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2000 Mathematics Subject Classification: Primary 47B36; Secondary 47A10
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Unbounded Jacobi Operators  
Jacobi Matrices 
Discrete Spectrum
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\documentclass[12pt]{amsart}
\usepackage{hyperref,latexsym}
\usepackage{enumerate}

\DeclareMathOperator{\re}{Re}
\DeclareMathOperator{\im}{Im}
\DeclareMathOperator{\dom}{Dom}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator*{\res}{Res}

\theoremstyle{plain}
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\newtheorem{corollary}[theorem]{Corollary}
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\newcommand{\abs}[1]{\left|#1\right|}
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\begin{document}

\title[Jacobi Matrices with Rapidly Growing Weights]
      {Jacobi Matrices with Rapidly Growing Weights Having Only
      Discrete Spectrum}

\date{}
\author{Luis O. Silva}
\address{Departamento de M{\'e}todos Matem{\'a}ticos y Num{\'e}ricos\\
         IIMAS, Universidad Nacional Aut{\'o}noma de M{\'e}xico\\
         Apdo. Postal 20-726 M{\'e}xico, D.F. 01000}
\email{silva@leibniz.iimas.unam.mx}
\thanks{Research partially supported by Universidad Nacional Aut\'onoma
        de M\'exico under Project PAPIIT-DGAPA IN 101902 and by CONACYT
          under Project P42553F}
\author{Julio H. Toloza}
\email{jtoloza@leibniz.iimas.unam.mx}
\subjclass[2000]{Primary 47B36; Secondary 47A10}
\keywords{Unbounded Jacobi Operators; Jacobi Matrices; Discrete Spectrum}

\begin{abstract}
  We establish sufficient conditions for self-adjointness on a
  class of unbounded Jacobi operators defined by matrices with
  main diagonal sequence of very slow growth and rapidly
  growing off-diagonal entries.  With some additional
  assumptions, we also prove that these operators have only
  discrete spectrum.
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $\{b_n\}_{n\in{\mathbb N}}$ be a real sequence whose
elements are given by
\begin{equation}
  \label{eq:weights}
  b_n = n^{\alpha} \left(1+\frac{c_n}{n}+h_n\right)\,,
  \quad \alpha>1\,, \quad 1+\frac{c_n}{n}+h_n\neq 0\,\quad
  \forall n\in\mathbb{N}\,,
\end{equation}
where $\{c_n\}_{n\in{\mathbb N}}$ is bounded with
$\liminf\abs{c_n}>0$, and $\{h_n\}_{n\in{\mathbb N}}$ is such
that $h_n=O(n^{-\gamma})$ when $n\to\infty$, for some $\gamma>1$.

Let $\{q_n\}_{n\in{\mathbb N}}$ be a real sequence that grows
not faster than logarithmically, viz.,
\begin{equation}
  \label{eq:diagonal}
  q_n=O(\log n)\quad\text{as }n\to\infty\,.
\end{equation}

In the Hilbert space $l_2(\mathbb{N})$, we define the Jacobi
operator $J$ as the minimal closed operator such that
\begin{equation}
  \label{eq:matrix-rep}
  (Je_n,e_n)=q_n\,,\quad (Je_n,e_{n+1})=(Je_{n+1},e_n)
  =b_n\,,\quad\forall n\in\mathbb{N}\,,
\end{equation}
where $\{e_n\}_{n\in{\mathbb N}}$ is the canonical basis in
$l_2(\mathbb{N})$.  Clearly, there are closed operators
satisfying (\ref{eq:matrix-rep}), thus $J$ is correctly
defined. This operator is unbounded and we have
defined it in such a way that the semi-infinite Jacobi matrix

\begin{equation}
  \label{eq:jm}
  \begin{pmatrix}
  q_1 & b_1 & 0  &  0  &  \cdots \\[1mm]
  b_1 & q_2 & b_2 & 0 & \cdots \\[1mm]
  0  &  b_2  & q_3  & b_3 &  \\
  0 & 0 & b_3 & q_4 & \ddots\\
  \vdots & \vdots &  & \ddots & \ddots
  \end{pmatrix}
\end{equation}
is the matrix representation of $J$ with respect to the basis
$\{e_n\}_{n\in{\mathbb N}}$ (we refer to \cite{MR1255973} for
a discussion on matrix representation of unbounded symmetric
operators).

For $\alpha>1$ fixed, different sequences
$\{c_n\}_{n\in{\mathbb N}}$, $\{h_n\}_{n\in{\mathbb N}}$ and
$\{q_n\}_{n\in{\mathbb N}}$ define different concrete
realizations of $J$. Throughout this paper, we use $J$ to denote
anyone of these realizations prescribed by 
(\ref{eq:weights})--(\ref{eq:matrix-rep}).

Since the off-diagonal entries (weights) of (\ref{eq:jm}) grow
fast ($\alpha>1$), $J$ is not always
self-adjoint. It is known, nevertheless, that
certain conditions on the weights ensure self-adjointness
when $\{q_n\}_{n\in{\mathbb N}}$ is a bounded sequence.
Indeed, the class of operators studied here is a
generalization of an example suggested by Kostyuchenko and
Mirzoev in \cite{MR1711874} as an illustration of a Jacobi
operator whose matrix representation does not satisfy the
Carleman criterion \cite{MR0184042} but nevertheless, under certain conditions,
can be self-adjoint.  The case of operators $J$ for which
$\{c_n\}_{n\in{\mathbb N}}$ is even-periodic and
$\{q_n\}_{n\in{\mathbb N}}\equiv\{h_n\}_{n\in{\mathbb N}}\equiv 0$
has been studied in \cite{MR1959871,MR1924991}.  There are
formulae in \cite{MR1959871} for the asymptotic behavior of
the generalized eigenvectors and it is proven there, by means
of the Subordinacy Theory \cite{MR915965,MR1179528}, that the
spectrum of $J$ is pure point in the self-adjoint case.

We carry out the spectral analysis of $J$ on basis
of the asymptotic behavior of its generalized eigenvectors.
These asymptotics are obtained by means of a Levinson-type
theorem for discrete linear systems. We use transfer matrices
to study the generalized eigenequation of $J$ and arrive to a
system that turns out to be asymptotically diagonal in the
sense that it satisfies the assumptions of the Benzaid-Lutz
theorem \cite{MR1002291}.  The fact that we can study the
generalized eigenequation through a discrete system, for which
the Benzaid-Lutz theorem yields the asymptotic behavior of the
solutions, is a remarkable property of this class of
operators.

We impose certain mild conditions over the average behavior of the
sequence $\{c_n\}_{n\in{\mathbb N}}$ in order to ensure
self-adjointness.  These conditions are far less restrictive that those
used in previous works and contain them as special cases.
Also, these conditions turn out to be sufficient for
the absence of continuous spectrum.  In passing, we note that
in \cite{MR1959871,MR1924991} a much stronger assumption of
periodicity of the sequence $\{c_n\}_{n\in{\mathbb N}}$ was
required to gain spectral information from the asymptotic
behavior of the generalized eigenvectors of operator $J$.  As
it will become apparent later, the sequences
$\{q_n\}_{n\in{\mathbb N}}$ and $\{h_n\}_{n\in{\mathbb N}}$,
with the asymptotic behavior specified above, play no role
concerning both self-adjointness and spectral properties of
$J$.  Unlike the methods used in \cite{MR1959871,MR1924991},
our main results do not make use of Subordinacy Theory.
Instead we mainly use general facts for Jacobi operators and
Spectral Theory.

To the knowledge of the authors, the spectral analysis of
Jacobi operators corresponding to matrices with rapidly
growing weights has been treated in few occasions
\cite{MR1959871,MR1977920,MR1924991}. In contrast, literature
on Jacobi matrices with slowly growing weights is more
abundant (for instance,
\cite{MR2078087,MR1707753,MR1764960,MR2077208,MR2091056}). In
particular, Jacobi operators with zero main diagonal and
$b_n\sim n^\alpha$, with $1/2<\alpha\leq 1$, has been studied
in \cite{MR1707753,MR1764960,MR2077208}. In these papers it
is proven that, depending on the concrete realization, this
kind of Jacobi operators may have either purely absolutely
continuous spectrum or absolutely continuous spectrum with a
gap where the spectrum is pure point.

This paper is organized as follows. In Section~\ref{sec:preliminaries},
we lay down the notation and some preparatory facts. In
Section~\ref{sec:asymt-behav-gener}, we derive the
asymptotic behavior of solutions of the discrete system associated to
the generalized eigenequation for $J$.
In Section~\ref{sec:self-adjointness}, we establish sufficient conditions
for $J$ to be either self-adjoint (Theorem~\ref{thm:self-adjointness})
or non self-adjoint (Theorem~\ref{thm:nonself-adjointness}), and give
examples. Finally, in Section~\ref{sec:absence-cont-spectr}, we characterize
the spectrum (Corollary~\ref{pp} and Theorem~\ref{thm:compactness-resolvent}).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminaries}
\label{sec:preliminaries}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we introduce the notation and main
concepts that will be used throughout this paper.

