assertions of the theorem one needs to analyze the asymptotic
behavior of the matrix-valued sequence
\begin{equation}
T(n)=U\left( \prod_{k=2}^{n}T_{k}\right) U^{-1}.  \label{3.6}
\end{equation}
To this end we introduce a sequence of invertible matrices $\left\{
U_{k}\right\} _{k\geq 2}$ to be chosen later and rewrite $(\ref{3.6})$ as
follows
\begin{equation}
T(n)=\left( UU_{n+1}\right) \cdot \left\{
\prod_{k=2}^{n}(U_{k+1}^{-1}T_{k}U_{k})\right\} \cdot
U_{2}^{-1}U^{-1}=\left( UU_{n+1}\right) \cdot \left( \prod_{k=2}^{n}%
\widetilde{T}_{k}\right) \cdot \left( UU_{2}\right) ^{-1},  \label{3.7}
\end{equation}
where $\widetilde{T}_{n}:$ $=U_{n+1}^{-1}T_{n}U_{n}.$ With Lemma 1 in mind,
we want to represent each $\widetilde{T}_{k}$ as a diagonal matrix plus a
small remainder. We show now that the sequence

\begin{equation}
U_{n}=A(\xi ^{n})A(\alpha _{n})\left\{ I+B(\beta _{n})\right\} ,  \label{3.8}
\end{equation}
with proper $\alpha _{n}$ and $\beta _{n}$ will do it. Indeed, by $(\ref{3.4}%
)$ and $(\ref{3.8})$ one has
\begin{equation}
\widetilde{T}_{n}=U_{n+1}^{-1}T_{n}U_{n}=\left\{ I+B(\beta _{n+1})\right\}
^{-1}\Omega _{n}\left\{ I+B(\beta _{n})\right\} ,  \label{3.9}
\end{equation}
where
\begin{equation*}
\Omega _{n}:=A^{-1}(\alpha _{n+1})A^{-1}(\xi ^{n+1})\left\{ I+A(\tilde{z_{n}}%
)-B(\tilde{z_{n}})\right\} A(\xi )A(\xi ^{n})A(\alpha _{n}).
\end{equation*}
We first treat $\Omega _{n}$. Using the properties of matrices $A$ and $B$,
we get
\begin{eqnarray}
\Omega _{n} &=&A(\xi ^{-n-1}\alpha _{n+1}^{-1})\cdot \left\{ I+A(\tilde{z_{n}%
})-B(\tilde{z_{n}})\right\} \cdot A(\xi ^{n+1}\alpha _{n})  \notag \\
&=&A\left( \frac{\alpha _{n}}{\alpha _{n+1}}\right) +A(\xi ^{-n-1}\alpha
_{n+1}^{-1})A(\tilde{z_{n}})A(\xi ^{n+1}\alpha _{n})  \notag \\
&&-A(\xi ^{-n-1}\alpha _{n+1}^{-1})B(\tilde{z_{n}})A(\xi ^{n+1}\alpha _{n})
\notag \\
&=&A\left( \frac{\alpha _{n}}{\alpha _{n+1}}\right) +A\left( \frac{\alpha
_{n}}{\alpha _{n+1}}\widetilde{z}_{n}\right) -B\left( \xi ^{n+1}\overline{%
\alpha }_{n+1}^{-1}\widetilde{z}_{n}\right) A(\xi ^{n+1}\alpha _{n})  \notag
\\
&=&A\left( \frac{\alpha _{n}}{\alpha _{n+1}}\left( 1+\widetilde{z}%
_{n}\right) \right) -B\left( \xi ^{2\left( n+1\right) }\frac{\alpha _{n}}{%
\overline{\alpha }_{n+1}}\widetilde{z}_{n}\right) .  \label{3.10}
\end{eqnarray}
Choose now
\begin{equation*}
\alpha _{n}\left( \theta \right) =\exp \sum_{k=2}^{n-1}z_{k}\left( \theta
\right) .
\end{equation*}
Due to the conditions on $p_{n},$ we have $\dfrac{\alpha _{n}}{\alpha _{n+1}}%
=\exp (-z_{n})=1-z_{n}+\tilde{O}(n^{-1}),\,\widetilde{z}_{n}=\left(
1+p_{n}\right) ^{-1}z_{n}=z_{n}+\tilde{O}(n^{-1}),$ and hence the first term
on the right in $(\ref{3.10})$ reads
\begin{equation}
A\left( \frac{\alpha _{n}}{\alpha _{n+1}}\left( 1+\widetilde{z}_{n}\right)
\right) =I+\tilde{O}(n^{-1}).  \label{3.11}
\end{equation}
Next, since $\dfrac{\overline{\alpha }_{n}}{\overline{\alpha }_{n+1}}=1-%
\overline{z}_{n}+\tilde{O}(n^{-1}),$
\begin{eqnarray*}
\xi ^{2\left( n+1\right) }\frac{\alpha _{n}}{\overline{\alpha }_{n+1}}%
\widetilde{z}_{n} &=&\widetilde{z}_{n}\dfrac{\overline{\alpha }_{n}}{%
\overline{\alpha }_{n+1}}\xi _{n}=\left( z_{n}+\tilde{O}(n^{-1})\right)
\left( 1-\overline{z}_{n}+\tilde{O}(n^{-1})\right) \xi _{n} \\
&=&z_{n}\xi _{n}+\tilde{O}(n^{-1}),
\end{eqnarray*}
where
\begin{equation}
\xi _{n}=\xi _{n}\left( \theta \right) :=\exp 2i\left( \left( n+1\right)
\theta +\func{Im}\sum_{k=2}^{n-1}z_{k}\left( \theta \right) \right) ,
\label{3.12}
\end{equation}
and the second term on the right side of $(\ref{3.10})$ becomes
\begin{equation}
B\left( \xi ^{2\left( n+1\right) }\frac{\alpha _{n}}{\overline{\alpha }_{n+1}%
}\widetilde{z}_{n}\right) =B\left( z_{n}\xi _{n}\right) +\tilde{O}(n^{-1}).
\label{3.13}
\end{equation}
Plugging (\ref{3.11}) and (\ref{3.13}) in $(\ref{3.10})$ yields
\begin{equation}
\Omega _{n}=I-B\left( z_{n}\xi _{n}\right) +\tilde{O}(n^{-1}),  \label{3.14}
\end{equation}
and combining (\ref{3.14}) with (\ref{3.9}), we get
\begin{eqnarray}
\widetilde{T}_{n} &=&\left\{ I+B(\beta _{n+1})\right\} ^{-1}\cdot \left\{
I-B\left( z_{n}\xi _{n}\right) +\tilde{O}(n^{-1})\right\} \cdot \left\{
I+B(\beta _{n})\right\}  \notag \\
&=&(1-|\beta _{n+1}|^{2})^{-1}\cdot \left\{ \Xi _{n}+\tilde{O}%
(n^{-1})\right\} ,  \label{3.15}
\end{eqnarray}
where
\begin{equation}
\Xi _{n}:=\left\{ I-B(\beta _{n+1})\right\} \cdot \left\{ I-B\left( z_{n}\xi
_{n}\right) \right\} \cdot \left\{ I+B(\beta _{n})\right\} .  \label{3.16}
\end{equation}
Multiplying (\ref{3.16}) out and using the properties of $A$ and $B$, after
tedious but direct computations one obtains

\begin{eqnarray}
\Xi _{n} &=&A\left( I+\overline{\beta }{_{n+1}z_{n}\xi _{n}-\overline{\beta }%
{_{n+1}}\beta _{n}-}\overline{{z}_{n}{\xi _{n}}}{\beta _{n}}\right)  \notag
\\
&&+B\left( \beta _{n}-\beta _{n+1}-z{_{n}\xi _{n}+\beta {_{n+1}}\beta _{n}}%
\overline{{z}_{n}{\xi _{n}}}\right)  \label{3.17}
\end{eqnarray}
By condition, the sequences $\left\{ p_{n-1}\pm p_{n}\right\} $ and $\left\{
{\xi _{n}}\left( \theta \right) \right\} $ are subject to the conditions of
Proposition 1 and hence the series
\begin{equation*}
\sum_{n\geq 1}{z_{n}\xi _{n}}\left( \theta \right) {=\frac{1}{2}}\sum_{n\geq
1}{\xi _{n}\left( \theta \right) (p_{n-1}-p_{n})+\frac{i}{2}\cot }\theta
\sum_{n\geq 1}{\xi _{n}}\left( \theta \right) {(p_{n-1}+p_{n})}
\end{equation*}
converges for almost all $\theta \in (0,\pi ).$ Choose now $\beta _{n}$ to
satisfy
\begin{equation}
\beta _{n}=\beta _{n}\left( \theta \right) =-\sum_{k\geq n}z_{k}\xi _{k}{%
\left( \theta \right) }.  \label{3.18}
\end{equation}