The elements of a scalar sequence will be denoted by Latin
lowercase letters. For sequences of vectors and matrices we will use Greek
lowercase letters and Latin capital letters, respectively.
Besides the Hilbert space $l_2(\mathbb{N})=l_2(\mathbb{N}, \mathbb{C})$,
we shall consider the space $l_2(\mathbb{N}, \mathbb{C}^d)$ consisting
of sequences of $d$-dimensional vectors
$\{\xi_n\}_{n\in{\mathbb N}}$ such that
$\sum_{n=1}^\infty\norm{\xi_n}_{\mathbb{C}^d}^2$ is
convergent, where $\norm{\cdot}_{\mathbb{C}^d}$ denotes some
norm in $\mathbb{C}^d$.  Should no confusion arise, we will
denote $\norm{\xi}_{l_2(\mathbb{C}^d)}$ (for the norm in
$l_2(\mathbb{N},\mathbb{C}^d)$) simply as $\norm{\xi}$.
Keeping the notation used in the Introduction, the canonical
basis in $l_2(\mathbb{N})$ is denoted by
$\{e_n\}_{n\in{\mathbb N}}$, while $\{\delta^{(k)}\}_{k=1}^d$
will stand for the canonical basis in $\mathbb{C}^d$.  Given a
scalar sequence $\{f_n\}_{n\in{\mathbb N}}$, the standard
asymptotic notation $O(f_n)$ ($o(f_n)$) as $n\to\infty$,
depending on the context, will be used to denote the $n$-th
element of a sequence of either scalars, vectors, or matrices,
whose norm behaves as $O(f_n)$ ($o(f_n)$) as $n\to\infty$.
All products of matrices will be taken in ``chronological''
order, i.\,e.,
\begin{equation*}
  \prod_{s=p}^mA_s=A_mA_{m-1}\dots A_p\,.
\end{equation*}

Our approach to the spectral analysis of $J$ is based on the
study of the asymptotics of solutions
of the generalized eigenvalue equation
\begin{equation}
  \label{eq:main-recurrence}
  b_{n-1}x_{n-1}(\zeta)+ q_nx_n(\zeta)+
  b_nx_{n+1}(\zeta)=
  \zeta x_n(\zeta)\,,\quad n>1\,,\ \zeta\in\mathbb{C}\,.
\end{equation}
Clearly, if a solution
$x(\zeta)=\{x_n(\zeta)\}_{n\in{\mathbb N}}$ satisfies the
``boundary condition''
\begin{equation}
  \label{eq:boundary}
  q_1x_1(\zeta)+b_1x_2(\zeta)=\zeta  x_1(\zeta)
\end{equation}
and turns out to be in $\dom (J)$, then $x(\zeta)$ is an
eigenvector of $J$ corresponding to the eigenvalue $\zeta$.
To simplify somewhat the notation, in what follows we will not
indicate the dependency of $x$ on $\zeta$.

Equation (\ref{eq:main-recurrence}) can be written as follows
\begin{equation}
  \label{eq:mre}
  \xi_{n+1}=B_n(\zeta ) \xi_n\,,
  \qquad n > 1\,,\quad\zeta\in\mathbb{R}\,,
\end{equation}
where $\xi_n =  \biggl(
       \begin{smallmatrix}
       x_{n-1} \\[2mm]
       x_n
       \end{smallmatrix}\biggr)$ and
        $B_n(\zeta )= \biggl(
         \begin{smallmatrix}   0      &   1
       \\[2mm]   - \frac{b_{n-1}}{b_n}      &
       \frac{\zeta-q_n}{b_n}
       \end{smallmatrix}\biggr)$ is the so-called transfer matrix.
From (\ref{eq:mre}) it follows that
\begin{equation}
  \label{eq:prod}
  \xi_n=
  \prod_{k=n_0}^{n-1}B_k(\zeta)\xi_{n_0}\,,
  \qquad n_0\ge 2\,.
\end{equation}

Since clearly we have $\|x\|^2<\|\xi\|^2\leq 2\|x\|^2$, the
asymptotic behavior (in norm) of solutions of (\ref{eq:mre})
governs the asymptotic behavior of those of
(\ref{eq:main-recurrence}) (and vice-versa). There are various
methods for obtaining the asymptotic behavior of solutions of
discrete linear systems. As already mentioned, the main tools
are the so-called discrete Levinson-type theorems. For the
purpose of the present work we use the Levinson-type theorem
proved by Benzaid and Lutz in \cite{MR1002291} since, as it
will be shown in the next section, system (\ref{eq:mre})
can be rewritten as a discrete linear system which satisfies
the conditions of that theorem. For the sake of completeness, we
state the Benzaid-Lutz theorem below.

\begin{definition}
  A sequence $Q=\{Q_n\}_{n= n_0}^\infty$ ($n_0\in\mathbb{N}$)
  of $d\times d$ diagonal complex matrices,
\begin{equation*}
  Q_n=\diag\{q_n^{(k)}\}_{k=1}^d\,,
\end{equation*}
is said to satisfy the Levinson condition for $k$ (denoted
$Q\in\mathcal{L}(k)$) if there exists a natural number $N\geq
n_0$ and a constant $M>1$ such that, $k$ being fixed, each $j$
($1\leq j\leq d$) falls into one and only one of the two
classes $I_1$ or $I_2$, where
\begin{enumerate}[(a)]
\item \label{item:jin-i_1}
$j\in I_1$ if
\begin{equation*}
\frac{\abs{\prod_{i=N}^nq_i^{(k)}}}
{\abs{\prod_{i=N}^nq_i^{(j)}}}
\to \infty\quad\text{ as } n\to\infty\,,\quad\text{ and }
\end{equation*}
\begin{equation*}
\frac{\abs{\prod_{i=n_1}^{n_2}q_i^{(k)}}}
{\abs{\prod_{i=n_1}^{n_2}q_i^{(j)}}} >\frac{1}{M}\,,
\quad\forall\, n_2\,, n_1\text{ such that } n_2 > n_1 \geq N\,.
\end{equation*}
\item \label{item:jin-i_2}
$j\in I_2$ if
\begin{equation*}
\frac{\abs{\prod_{i=n_1}^{n_2}q_i^{(k)}}}
{\abs{\prod_{i=n_1}^{n_2}q_i^{(j)}}} < M\,,
\quad\forall\, n_2\,, n_1\text{ such that } n_2 > n_1 \geq N\,.
\end{equation*}
\end{enumerate}
\end{definition}

\begin{theorem}[Benzaid-Lutz]
  \label{thm:bl}
  Let $Q=\{Q_n\}_{n= n_0}^\infty$ be a sequence of
$d\times d$ diagonal complex matrices,
\begin{equation*}
  Q_n=\diag\{q_n^{(k)}\}_{k=1}^d\,.
\end{equation*}
Consider also another sequence $R=\{R_n\}_{n= n_0}^\infty$
($n_0\in\mathbb{N}$)  of
$d\times d$ complex matrices. Suppose that the following conditions hold:
\begin{enumerate}[i.]
\item The sequence $Q\in\mathcal{L}(k)$, for all
  $k=1,\dots,d$,\label{N}
\vspace{1mm}
\item $\displaystyle{\sum\limits_{n=n_0}^\infty
\frac{\norm{R_n}}{|q_n^{(k)}|}<\infty}$, \label{R-condition-2}
\vspace{1mm}
\item $\det(Q_n+R_n)\ne 0$ for all $n\ge n_0$.
\end{enumerate}
Then one can find a natural number $n_1\ge n_0$ such that
there exists a basis $\varphi^{(k)}$ ($k=1\dots, d$) in the
space of solutions of the system
\begin{equation*}
  \varphi_{n+1}=(Q_n+R_n)\varphi_n\,,
  \qquad n\ge n_1\,,
\end{equation*}
such that
 \begin{equation*}
    \norm{\frac{\varphi_n^{(k)}}{\prod_{i=n_1}^{n-1}
     q_i^{(k)}}
     - \delta^{(k)}}\to 0,\quad\text{ as
     }n\to\infty\,,\quad\text{for }k=1,\dots,d\,.
  \end{equation*}
\end{theorem}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Asymptotic behavior of solutions}
\label{sec:asymt-behav-gener}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let us define the matrices
\begin{equation}
% \label{eq:matrix-A}
  A_m(\zeta)= B_{2m+1}(\zeta)B_{2m}(\zeta)\,,
  \quad \forall m\in\mathbb{N}\,.
\end{equation}
Clearly, system (\ref{eq:mre}) is equivalent  to
\begin{equation}
  \label{eq:mreA}
  \eta_{m+1}=A_m(\zeta)\eta_m\,,
\quad \forall m\in\mathbb{N}\,,
\end{equation}
where $\eta_{m}=\xi_{2m}$.
A straightforward computation shows that
\begin{equation}
% \label{poorguy}
B_{n+1}(\zeta)B_{n}(\zeta) =
- I + \begin{pmatrix}
      \frac{c_{n}-c_{n-1}+\alpha}{n} & 0\\
      0 & \frac{c_{n+1}-c_{n}+\alpha}{n}
      \end{pmatrix}
+ O(n^{-1-\epsilon})
\end{equation}
as $n\to\infty$, where $I$ denotes the identity matrix and
$\epsilon>0$.  We notice that the term $O(n^{-1-\epsilon})$
contains all the dependencies on the parameter $\zeta$ and 
sequences $\{q_n\}_{n\in{\mathbb N}}$ and
$\{h_n\}_{n\in{\mathbb N}}$ . Up to the same order in $n$, we
may write
\[
  B_{n+1}(\zeta)B_{n}(\zeta) =
  -   \begin{pmatrix}
      \exp\frac{c_{n-1}-c_{n}-\alpha}{n} & 0\\
      0 & \exp\frac{c_{n}-c_{n+1}-\alpha}{n}
      \end{pmatrix}\left[I+O(n^{-1-\epsilon})\right],
\]
where again the dependency on $\zeta$ is contained exclusively in the
term $O(n^{-1-\epsilon})$. Thus, $A_m(\zeta)$ admits the
decomposition
\[
A_m(\zeta) = G_m + R_m(\zeta)\,,
\]
where, for notational convenience, we define
\begin{equation*}
G_m:= -
  \begin{pmatrix}
  \exp g_m^{(1)} & 0\\
  0 & \exp g_m^{(2)}
  \end{pmatrix}
\end{equation*}
with
\[
g_m^{(1)} := \frac{c_{2m-1}-c_{2m}-\alpha}{2m}\quad\mbox{and}\quad
g_m^{(2)} := \frac{c_{2m}-c_{2m+1}-\alpha}{2m}\,,
\]
and the residual term is given by
\begin{equation*}
  R_m(\zeta):= G_m\,O(m^{-1-\epsilon})\,.
\end{equation*}
The r.\ h.\ s.\ of the last identity must be understood as the
product of $G_m$ times a matrix that depends on $\zeta$ and
whose norm behaves as specified by the last factor as
$m\to\infty$.