The sequence $\beta _{n}$ is well-defined and due to (\ref{2.8})
\begin{equation}
\beta _{n}=O(n^{-\frac{1}{6}}),\,\quad \text{a. e. }\mathrm{\ \,\,}\,\theta
\in (0,\pi ).  \label{3.19}
\end{equation}
Since $\beta _{n}-\beta _{n+1}=z_{n}\xi _{n}$ and ${\beta {_{n+1}}\beta _{n}}%
\overline{{z}_{n}{\xi _{n}}}=\tilde{O}(n^{-1}),$ (\ref{3.17}) reduces to
\begin{equation*}
\Xi _{n}=A\left( I{-}\left| {\beta {_{n+1}}}\right| ^{2}{-}\overline{{z}_{n}{%
\xi _{n}}}{\beta _{n}}\right) +\tilde{O}(n^{-1}),
\end{equation*}
and it follows from (\ref{3.15}) that for a. e.$\mathrm{\,}\,\,\theta \in
(0,\pi )\ $
\begin{equation}
\widetilde{T}_{n}=A\left( 1+a_{n}\right) +\tilde{O}(n^{-1}),\text{ }\ \
\text{where}\ \ \ \ a_{n}:=-\overline{{z}_{n}{\xi _{n}}}{\beta _{n}}.\ \
\label{3.20}
\end{equation}
Since $a_{n}=\tilde{O}(n^{-2/3})O(n^{-1/6})=\tilde{O}(n^{-5/6}),$ the limit
\begin{equation*}
\lim_{n\rightarrow \infty }\left\{ \ln \prod_{k=2}^{n}\left( 1+a_{k}\right)
-\sum_{k=2}^{n}a_{k}\right\} =:C,
\end{equation*}
clearly exists, and applying Lemma 1 to $\left\{ \widetilde{T}_{n}\right\}
\, $defined by (\ref{3.20}) yields
\begin{eqnarray}
\prod_{k=2}^{n}\widetilde{T}_{k} &=&A\left( \prod_{k=2}^{n}\left(
1+a_{k}\right) \right) \left( M+\tilde{O}\left( 1\right) \right)  \notag \\
&=&A\left( \exp \sum_{k=2}^{n}a_{k}\right) \left( M+\tilde{O}\left( 1\right)
\right) ,  \label{3.21}
\end{eqnarray}
where $M$ \ is some matrix. Observe now that for $U_{n+1}$ defined by (\ref
{3.8}) we have:
\begin{equation}
U_{n+1}=A\left( {\xi ^{n+1}\alpha _{n+1}}\right) +B\left( {\beta _{n+1}}%
\overline{{\xi ^{n+1}\alpha _{n+1}}}\right) =A\left( {\xi ^{n+1}\alpha _{n+1}%
}\right) +\tilde{O}\left( 1\right) ,  \label{3.22}
\end{equation}
\newline
for $\,$a.e. $\theta \in (0,\pi ).$ Combining (\ref{3.7}), (\ref{3.21}), and
(\ref{3.22}) we have
\begin{eqnarray}
T(n) &=&UA\left( {\xi ^{n+1}\alpha _{n+1}}\right) A\left( \exp
\sum_{k=2}^{n}a_{k}\right) M\left( UU_{2}\right) ^{-1}+\tilde{O}\left(
1\right)  \notag \\
&=&U\cdot A\left( {\xi ^{n+1}\alpha _{n+1}}\exp \sum_{k=2}^{n}a_{k}\right)
\cdot M+\tilde{O}\left( 1\right)  \notag \\
&=&U\cdot A\left( \exp \left\{ i\left( n+1\right) \theta
+\sum_{k=2}^{n}z_{k}-\sum_{k=2}^{n}\overline{{z}_{k}{\xi _{k}}}{\beta _{k}}%
\right\} \right) \cdot M+\tilde{O}\left( 1\right) .  \label{3.23}
\end{eqnarray}
Let us now treat the sums on the right in (\ref{3.23}):
\begin{eqnarray}
\sum_{k=2}^{n}z_{k} &=&\sum_{k=2}^{n}\frac{{p_{k-1}-p_{k}}}{2}{+{i\cot
\theta }\sum_{k=2}^{n}\frac{{p_{k-1}+p_{k}}}{2}}  \notag \\
{} &=&\frac{{p_{1}-p_{n}}}{2}+{{i\cot \theta }\sum_{k=1}^{n}{p_{k}-i\frac{{%
p_{1}+p_{n}}}{2}\cot \theta }}  \notag \\
&=&{{i\cot \theta \cdot }\sum_{k=1}^{n}{p_{k}+}}C\left( \theta \right) +%
\tilde{O}\left( 1\right) .  \label{3.24}
\end{eqnarray}
We show that the other sum on the right side of (\ref{3.23}) actually
converges. Indeed, by (\ref{3.18}) we have
\begin{eqnarray}
-\sum_{k=2}^{n}\overline{{z}_{k}{\xi _{k}}}{\beta _{k}} &=&{}\sum_{k=2}^{n}%
\overline{{z}_{k}{\xi _{k}}}{\sum_{m=k}^{n}{z}_{m}{\xi _{m}+}}\sum_{k=2}^{n}%
\overline{{z}_{k}{\xi _{k}}}\cdot \sum_{k\geq n}{z}_{k}{\xi _{k}}  \notag \\
&=&\sum_{m\geq k\geq 2}^{n}{{{\xi _{m}}}\overline{{\xi }}{{_{k}}}}\,%