\begin{lemma}
\label{lem:asympt-sol-vec-sys}
System (\ref{eq:mreA}) satisfies the assumptions of
Theorem~\ref{thm:bl}.  Therefore, one can find
$n_1\in\mathbb{N}$ for which there exists a basis $\eta^{(k)}$
($k=1,2$) in the space of solutions of (\ref{eq:mreA}) such
that, for $k=1$ and $k=2$,
\begin{equation}
\label{eq:basis-asympt-sol}
\left\|\eta_m^{(k)}/\prod\limits_{s=n_1}^{m-1}(-1)
        \exp g_s^{(k)}
        - \delta^{(k)}\right\|\to 0,
\end{equation}
as $m\to\infty$.
\end{lemma}

\begin{proof}
  We shall verify the fulfillment of conditions (i), (ii) and
  (iii) of Theorem~\ref{thm:bl}, in that order.

  Suppose that $\{G_m\}_{m\in{\mathbb
      N}}\not\in\mathcal{L}(1)$, i.\,e., it does not satisfy
  the Levinson condition for $k=1$.  Then, it must occur that
  \begin{equation*}
    \liminf_{m\to\infty}\sum_{s=1}^{m}
    \left(\frac{c_{2s-1}-c_{2s}}{2s}-
      \frac{c_{2s}-c_{2s+1}}{2s}\right)\ne+\infty
  \end{equation*}
and
\begin{equation*}
\limsup_{m\to\infty}\sum_{s=1}^{m}
\left(\frac{c_{2s-1}-c_{2s}}{2s}-
\frac{c_{2s}-c_{2s+1}}{2s}\right)=+\infty\,.
\end{equation*}
Let $\{\widetilde{m}_i\}_{i\in\mathbb{N}}$ be the sequence
that contains all the sequences $\{m_i\}_{i\in\mathbb{N}}$ for
which
\begin{equation*}
   \lim_{i\to\infty}\sum_{s=1}^{m_i}
    \left(\frac{c_{2s-1}-c_{2s}}{2s}-
      \frac{c_{2s}-c_{2s+1}}{2s}\right)=+\infty\,.
\end{equation*}
Clearly, $\{\widetilde{m}_i\}_{i\in\mathbb{N}}$ is such that:
\begin{enumerate}
\item For any $i_0\in\mathbb{N}$, there exists an $i>i_0$ such
  that $\widetilde{m}_{i}-1>\widetilde{m}_{i-1}$.

\item For all sequences $\{m_i\}_{i\in\mathbb{N}}$ with the
  property that
  $m_i\not\in\{\widetilde{m}_n\}_{n\in\mathbb{N}}$ for every
  $i$ greater than some $i_0$, there exists $M_0$ (independent
  of the sequence) such that
  \begin{equation*}
   \sup_{i\in\mathbb{N}}\,\left\{
   \sum_{s=1}^{m_{i}}\frac{c_{2s-1}+c_{2s+1}-2c_{2s}}{2s}\right\}<M_0\,.
  \end{equation*}
\end{enumerate}
Pick any $M>0$ and choose $i$ (of property (1)) such that
\begin{equation*}
  \sum_{s=1}^{m_{i}}\frac{c_{2s-1}+c_{2s+1}-2c_{2s}}{2s}>M\,.
\end{equation*}
However, since $m_{i}-1\not\in\{\widetilde{m}_i\}_{i\in\mathbb{N}}$,
we also have
\begin{equation*}
  \sum_{s=1}^{m_{i}-1}\frac{c_{2s-1}+c_{2s+1}-2c_{2s}}{2s}<M_0\,.
\end{equation*}
Thus,
\begin{equation*}
  M-M_0<\frac{c_{2m_i-1}+c_{2m_i+1}-2c_{2m_i}}{2m_i}
  < \frac{v_0}{2m_i}
\end{equation*}
valid for some constant $v_0$ (because $\{c_n\}_{n\in{\mathbb N}}$ is bounded).
Clearly, the last inequality makes no sense since
$M-M_0$ can be done as big as we want. We therefore conclude that condition
(i) of Theorem~\ref{thm:bl} is satisfied for $k=1$. By a similar reasoning,
the same assertion holds for $k=2$.

Now, we have
\begin{equation*}
  \frac{v_1}{2m}\le g_m^{(k)}\le\frac{v_2}{2m}
\end{equation*}
for $k=1, 2$ and some constants $v_1$ and $v_2$. It follows that
\[
\frac{\norm{R_m(\zeta)}}{\exp g_m^{(k)}}
\leq
r_1 m^{-1-\epsilon}\frac{\norm{G_m}}{\exp g_m^{(k)}}
\leq
r_2 m^{-1-\epsilon}\exp \frac{v_2-v_1}{2m}
\]
for some positive constants $r_1$ and $r_2$. This implies that
the l.\ h.\ s.\ of the inequality above is summable, hence
condition (ii) of Theorem \ref{thm:bl} holds.

It remains to verify condition (iii) of Theorem \ref{thm:bl}.
Since the determinant of each transfer matrix is different from zero,
the same is true for the determinant of each $A_m$ (which equals
$G_m + R_m$).
\end{proof}

Up to a normalization constant, statement (\ref{eq:basis-asympt-sol})
is equivalent to saying that
  \begin{equation}
  \label{asymp}
  \eta_{m+1}^{(k)} = (-1)^{m}
                 \exp\left(\sum\limits_{s=1}^{m}g_s^{(k)}\right)
                 \left[\delta^{(k)} + \tau_m^{(k)}\right]\,,
  \end{equation}
where $\tau_m^{(k)}$ is $o(1)$ as $m\to\infty$. We will use this expression
in the sequel.

Solutions of~(\ref{eq:mreA}) obviously depend on $\zeta$ and
the specific form of the sequences $\{q_n\}_{n\in{\mathbb N}}$ 
and $\{h_n\}_{n\in{\mathbb N}}$.  However, equation
(\ref{eq:basis-asympt-sol}) (or (\ref{asymp})) shows that
the main asymptotic term does not.

\begin{remark}
  We have the following consequence of
  Lemma~\ref{lem:asympt-sol-vec-sys}: Let
  $\{m_j\}_{j\in{\mathbb N}}$ be a monotone growing sequence
  of natural numbers. Then we have, from (\ref{asymp}),
\[
\xi_{2m_j+2}^{(k)}=(-1)^{m_j}\exp\left(\sum_{s=1}^{m_j} g_s^{(k)}\right)
                 \left[\delta^{(k)} + \tau_{m_j}^{(k)}\right]\,.
\]
Moreover,
\[
\xi_{2m_j+2p+2}^{(k)}=(-1)^{m_j+p}
        \exp\left(\sum_{s=1}^{m_j+p} g_s^{(k)}\right)
        \left[\delta^{(k)} + \tau_{m_j+p}^{(k)}\right]
\]
for $p\in{\mathbb N}$. Thus, by splitting the exponential factor and
working out residual terms, we obtain
\begin{equation}
\label{schubert}
\xi_{2m_j+2p+2}^{(k)}=(-1)^p\exp\left(\sum_{s=m_j+1}^{m_j+p}g_s^{(k)}\right)
                      \left[\xi_{2m_j+2}^{(k)}+\phi_{j,p}^{(k)}\right]\,,
\end{equation}
where the norm of $\phi_{j,p}^{(k)}$ equals the product of the norm of
$\xi_{2m_j+2}^{(k)}$ times a function $o(1)$ as 
$j\rightarrow\infty$, uniformly in $p$.
\end{remark}

We close this section noticing the fact that, in order to
obtain the asymptotic behavior of solutions of
(\ref{eq:mreA}), it was sufficient to consider those conditions
on $\{c_n\}_{n\in{\mathbb N}}$, $\{h_n\}_{n\in{\mathbb N}}$, and
$\{q_n\}_{n\in{\mathbb N}}$ given in the Introduction. 
However, without further assumptions on
$\{c_n\}_{n\in{\mathbb N}}$, these asymptotics yields no clue
about the decaying behavior of these solutions. This
information is necessary to determine whether the operator is
self-adjoint or not and, in the latter case, to classify the
spectrum (by using, for instance, Subordinacy Theory).
Theses issues will be attended, in the following sections, by
requiring certain regularity on average of
$\{c_n\}_{n\in{\mathbb N}}$ over sequences
$\{m_j\}_{j\in{\mathbb N}}$. This will give rise to nice
asymptotic behavior for the norm of $\xi_{2m_j+2}^{(k)}$ from
which, the asymptotic behavior of the norm of
$\xi_{2m+2}^{(k)}$ will be interpolated by resorting to
(\ref{schubert}).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Self-adjointness}
\label{sec:self-adjointness}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

It is well known that, for Jacobi matrices, we have the discrete
analogue of Weyl's limit point--limit circle theory for
self-adjoint extensions of differential operators. In
particular, it is true that if two linearly independent
solutions of (\ref{eq:main-recurrence}) are in
$l_2(\mathbb{N})$, then $J$ is not self-adjoint (limit circle
case). If, however, one solution of
(\ref{eq:main-recurrence}) is not in $l_2(\mathbb{N})$ then
$J=J^*$ (limit point case) \cite{MR0184042,MR1627806}.

Next, we establish sufficient conditions for $J$ to be
self-adjoint.
\begin{theorem}
\label{thm:self-adjointness}
Define $w^{(1)}=\min\{0,\inf_{s}\{c_{2s-1}-c_{2s}-\alpha\}\}$
and $w^{(2)}=\min\{0,\inf_{s}\{c_{2s}-c_{2s+1}-\alpha\}\}$.
Fix the value of $k$.  Consider a monotone growing sequence
$\{m_j\}_{j\in{\mathbb N}}$ of natural numbers that obeys
  \begin{equation}
  \sum\limits_{j=1}^\infty\frac{m_{j+1}-m_j}{m_{j}}
  \left(\frac{m_{j+1}}{m_j}\right)^{w^{(k)}}=\infty\,.
  \label{cond-2-self-adjoint}
  \end{equation}
  Suppose now that, for this sequence, the following holds:
  \begin{equation}
  \sum\limits_{s=1}^{m_j}g_s^{(k)}
    \ge -\frac{1}{2}\log(m_j) + O(1),\quad\mbox{when $j\to\infty$}\,.
    \label{cond-1-self-adjoint}
  \end{equation}
Then, there exists a solution of (\ref{eq:main-recurrence})
that is not in $l_2(\mathbb{N})$ and, therefore, $J=J^*$.
\end{theorem}

\begin{proof}
  We prove the assertion under the assumption that $k=1$. The
  case $k=2$ follows analogously.  We first note that
  (\ref{cond-1-self-adjoint}) and (\ref{asymp}) imply
  \begin{equation}
  \label{pim}
  \norm{\xi_{2m_j+2}^{(1)}}^2 \geq r_1 m_{j}^{-1}\,,\quad\forall j\geq j_0,
  \end{equation}
  where $r_1$ is some positive constant and $j_0$ is
  sufficiently big. Moreover, from (\ref{schubert}) we obtain
  \begin{eqnarray}
  \norm{\xi^{(1)}_{2m_j+2p+2}}^2
  &\geq&
  r_2\exp\left(2\sum\limits_{s=m_j+1}^{m_j+p}g_s^{(1)}\right)
  \norm{\xi^{(1)}_{2m_j+2}}^2\nonumber\\
  &\geq&
  r_3\left(\frac{m_j+p}{m_j+1}\right)^{w^{(1)}}
  \norm{\xi^{(1)}_{2m_j+2}}^2\,,\label{pot}
  \end{eqnarray}
  for some positive constants $r_2$, $r_3$ and
  $0<p<m_{j+1}-m_j$. Inequality (\ref{pot}) follows from the
  fact that
  \[
  \sum_{s=m_j+1}^{m_j+p}g_s^{(1)}
  \geq
  \frac{w^{(1)}}{2}\sum_{s=m_j+1}^{m_j+p}\frac{1}{s}
  =
  \frac{w^{(1)}}{2}\log\left(\frac{m_j+p}{m_j+1}\right) + O(1)\,,
  \]
  as $j\to\infty$.

  Now,
  \begin{eqnarray*}
  \norm{\xi^{(1)}}^{2}
  &\geq&
  \sum_{m=0}^{\infty}\norm{\xi^{(1)}_{2m+2}}^{2}\\
  &\geq&
  \sum_{j=j_0}^{\infty}
  \left(\norm{\xi^{(1)}_{2m_j+2}}^{2} +
  \sum_{p=1}^{m_{j+1}-m_j-1}\norm{\xi^{(1)}_{2m_j+2p+2}}^{2}\right)\\
  &\geq&
  r_1r_3\sum\limits_{j=j_0}^\infty\frac{m_{j+1}-m_j-1}{m_{j+1}}
  \left(\frac{m_{j+1}-1}{m_j+1}\right)^{w^{(1)}},
  \end{eqnarray*}
  where the last inequality follows from (\ref{pim}) and
  (\ref{pot}). Clearly, $\xi^{(1)}$ is not in
  $l_2(\mathbb{N},\mathbb{C}^2)$ if
  (\ref{cond-2-self-adjoint}) is fulfilled. Since
  $\norm{\xi}\asymp\left\|x\right\|$, it follows that
  $x^{(1)}$ does not belong to $l_2(\mathbb{N})$ (so $J$ is in
  the limit point case).
\end{proof}

\begin{remark}
\label{divergent}
  For $w^{(k)}=0$, condition (\ref{cond-2-self-adjoint})
  seems to be met for any monotone growing sequence
  $\{m_j\}_{j\in{\mathbb N}}$. In any case, we could not find 
  counterexamples.
  However, this condition is more stringent for
  $w^{(k)}<0$, although it is satisfied for sequences of fairly fast growth,
  for instance, $m_j\sim\exp j$ (for any $w^{(k)}<0$)
  and also $m_j\sim j^j$ (for $-2\leq w^{(k)}<0$). Clearly,
  sequences of polynomial growth, like $m_j\sim j^l$ with $l\geq 1$,
  also satisfy (\ref{cond-2-self-adjoint}) for any $w^{(k)}\leq 0$.
\end{remark}

\begin{remark}
\label{614}
Theorem~\ref{thm:self-adjointness} applies quite easily to the
particular case where the sequence $\{c_n\}_{n\in{\mathbb N}}$
is even-periodic (say, of period $2L$). In this case
$\abs{\sum_{n=1}^{2L}(-1)^nc_n}\geq (\alpha-1)L$ implies
(\ref{cond-1-self-adjoint}) and therefore self-adjointness. As
already mentioned, this model, with 
$\{q_n\}_{n\in{\mathbb N}}\equiv\{h_n\}_{n\in{\mathbb N}}\equiv 0$, 
was discussed in \cite{MR1959871,MR1924991}.
\end{remark}

\begin{example}
\label{ex:self-adjointness}
As a more involved example of application of
Theorem~\ref{thm:self-adjointness}, we now proceed with the
construction of a self-adjoint operator $J$ with no periodicity
in $\{c_n\}_{n\in{\mathbb N}}$.  Let the sequence
$\{c_n\}_{n\in{\mathbb N}}$ be given by $c_1$, $c_2$, and
$c_3$ as follows:
\begin{equation}
  \label{eq:model-dislocations}
\underbrace{
\underbrace{
c_1\,c_2\,c_1\,c_2\cdots c_1\,c_2\,c_1\,c_2}_{2p_1}
c_2\,c_3\,c_2\,c_3\cdots c_2\,c_3\,c_2\,c_3}_{2p_2}
c_1\,c_2\,c_1\,c_2\cdots
\end{equation}
(we pray allowance for the abuse of notation), where
$\{p_j\}_{j\in{\mathbb Z}^{+}}$ is a sequence with polynomial
growth of the form $p_j=Cj^a$, with $a\in{\mathbb
  N}\setminus\{1\}$ and $C>0$. Define $m_j=p_{2j}$; this
sequence clearly satisfies (\ref{cond-2-self-adjoint}).  Next,
we show that, for a suitable choice of $c_1$ and $c_3$,
inequality (\ref{cond-1-self-adjoint}) holds so $J=J^*$.