\overline{{z}}_{k}{{z}_{m}{-}\beta _{n}}\sum_{k=2}^{n}\overline{{z}_{k}{\xi
_{k}}}=\sum_{m\geq k\geq 2}^{n}{{\xi _{m}}}\overline{{\xi }}{{_{k}}}%
\overline{{z}}_{k}{{z}_{m}{+}}\tilde{O}\left( n^{-1/6}\right) .  \label{3.25}
\end{eqnarray}
Since $z_{n}=x_{n}^{\left( -\right) }+ix_{n}^{\left( +\right) }\cot \theta $%
, where $x_{n}^{\left( \pm \right) }={p_{n-1}\pm p_{n},\,}$the sum on the
right side of (\ref{3.25}) breaks down into four${\,\ }$series
\begin{equation*}
\sum_{m\geq k\geq 2}^{n}{{\xi _{m}(\theta )}}\overline{{\xi }}{{_{k}{(\theta
)}}x_{m}^{\left( \pm \right) }}x_{k}^{\left( \pm \right) }
\end{equation*}
which by Proposition 1 converge for a.e. $\theta \in \left( 0,\pi \right) \,$%
\ and (\ref{3.25}) takes form
\begin{equation*}
-\sum_{k=2}^{n}\overline{{z}_{k}{\xi _{k}}}{\beta _{k}=C}\left( {\theta }%
\right) {+}\tilde{O}\left( n^{-1/6}\right) ,\text{ a.e. }\theta \in \left(
0,\pi \right) .\text{ }
\end{equation*}
Combing this with (\ref{3.24}) and inserting into (\ref{3.23}), we finally
obtain that for a.e. $\theta \in \left( 0,\pi \right) $%
\begin{equation}
T\left( n\right) =U\cdot A\left( \exp i\left\{ n\theta +\cot \theta \cdot
\sum_{k=1}^{n}p_{k}\right\} \right) \cdot M\left( \theta \right) +\tilde{O}%
\left( 1\right) .  \label{3.26}
\end{equation}
It is not hard to see that the matrix $M$ is invertible (after, if needed,
dropping a finite number of rows and columns in the original matrix (\ref
{1.3})). This means that by choosing initial conditions $\left(
u_{1},u_{2}\right) \,\ $we can always attain
\begin{equation*}
M\left(
\begin{array}{c}
u_{1} \\
u_{2}
\end{array}
\right) =\left(
\begin{array}{c}
1 \\
0
\end{array}
\right) ,
\end{equation*}
and it follows from (\ref{3.3}) and (\ref{3.26}) that for a.e. $\theta \in
\left( 0,\pi \right) $
\begin{eqnarray*}
\left(
\begin{array}{c}
u_{n} \\
u_{n+1}
\end{array}
\right) &=&U\cdot A\left( \exp i\left\{ n\theta +\cot \theta \cdot
\sum_{k=1}^{n}p_{k}\right\} \right) \left(
\begin{array}{c}
1 \\
0
\end{array}
\right) +\tilde{O}\left( 1\right) \\
&=&U\left(
\begin{array}{c}
1 \\
0
\end{array}
\right) \cdot \exp i\left\{ n\theta +\cot \theta \cdot
\sum_{k=1}^{n}p_{k}\right\} +\tilde{O}\left( 1\right) ,n\rightarrow \infty .
\end{eqnarray*}
The latter immediately implies (\ref{1.7}) and the theorem is proven.

\begin{corollary}
Under the conditions of Theorem 1, the absolutely continuous spectrum of $J$
fills the interval $\left( -2,2\right) .$

\begin{remark}
It is not difficult to prove Theorem 1 under conditions $p_{n-1}\pm
p_{n}=o\left( n^{-2/3}\right) .$ Moreover $\left[ 1\right] $ makes one
believe that the theorem still holds if $p_{n-1}\pm p_{n}=\tilde{O}\left(
n^{-1/2}\right) $. \ As it was discussed in Introduction, the conclusion of
Corollary 1 remains valid $\left[ 6\right] $ under the only condition $%
\left\{ p_{n}\right\} \in l_{2}$ (see (\ref{1.2})) and one may cautiously
hope that so does asymptotics (\ref{1.7}). It seems to be an open problem
even in the case of Schr\"{o}dinger operators.
\end{remark}
\end{corollary}

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\end{document}
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