Set $k=1$, then
\begin{equation}
\label{dislocation}
\sum_{s=1}^{m_j}g_s^{(1)} =
  \frac{1}{2}\sum_{n=1}^{j}
  \sum_{s=m_{n-1}+1}^{m_n}\frac{c_{2s-1}-c_{2s}}{s}
  - \frac{\alpha}{2}\log(m_j) + O(1)\,.
\end{equation}
The sequences $\{m_j\}_{j\in{\mathbb Z}^{+}}$ and
$\{c_n\}_{n\in{\mathbb N}}$ are such that
\begin{equation*}
\sum_{s=m_{n-1}+1}^{m_n}\!\frac{c_{2s-1}-c_{2s}}{s}\ =\!
        \sum_{s=p_{2n-2}+1}^{p_{2n-1}}\!\frac{c_1-c_2}{s}\ +\!
        \sum_{s=p_{2n-1}+1}^{p_{2n}}\!\frac{c_2-c_3}{s}\,.
\end{equation*}
Furthermore,

\begin{eqnarray*}
\lefteqn{
\sum_{s=p_{2n-2}+1}^{p_{2n-1}}\!\frac{c_1-c_2}{s}\ +\!
\sum_{s=p_{2n-1}+1}^{p_{2n}}\!\frac{c_2-c_3}{s}}\hspace*{10mm}\\
&=&
\frac{1}{2}\Biggl(
        \sum_{s=p_{2n-2}+1}^{p_{2n-1}}\frac{c_1-c_2}{s}\ +\!
        \sum_{s=p_{2n-2}+1}^{p_{2n-1}}
        \frac{c_2-c_3}{s+p_{2n-1}-p_{2n-2}}\\[1mm]
& &
\phantom{\frac{1}{2}\Biggl(}+
        \sum_{s=2p_{2n-1}-p_{2n-2}+1}^{p_{2n}}\frac{c_2-c_3}{s}\ +\!
        \sum_{s=p_{2n-1}+1}^{p_{2n}}
        \frac{c_1-c_2}{s-p_{2n-1}+p_{2n-2}}\\[1mm]
& &
\phantom{\frac{1}{2}\Biggl(}-
        \sum_{s=p_{2n-1}+1}^{p_{2n}-p_{2n-1}+p_{2n-2}}\frac{c_1-c_2}{s}\ +\!
        \sum_{s=p_{2n-1}+1}^{p_{2n}}\!\frac{c_2-c_3}{s}
\Biggr).
\end{eqnarray*}
Thus,
\begin{eqnarray*}
\lefteqn{
\sum_{s=m_{n-1}+1}^{m_n}\!\frac{c_{2s-1}-c_{2s}}{s}}\\
&=&
\frac{c_1-c_3}{2}\sum_{s=m_{n-1}+1}^{m_n}\frac{1}{s}\\
& &
+\ \frac{c_2-c_3}{2}\left(\sum_{s=p_{2n-2}+1}^{p_{2n-1}}\frac{1}{s^2}
        \left(\frac{p_{2n-2}\!\!-\!p_{2n-1}}
        {1+\frac{p_{2n-1}-p_{2n-2}}{s}}\right) +
        \sum_{s=2p_{2n-1}-p_{2n-2}+1}^{p_{2n}}\frac{1}{s}\right)\\
& &
+\ \frac{c_1-c_2}{2}\left( \sum_{s=p_{2n-1}+1}^{p_{2n}}\frac{1}{s^2}
        \left(\frac{p_{2n-1}\!\!-\!p_{2n-2}}
        {1+\frac{p_{2n-2}-p_{2n-1}}{s}}\right)+
        \sum_{s=p_{2n-1}+1}^{p_{2n}-p_{2n-1}+p_{2n+2}}\frac{1}{s}\right).
\end{eqnarray*}
It is now easy to verify that
\begin{equation}
  \label{eq:self-adjoint-example-g1}
  \sum_{1}^{m_j}g_s^{(1)}=
  \frac{c_1-c_3-2\alpha}{4}\log(m_j) +O(1)\,,
\end{equation}
since, as a consequence of the polynomial growth of $p_n$, we have
\begin{equation*}
  (p_{2n-1} - p_{2n-2})\sum_{s=m_{n-1}+1}^{m_n}\frac{1}{s^2}
  =O(n^{-2})
\end{equation*}
and
\begin{equation*}
  \sum_{s=2p_{2n-1}-p_{2n-2}+1}^{p_{2n}}\frac{1}{s}\ +\!
  \sum_{s=p_{2n-1}+1}^{p_{2n}-p_{2n-1}+p_{2n+2}}\frac{1}{s}=O(n^{-2})
\end{equation*}
as $n\to\infty$.
From (\ref{eq:self-adjoint-example-g1}) it follows that if
\begin{equation*}
  c_1-c_3\ge 2(\alpha-1)\,,
\end{equation*}
then Theorem~\ref{thm:self-adjointness} holds. Since a similar
result must follow when $c_1$ and $c_3$ are switched over (or
from doing an analogous computation for $k=2$), we conclude
that the inequality
\begin{equation*}
  \abs{c_1-c_3}\ge 2(\alpha-1)
\end{equation*}
is sufficient for self-adjointness. We note the fact that the
value of $c_2$ plays no role in this example. The computation
done above shows also that this construction may be
generalized to the cases where $\{p_j\}_{j\in{\mathbb Z}^+}$
grows faster that polynomially, with probably different
conditions imposed upon $c_1$, $c_2$ and $c_3$ to ensure
self-adjointness.
\end{example}

We now establish sufficient conditions for $J$ to be
non-selfadjoint.

\begin{theorem}
\label{thm:nonself-adjointness}
Define $v$ as the largest among $0$,
$\sup_s\{c_{2s}-c_{2s+1}-\alpha\}$ and
$\sup_s\{c_{2s-1}-c_{2s}-\alpha\}$.  Consider a monotone
growing sequence $\{m_j\}_{j\in{\mathbb N}}$ that satisfies
  \begin{equation}
  \label{non-self}
  \sum_{j=1}^{\infty}\frac{m_{j+1}-m_j}{m_j^\beta}
  \left(\frac{m_{j+1}}{m_j}\right)^{v}<\infty\,,
  \end{equation}
for some $\beta>1$. Now suppose that the inequality
  \begin{equation}
  \label{eq:nonself-adj-cond}
  \sum\limits_{s=1}^{m_j}g_s^{(k)}
  \leq -\frac{\beta}{2}\log(m_j) + O(1)\,,
  \quad\text{when $j\to\infty$},
  \end{equation}
  holds for both $k=1$ and $k=2$. Then all solutions of
  (\ref{eq:main-recurrence}) are in $l_2(\mathbb{N})$ and,
  therefore, $J\ne J^*$.
\end{theorem}
\begin{proof}
Firstly, (\ref{schubert}) and (\ref{eq:nonself-adj-cond}) imply
\begin{equation*}
  \norm{\xi_{2+2m_{j}}^{(k)}}^2  <
  r_0 m_{j}^{-\beta}\,,\quad\forall j\geq j_1,
\end{equation*}
where $r_0$ is some positive constant. Without loss we may assume $j_1=1$ to
avoid unnecessary complications. Secondly, (\ref{schubert}) implies
\begin{eqnarray*}
  \norm{\xi^{(k)}_{2m_j+2p+2}}^2
  &\le&
  r_1\exp\left(2\sum\limits_{s=m_j+1}^{m_j+p}g_s^{(k)}\right)
  \norm{\xi^{(k)}_{2m_j+2}}^2\\
  &\leq&
  r_2\left(\frac{m_j+p}{m_j+1}\right)^{v}\norm{\xi^{(k)}_{2m_j+2}}^2\,,
\end{eqnarray*}
for $0<p<m_{j+1}-m_j$ and some positive constants $r_1$ and $r_2$, where
the last inequality follows from an argument analogous to the one yielding
(\ref{pot}).
Finally, we note that the transfer matrices $B_n(\zeta)$ are uniformly
norm-bounded with respect to the index $n$. Let $r_3>0$ denote the
square of such a bound.

Now, bearing these facts in mind, we have
\begin{eqnarray*}
\norm{\xi^{(k)}}^2
&\leq&
r_3 \sum_{m=m_1}^{\infty}\norm{\xi_{2m+2}^{(k)}}^2\\
&=&
r_3 \sum_{j=1}^{\infty}\sum_{p=0}^{m_{j+1}-m_j-1}
\norm{\xi_{2m_j+2p+2}^{(k)}}^2\\
&\leq&
r_3 \sum_{j=1}^{\infty}\norm{\xi_{2m_j+2}^{(k)}}^2
    \left[1+r_2\sum_{j=1}^{m_{j+1}-m_j-1}
    \left(\frac{m_j+p}{m_j+1}\right)^{v}\right]\\
&\leq&
r_1r_3\sum_{j=1}^{\infty}\frac{1}{m_j^\beta} +
      r_1r_2r_3\sum_{j=1}^{\infty}\frac{1}{m_j^\beta}
      \sum_{p=1}^{m_{j+1}-m_j-1}
      \left(\frac{m_j+p}{m_j+1}\right)^{v}\,.
\end{eqnarray*}
The first term in the last inequality is clearly convergent,
since $m_j\geq j$ and $\beta>1$. The second term is also
convergent since (\ref{non-self}) is fulfilled.

Thus, we have proved that $\xi^{(1)}$ and $\xi^{(2)}$ belong
to $l_2(\mathbb{N},\mathbb{C}^2)$. It follows (by the
argumentation given in the proof of
Theorem~\ref{thm:self-adjointness}) that $x^{(1)}$ and
$x^{(2)}$ belong to $l_2(\mathbb{N})$. Moreover, they are
linearly independent as a consequence of the linear
independency of $\xi^{(1)}$ and $\xi^{(2)}$.  The lack of
self-adjointness now follows immediately.
\end{proof}

\begin{remark}
  Theorem \ref{thm:nonself-adjointness} does not
  allow to obtain the converse of Theorem
  \ref{thm:self-adjointness}.
\end{remark}

\begin{remark}
\label{all-of-them}
A fairly large set of sequences $\{m_j\}_{j\in{\mathbb N}}$ satisfies
condition (\ref{non-self}), ranging from those of linear growth, i.\ e.
$m_j\sim j$, up to those of very fast growth like
$m_j\sim j^j$.
\end{remark}

As we have mentioned previously, Jacobi operators of the form
proposed in the present work has been already studied for the
case of even-periodic sequences $\{c_n\}_{n\in{\mathbb N}}$.  No
results are known (to the authors anyway) concerning the
odd-periodic case, although it has been suggested that, in
this case, these operators would not be self-adjoint
\cite{MR1924991}.  Theorem~\ref{thm:nonself-adjointness}
allows us to prove that assertion easily.

\begin{corollary}
\label{corollary:non-self}
Let $\{c_n\}_{n\in{\mathbb N}}$ be an odd-periodic sequence.
Then the operator $J$,
given by the matrix representation (\ref{eq:jm}) with entries defined by
(\ref{eq:weights}) and (\ref{eq:diagonal}), is not self-adjoint.
\end{corollary}

\begin{proof}
Let $T$ be the period of the sequence. Define $m_j=jT$ for $j\in{\mathbb N}$.
Clearly, $\{m_j\}_{j\in{\mathbb N}}$ satisfies
(\ref{non-self}) for any $\beta>1$ and $v\geq 0$. Thus, it only remains
to verify (\ref{eq:nonself-adj-cond}) for $k=1,2$.

We have,
\begin{eqnarray}
\sum_{s=1}^{m_j}g_s^{(1)}
&=&
\frac{1}{2}\sum_{s=1}^{m_j}\frac{c_{2s-1}-c_{2s}-\alpha}{s}\nonumber\\
&=&
\frac{1}{2}\sum_{n=1}^{j}\sum_{s=m_{n-1}+1}^{m_n}\frac{c_{2s-1}-c_{2s}}{s}
- \frac{\alpha}{2}\sum_{s=1}^{m_j}\frac{1}{s}\nonumber\\
&=&
\frac{1}{2}\sum_{n=1}^{j}\sum_{s=1}^{T}\frac{c_{2s-1}-c_{2s}}{s+(j-1)T}
- \frac{\alpha}{2}\log(m_j) + O(1)\,,\label{per}
\end{eqnarray}
where the first term in the last equality follows from the periodicity of
$\{c_n\}_{n\in{\mathbb N}}$. Now, for $j\geq 2$,
\begin{eqnarray*}
\sum_{s=1}^{T}\frac{c_{2s-1}-c_{2s}}{s+(j-1)T}
&=&
\sum_{s=1}^{T}\frac{c_{2s-1}-c_{2s}}{(j-1)T}\\
& &
+\ \sum_{s=1}^{T}(c_{2s-1}-c_{2s})
   \left[\frac{1}{s+(j-1)T}-\frac{1}{(j-1)T}\right].
\end{eqnarray*}
The first term above equals zero because of the
odd-periodicity of the sequence, while the second term is
$O(j^{-2})$. Therefore, the first term in (\ref{per}) is also
$O(1)$ thus yielding the expected inequality (with
$\beta=\alpha$) for $\sum_{s=1}^{h_m}g_s^{(1)}$. A similar
computation shows that also $\sum_{s=1}^{h_m}g_s^{(2)}$
fulfills the same inequality.
\end{proof}

The proof of Corollary~\ref{corollary:non-self} tells us that
an odd-periodic sequence $\{c_n\}_{n\in{\mathbb N}}$ produces a
non-self-adjoint operator because one can find a suitable
sequence $\{m_j\}_{j\in{\mathbb N}}$ for which, the contribution of 
$c_{2s-1}-c_{2s}$
to the l.\ h.\ s.\ of (\ref{eq:nonself-adj-cond}) is nearly canceled
(and the same occurs for $c_{2s}-c_{2s+1}$). This idea may
be used to construct non-selfadjoint Jacobi operators out of
certain non-periodic sequences.
As a matter of fact, the following example is defined essentially in
the same way as the one discussed previously, having exactly one
modification.

\begin{example}
\label{ex:non-selfadjointness}
Consider a sequence $\{c_n\}_{n\in{\mathbb N}}$ defined by two given numbers
$c_1$ and $c_2$ arranged as follows:
\begin{equation}
  \label{non-self-adjiont-model}
\underbrace{
\underbrace{
c_1\,c_2\,c_1\,c_2\cdots c_1\,c_2\,c_1\,c_2}_{2p_1}
c_2\,c_1\,c_2\,c_1\cdots c_2\,c_1\,c_2\,c_1}_{2p_2}
c_1\,c_2\,c_1\,c_2\cdots\,,
\end{equation}
where $\{p_j\}_{j\in{\mathbb Z}^+}$ is a sequence
with polynomial growth of the form $p_j=Cj^a$,
with $a\in{\mathbb N}\setminus\{1\}$ and $C>0$. That is, we
use the sequence defined by (\ref{eq:self-adjoint-example-g1})
with $c_3=c_1$. Alternatively, we may think
of this sequence as one obtained from a two-periodic sequence
by inserting dislocations at sites $p_j+1$. We claim that the
Jacobi operator defined by (\ref{non-self-adjiont-model}) is not
self-adjoint.  To prove it, we first define
$m_j=p_{2j}$.  This sequence satisfies
(\ref{non-self}) for any $\beta>1$, as a straightforward computation shows.
Next, we must verify (\ref{eq:nonself-adj-cond}) for $k=1,2$.  For
$k=1$, (\ref{dislocation}) holds true. The computation done
in Example 1 shows that
\begin{eqnarray*}
\lefteqn{
\sum_{s=m_{n-1}+1}^{m_n}\!\frac{c_{2s-1}-c_{2s}}{s}}\\
&=&
\frac{c_2-c_1}{2}\left(\sum_{s=p_{2n-2}+1}^{p_{2n-1}}\frac{1}{s^2}
        \left(\frac{p_{2n-2}\!\!-\!p_{2n-1}}
        {1+\frac{p_{2n-1}-p_{2n-2}}{s}}\right) +
        \sum_{s=2p_{2n-1}-p_{2n-2}+1}^{p_{2n}}\frac{1}{s}\right)\\
& &
+\ \frac{c_1-c_2}{2}\left( \sum_{s=p_{2n-1}+1}^{p_{2n}}\frac{1}{s^2}
        \left(\frac{p_{2n-1}\!\!-\!p_{2n-2}}
        {1+\frac{p_{2n-2}-p_{2n-1}}{s}}\right)+
        \sum_{s=p_{2n-1}+1}^{p_{2n}-p_{2n-1}+p_{2n+2}}\frac{1}{s}\right).
\end{eqnarray*}
By what has been shown before, the r.\ h.\ s.\ of this equality is
$O(n^{-2})$ as $n\to\infty$. This, together with (\ref{dislocation}), yields
\begin{equation*}
   \sum_{s=1}^{m_j}g_s^{(1)}=
  -\frac{\alpha}{2}\log(m_j) +O(1)\,.
\end{equation*}
In a similar fashion, we can show that the same equality
is true for $k=2$. Hence, we have verified that this example fulfills the
hypotheses of Theorem~\ref{thm:nonself-adjointness}.

We note that the case $p_m\sim m$ can not be included here
since the summability of
\begin{equation*}
  \sum_{j=1}^\infty\sum_{s=m_{j-1}+1}^{m_j}\!\frac{c_{2s-1}-c_{2s}}{s}
\end{equation*}
is no longer ensured. This is to be expected because, for
$p_m\sim m$, the sequence $\{c_n\}_{n\in{\mathbb N}}$ is again
even-periodic so the associated Jacobi operator may be
self-adjoint (see Remark~\ref{614}).
\end{example}

Other examples of non-self-adjoint Jacobi operators (of the class discussed in
this work) could be devised along similar ideas. We think
that the example given above is of some interest because it is
derived from an operator defined by a even-periodic sequence
and therefore self-adjoint (provided that $|c_1-c_2|\geq
\alpha-1$, although this condition plays no role in our example)
\cite{MR1959871,MR1924991}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Discreteness of spectrum}
\label{sec:absence-cont-spectr}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we show that, if the assumptions of
Theorem~\ref{thm:self-adjointness} are fulfilled and
$\{m_j\}_{j\in{\mathbb N}}$ obeys certain further conditions,
then $J$ has only discrete spectrum.

To simplify proofs, we will assume that the conditions of
Theorem~\ref{thm:self-adjointness} are satisfied for $k=1$ (so
$\xi^{(1)}$ is not in $l_2(\mathbb{N},\mathbb{C}^2)$).

We first prove an easy consequence of inequality
(\ref{cond-1-self-adjoint}).
\begin{lemma}
\label{lemma-bullshit}
If (\ref{cond-1-self-adjoint}) is true for $k=1$, then
  \begin{equation}
    \label{eq:silly-ineq}
    \sum_{s=1}^{m_j}g_s^{(2)}
    < -\frac{\alpha}{2}\log(m_j) + O(1)\,,
    \quad\text{when }m\to\infty\,.
  \end{equation}
  An analogous assertion holds if the roles of $k=1$ and $k=2$
  are interchanged.
\end{lemma}

\begin{proof}
We have
\begin{equation}
\label{bullshit}
\sum_{s=1}^{m_j}g_s^{(2)}
=
-\sum_{s=1}^{m_j}g_s^{(1)} + \sum_{s=1}^{m_j}\frac{c_{2s-1}-c_{2s+1}}{2s}
-\alpha\sum_{s=1}^{m_j}\frac{1}{s}\,.
\end{equation}
Furthermore,
\[
\sum_{s=1}^{m_j}\frac{c_{2s-1}-c_{2s+1}}{2s}
=
\sum_{s=2}^{m_j}\frac{c_{2s-1}}{2}\left(\frac{1}{s}-\frac{1}{s-1}\right)
+\frac{c_1}{2}-\frac{c_{2m_j+1}}{2m_j}\,.
\]
Thus, the second term in (\ref{bullshit}) is $O(1)$ because
the sequence $\{c_n\}_{n\in{\mathbb N}}$ is bounded and $(s^{-1}
- (s-1)^{-1})$ is $O(s^{-2})$. Since moreover the last term in
(\ref{bullshit}) equals $\alpha\log(m_j)$ up to a term $O(1)$, we
obtain
\begin{eqnarray*}
\sum_{s=1}^{m_j}g_s^{(2)}
&=&
-\sum_{s=1}^{m_j}g_s^{(1)} - \alpha\log(m_j) + O(1)\\
&\leq&
\frac{1}{2}\log(m_j) - \alpha\log(m_j) + O(1)\\
&<&
- \frac{\alpha}{2}\log(m_j) + O(1)\,,
\end{eqnarray*}
where the last inequality holds since $\alpha>1$.
\end{proof}

\begin{theorem}
\label{arafat}
Suppose that the hypotheses of
Theorem~\ref{thm:self-adjointness} are satisfied for $k=1$
(then $x^{(1)}$ is not in $l_2(\mathbb{N})$). In addition, assume
that $\{m_j\}_{j\in{\mathbb N}}$ satisfies
\begin{equation}
\label{one-more}
\sum_{j=1}^{\infty}\frac{m_{j+1}-m_j}{m_j^\alpha}
\left(\frac{m_{j+1}}{m_j}\right)^{\check{v}}<\infty\,,
\end{equation}
with $\check{v}=\max\{0,\,\sup_s\{c_{2s}-c_{2s+1}-\alpha\}\}$.
Then $x^{(2)}$ is in $l_2(\mathbb{N})$. The assertion holds also if
we interchange $k=1$ and $k=2$ and take
$\check{v}=\max\{0,\,\sup_s\{c_{2s-1}-c_{2s}-\alpha\}\}$.
\end{theorem}

\begin{proof}
  By Lemma~\ref{lemma-bullshit} we have (\ref{eq:silly-ineq})
  and then, taking into account (\ref{one-more}) and reasoning
  as in Theorem ~\ref{thm:nonself-adjointness}, we obtain that
  $\xi^{(2)}$ is in $l_2(\mathbb{N},\mathbb{C}^2)$.
\end{proof}

\begin{remark}
\label{last}
  In view of Remarks~\ref{divergent} and \ref{all-of-them}, the set of
  sequences $\{m_j\}_{j\in{\mathbb N}}$ that fulfill both
  (\ref{cond-2-self-adjoint}) and (\ref{one-more}) is far from being empty.
\end{remark}

\begin{corollary}
\label{pp}
  $J$ has only pure point spectrum
  whenever the assumptions of Theorem~\ref{arafat} are met.
\end{corollary}

\begin{proof}
  Under the assumptions of the previous theorem we have that
  $x^{(1)}\not\in l_2(\mathbb{N})$ and $x^{(2)}\in
  l_2(\mathbb{N})$. This behavior of the basis
  $\{x^{(1)},\,x^{(2)}\}$ in the space of solutions of
  (\ref{eq:main-recurrence}) is the same for any
  $\zeta\in\mathbb{C}$. Thus, we always have a subordinate
  solution for the generalized eigenequation of the self-adjoint
  operator $J$. By applying Subordinacy Theory
  \cite[Theorem 3]{MR1179528}, we conclude that the spectrum is
  pure point.
\end{proof}

\begin{remark}
  Notice that Theorem~\ref{thm:self-adjointness} does not
  necessarily imply Theorem~\ref{arafat}. However, in account of what is
  said in Remark~\ref{last}, the implication is true for a large set
  of Jacobi operators $J$.
\end{remark}

We notice that (\ref{cond-2-self-adjoint}) along with
(\ref{eq:silly-ineq}) imply a certain correlation between the
slow decay of $x^{(1)}$ and the fast decay of $x^{(2)}$ (or
vice-versa). We shall use this to show that the resolvent $(J-\zeta I)^{-1}$
is Hilbert-Schmidt. Unfortunately,
(\ref{cond-1-self-adjoint}) is not good enough for that purpose.
The next technical result accounts for what is needed.

\begin{lemma}
\label{life-saver}
Suppose that for a certain sequence
$\{m_j\}_{j\in{\mathbb{N}}}$ one of the following holds:
\begin{enumerate}[i.]
\item For any natural $i$ greater than some $i_0\in\mathbb{N}$,
  \begin{equation*}
    \sum_{s=m_i+1}^{m_{i+1}}g_s^{(1)}
       \ge -\frac{1}{2}\log\left(\frac{m_{i+1}}{m_i}\right) +
       f_i\,,
  \end{equation*}
 where $\sum_{i=j}^lf_i$ has an upper bound for any $j$ and $l$
 ($i_0\le j<l$).
\item For any natural numbers $j$ and $l$, where $i_0\le j<l$ and
  $i_0\in\mathbb{N}$,
  \begin{equation*}
    \sum_{s=m_j+1}^{m_l}\frac{c_{2s}-c_{2s+1}}{s}
  \end{equation*}
has an upper bound.
\end{enumerate}
Then,
\[
\exp\left(\sum_{s=m_j+1}^{m_l}g_s^{(2)}\right)
\leq r\left(\frac{m_l}{m_j}\right)^{-\frac{\alpha}{2}}\,,
\]
for some constant $r$. A similar result holds if the roles of
$k=1$ and $k=2$ are interchanged and $c_{2s}-c_{2s+1}$ is
substituted by $c_{2s-1}-c_{2s}$ in (ii)
\end{lemma}
\begin{proof}
  Let us assume (i) first. From (\ref{bullshit}), it follows that
  \begin{equation*}
    \exp\!\left(\sum_{s=m_j+1}^{m_l}\!g_s^{(2)}\right)\!=
    \exp\!\left(-\!\!\!\sum_{s=m_j+1}^{m_l}\!g_s^{(1)} +\!\!\!
      \sum_{s=m_j+1}^{m_l}\!\!\frac{c_{2s-1}-c_{2s+1}}{2s}
      -\alpha\!\!\!\!\sum_{s=m_j+1}^{m_l}\!\frac{1}{s}\right)
  \end{equation*}
As in the proof of Lemma~\ref{lemma-bullshit}, one has
\begin{equation*}
  \sum_{s=m_j+1}^{m_l}\frac{c_{2s-1}-c_{2s+1}}{2s}=
\sum_{s=m_j+2}^{m_l}\frac{c_{2s-1}}{2}
\left(\frac{1}{s}-\frac{1}{s-1}\right)
+\frac{c_{2m_j+1}}{2m_j+2}-\frac{c_{2m_l+1}}{2m_l}\,,
\end{equation*}
where the first term above is clearly $O(m_j^{-1})$.
Therefore, there is a constant $r_1$ such that, for any pair $j<l$,
\begin{equation*}
  \exp\left(\sum_{s=m_j+1}^{m_l}\!\!\frac{c_{2s-1}-c_{2s+1}}{2s}\right)
  <r_1\,.
\end{equation*}
We thus conclude that there exists a constant $r_2$ such that
\begin{equation*}
   \exp\left(\sum_{s=m_j+1}^{m_l}g_s^{(2)}\right)\le r_2
   \left(\frac{m_l}{m_j}\right)^{-\frac{\alpha}{2}}\exp
   \left(\sum_{i=j}^lf_i\right)
\end{equation*}
which proves our assertion.

In the case where (ii) holds, we have the result straightforwardly.
\end{proof}

\begin{remark}
\label{block}
Condition (i) of last lemma clearly implies
(\ref{cond-1-self-adjoint}). 
\end{remark}

The results proven above allow us to give conditions ensuring
the discreteness of $J$'s spectrum when it is self-adjoint.

\begin{theorem}
\label{thm:compactness-resolvent}
Consider a sequence $\{m_j\}_{j\in{\mathbb N}}$ that complies with
(\ref{cond-2-self-adjoint}) and
\begin{equation}
\label{last-one}
\sum_{j=1}^{\infty}
\frac{m_{j+1}-m_j}{m_j^{\alpha}}\left(\frac{m_{j+1}}{m_j}\right)^{\hat{v}}
<\infty\,,
\end{equation}
where $\hat{v}=\max
\left\{0,\sup_s\{c_{2s}-c_{2s+1}-\alpha\}\,,\,
  \sup_s\{c_{2s-1}-c_{2s}-\alpha\}\right\}$.  Suppose moreover
the fulfillment of either condition (i) of
Lemma~\ref{life-saver}, or condition (ii) of the same lemma and
(\ref{cond-1-self-adjoint}).  Then the resolvent
\begin{equation*}
  (J-\zeta I)^{-1}\,,\qquad \zeta\in\mathbb{C}\setminus\sigma(J),
\end{equation*}
is Hilbert-Schmidt and, therefore, the spectrum of $J$ is discrete.
\end{theorem}

\begin{proof}
We first note that (\ref{last-one}) implies (\ref{one-more}) since 
$\hat{v}\geq\check{v}$. Thus,
the hypotheses of the theorem (along with Remark~\ref{block}) imply 
that both Theorems~\ref{thm:self-adjointness} and \ref{arafat} hold true.
Thence, there exists a basis
$\{x^{(1)}\,,x^{(2)}\}$ in the space of solutions of 
(\ref{eq:main-recurrence}), with one element in $l_2(\mathbb{N})$ and 
the other outside it. Let $x^{(2)}$ be the element in $l_2(\mathbb{N})$.

In what follows, $r_1, r_2,\ldots,r_{12}$ will stand for certain suitable
constants that will appear in estimations along the way.

Let us denote by $P_n(\zeta)$ the $n$-th orthogonal polynomial
of the first kind associated to $J$. Define the sequence
$u=\{u_n\}_{n\in{\mathbb N}}$ in $l_2(\mathbb{N})$ by
\begin{equation*}
 u=(J-\zeta I)^{-1}e_1\,.
\end{equation*}
According to \cite{MR2085852,MR1711536}, the Green function is
given by
\begin{equation*}
G_{m\,n}(\zeta)=
 \begin{cases}
   P_m(\zeta)u_n(\zeta)&\text{ when } m\le n\,,\\[2mm]
   P_n(\zeta)u_m(\zeta)&\text{ otherwise }.
 \end{cases}
\end{equation*}

The sequence $\{P_n(\zeta)\}_{n\in{\mathbb N}}$ is
not square-summable because $J=J^*$ \cite{MR0184042,MR1627806}.
Since it belongs to the space of solutions of
(\ref{eq:main-recurrence}), it may be expressed as a linear combination
of the basis $\{x^{(1)}\,,x^{(2)}\}$.
However, the solution $u=\{u_n\}_{n\in{\mathbb N}}$ of
(\ref{eq:main-recurrence}) must be proportional to $x^{(2)}$. Noticing that
$\norm{\xi_m}^2=\abs{x_{m-1}}^2+\abs{x_m}^2>\abs{x_m}^2$, we have
(dependencies on $\zeta$ are dropped from now on)
\[
\abs{G_{mn}} <
    \begin{cases}
     r_0\norm{\xi_{m}^{(1)}}\norm{\xi_{n}^{(2)}} +
     r_1\norm{\xi_{m}^{(2)}}\norm{\xi_{n}^{(2)}}&\text{ when } m\le n\,,
     \\[4mm]
     r_0\norm{\xi_{m}^{(2)}}\norm{\xi_{n}^{(1)}} +
     r_1\norm{\xi_{m}^{(2)}}\norm{\xi_{n}^{(2)}}&\text{ otherwise }.
     \end{cases}
\]
Since $\xi_m=B_{m-1}\xi_{m-1}$ and $B_n$ is uniformly norm-bounded in $n$,
we have $\norm{\xi_{2m+1}}\leq r_2 \norm{\xi_{2m}}$. It thus follows that
\begin{eqnarray}
\sum_{m=m_0}^{\infty}\sum_{n=n_0}^{\infty}\abs{G_{mn}}^2\!\!\!
&<&\!\!\!
\sum_{k_0\le k\le j}^\infty\!
        \left(r_3\norm{\xi_{2k+2}^{(1)}}\norm{\xi_{2j+2}^{(2)}} +
              r_4\norm{\xi_{2k+2}^{(2)}}\norm{\xi_{2j+2}^{(2)}}\right)^2
              \nonumber
\\
& &\!\!
+\!\sum_{j_0\le j<k}^\infty\!
        \left(r_3\norm{\xi_{2k+2}^{(2)}}\norm{\xi_{2j+2}^{(1)}} +
              r_4\norm{\xi_{2k+2}^{(2)}}\norm{\xi_{2j+2}^{(2)}}\right)^2\!\!,
             \label{tired}
\end{eqnarray}
for some large $m_0$, $n_0$, $k_0$ and $j_0$.

Let us consider any term in the first sum above. Since $k=m_q+p$ and
$j=m_t+l$ for some $m_q$, $m_t$, $0\leq p < m_{q+1} - m_q$ and
$0\leq l < m_{t+1} - m_t$, these terms may be written as
\begin{equation}
\label{first-sum}
r_3\norm{\xi_{2m_q+2p+2}^{(1)}}\norm{\xi_{2m_t+2l+2}^{(2)}} +
              r_4\norm{\xi_{2m_q+2p+2}^{(2)}}\norm{\xi_{2m_t+2l+2}^{(2)}}.
\end{equation}
From (\ref{schubert}), it follows that
\begin{eqnarray*}
\norm{\xi_{2m_q+2p+2}^{(k)}}
&\leq&
r_5\exp\left(\sum_{s=m_q+1}^{m_q+p}g_s^{(k)}\right)\norm{\xi_{2m_q+2}^{(k)}}
\\
&\leq&
r_6\left(\frac{m_q+p}{m_q}\right)^\frac{\hat{v}}{2}\norm{\xi_{2m_q+2}^{(k)}}
\end{eqnarray*}
holds for $k=1,2$. Thus, (\ref{first-sum}) is bounded from above by
\[
r_7\left(\frac{m_q+p}{m_q}\right)^\frac{\hat{v}}{2}
  \left(\frac{m_t+l}{m_t}\right)^\frac{\hat{v}}{2}
  \left(\norm{\xi_{2m_q+2}^{(1)}}\norm{\xi_{2m_t+2}^{(2)}} +
         \norm{\xi_{2m_q+2}^{(2)}}\norm{\xi_{2m_t+2}^{(2)}}\right).
\]

Let us now look at the sum between parentheses in the last expression.
On one hand, since Lemma~\ref{lemma-bullshit} holds, we have
\[
\norm{\xi_{2m_q+2}^{(2)}}\norm{\xi_{2m_t+2}^{(2)}}
\leq r_8\, m_q^{-\frac{\alpha}{2}}m_t^{-\frac{\alpha}{2}}.
\]
On the other hand, since $k\leq j$ implies $m_q\leq m_t$,
\begin{eqnarray*}
\norm{\xi_{2m_q+2}^{(1)}}\norm{\xi_{2m_t+2}^{(2)}}
&\leq&
r_9 \exp\left(\sum_{s=1}^{m_q}g_s^{(1)}+\sum_{s=1}^{m_q}g_s^{(2)}\right)
    \exp\left(\sum_{s=m_q+1}^{m_t}g_s^{(2)}\right)
\\
&\leq&
r_9 \exp\left[-\alpha\log(m_q)+O(1)\right]
    \left(\frac{m_t}{m_q}\right)^{-\frac{\alpha}{2}}
\\
&\leq&
r_{10} \, m_q^{-\frac{\alpha}{2}}m_t^{-\frac{\alpha}{2}}\,,
\end{eqnarray*}
where the second inequality above follows from the proof
of Lemma~\ref{lemma-bullshit} along with Lemma~\ref{life-saver}.
Therefore, we have
\[
r_3\norm{\xi_{2m_q+2p+2}^{(1)}}\norm{\xi_{2m_t+2l+2}^{(2)}} +
              r_4\norm{\xi_{2m_q+2p+2}^{(2)}}\norm{\xi_{2m_t+2l+2}^{(2)}}
\leq
r_{11} \, m_q^{-\frac{\alpha}{2}}m_t^{-\frac{\alpha}{2}}.
\]
Similar estimates show that the same bound holds for the terms of
the second sum in (\ref{tired}). By rearranging sums, we finally obtain
\begin{eqnarray*}
\lefteqn{
\sum_{m=m_0}^{\infty}\sum_{n=n_0}^{\infty}\abs{G_{mn}}^2}\\
&<&
r_{11}  \sum_{m=q_0}^{\infty}\!\sum_{p=0}^{m_{q+1}-m_q-1}
        m_q^{-\alpha}\left(\frac{m_q+p}{m_q}\right)^{\hat{v}}
        \sum_{t=t_0}^{\infty}\!\sum_{l=0}^{m_{t+1}-m_t-1}
        m_t^{-\alpha}\left(\frac{m_t+l}{m_t}\right)^{\hat{v}}
\\
&<&
r_{12}  \sum_{q=q_0}^{\infty}
  \frac{m_{q+1}-m_q}{m_q^{\alpha}}\left(\frac{m_{q+1}}{m_q}\right)^{\hat{v}}
        \sum_{t=t_0}^{\infty}
   \frac{m_{t+1}-m_t}{m_t^{\alpha}}\left(\frac{m_{t+1}}{m_t}\right)^{\hat{v}}\,.
\end{eqnarray*}
Since (\ref{last-one}) holds, we conclude that
$\mbox{Tr}\left(G^* G\right)<\infty$.
\end{proof}

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