Spectral analysis of the Kronig-Penney model with Wigner-von Neumann perturbations
Vladimir Lotoreichik
The spectrum of the self-adjoint Schr\"odinger operator associated with the Kronig-Penney model on the half-line has a band-gap structure: its absolutely continuous spectrum consists of intervals (bands) separated by gaps. We show that if one changes strengths of interactions or locations of interaction centers by adding an oscillating and slowly decaying sequence which resembles the classical Wigner-von Neumann potential, then this structure of the absolutely continuous spectrum is preserved. At the same time in each spectral band precisely two critical points appear. At these points ``instable'' embedded eigenvalues may exist. We obtain locations of the critical points and discuss for each of them the possibility of an embedded eigenvalue to appear. We also show that the spectrum in gaps remains pure point.
Fig1.tex
 
\begin{figure}
\label{fig1}
\begin{tikzpicture}[domain=0:7.8]
    \draw[->] (0.0,0) -- (8.0,0) node[right] {$k$};
    \draw[->] (0,-2.0) -- (0,4.2) node[above] {$L_\delta(k)$};
\draw [blue, domain= 0.01:8, smooth] plot (\x, {(4.0 *  sin((\x * 1.5) r ) / (2.0 * \x) + cos((\x * 1.5)  r))});
\draw (4.0, 1.6)  node[above] {$L_{\delta}(k) = \alpha_0 \frac{\sin(kd)}{2k} + \cos(kd)$};
 \draw (0.0,-1) node[left] {$-1$};
 \draw  (0.0,1) node[left] {$1$};
 \draw (0.0,0.6) node[left] {$\cos(\omega)$};
 \draw  (0.0,-0.6) node[left] {$-\cos(\omega)$};


\draw [black, domain= 0.01:8, smooth, dashed] plot (\x, {0.6});
\draw [black, domain= 0.01:8, smooth, dashed] plot (\x, {-0.6});
\draw [black, domain= 0.01:8, smooth] plot (\x, {1.0});
\draw [black, domain= 0.01:8, smooth] plot (\x, {-1.0});

    
%band 1   
\draw[connection] (1.3,1.00) -- (1.3,0.0) node[below] {
%{\small $\begin{smallmatrix}\lambda_1^+ \lambda_{1,\mathrm cr}^+\end{smallmatrix}$}
};

\draw[connection] (2.1,-1.00) -- (2.1,0.0) node[above] {
%{\small $\begin{smallmatrix}\lambda_1^-\end{smallmatrix}$}
};
\draw[very thick, color=black] (1.3,0.01) -- (2.1, 0.01);


\draw[connection] (1.45,\cw) -- (1.45,0); 
\draw[color = red, very thick] (1.45,0.0) node[] {\small \bf x};

\draw[connection] (1.85,-\cw) -- (1.85,0);
\draw[color = red, very thick] (1.85,0.0) node[] {\small \bf x};


%band 2   
\draw[connection] (2.9,-1.00) -- (2.9,0.0) node[below] {
%{\small $\begin{smallmatrix}\lambda_1^+ \lambda_{1,\mathrm cr}^+\end{smallmatrix}$}
};

\draw[connection] (4.2,1.00) -- (4.2,0.0) ;
\draw[very thick, color=black] (2.9,0.01) -- (4.2, 0.01);


\draw[connection] (3.1,-\cw) -- (3.1,0); 
%$\begin{smallmatrix}\lambda^+_{1,\rm cr}\end{smallmatrix}$};
\draw[color = red, very thick] (3.1,0.0) node[] {\small \bf x};

\draw[connection] (3.9,\cw) -- (3.9,0);
\draw[color = red, very thick] (3.9,0) node[] {\small \bf x};


%band 3 
\draw[connection] (4.7,1.00) -- (4.7,0.0);

\draw[connection] (6.3,-1.00) -- (6.3,0.0);
\draw[very thick, color=black] (4.7,0.01) -- (6.3, 0.01);


\draw[connection] (5.9,-\cw) -- (5.9, 0); 
\draw[color = red, very thick] (5.9,0) node[] {\small \bf x};

\draw[connection] (5.05,\cw) -- (5.05,0);
\draw[color = red, very thick] (5.05,0) node[] {\small \bf x};

%band 4 
\draw[connection] (6.6,-1.00) -- (6.6,0.0);;


\draw[connection] (6.3,-1.00) -- (6.3,0.0);
\draw[very thick, color=black] (6.6,0.01) -- (7.85, 0.01);


\draw[connection] (7.1,-0.6) -- (7.1,  0); 
\draw[color = red, very thick] (7.1,0) node[] {\small \bf x};
\end{tikzpicture}
\caption{{\em The curve} is the graph of $L_\delta$; {\em bold intervals} are bands of 
the absolutely continuous spectrum; {\em crosses} denote the critical points $\{\lambda_{n,\rm cr}^\pm\}_{n\in\dN}$.}
\end{figure}

Lotoreichik-Simonov_-_2011-12-20__sent_to_arXiv_.tex
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\begin{document}

\title[Wigner von-Neumann perturbation of the Kronig-Penney model]{Spectral analysis of the Kronig-Penney model with Wigner-von Neumann perturbations}


\author{Vladimir Lotoreichik}
\address{Department of Mathematics,
Saint-Petersburg State University of  Information Technologies, Mechanics and Optics,
Kronverkskiy, 49, Saint-Petersburg, Russia}
\address{
Department of Computational Mathematics,
Graz University of Technology,
Steyrergasse 30, 8010, Graz, Austria
}
\email{vladimir.lotoreichik@gmail.com}
%\address{Department of Mathematics,
%Saint-Petersburg State University of  Information Technologies, Mechanics and Optics,
%Kronverkskiy, 49, Saint-Petersburg, Russia}


\author{Sergey Simonov}
\address{Chebyshev Laboratory, Department of Mathematics and Mechanics, Saint-Petersburg State University, 14th Line, 29b, Saint-Petersburg, 199178, Russia}
\address{Institut f\"ur Analysis und Scientific Computing, Technische Universit\"at Wien, Wiedner Hauptstrasse, 8--10, Wien, A--1040, Austria}
\email{sergey.a.simonov@gmail.com}


\begin{abstract}
The spectrum of the self-adjoint Schr\"odinger operator associated with the Kronig-Penney model  
on the half-line has a band-gap structure:
its absolutely continuous spectrum consists of intervals (bands) separated by gaps. 
We show that if one changes strengths of interactions or locations of interaction centers 
by adding an oscillating and slowly decaying sequence which resembles the classical 
Wigner-von Neumann potential, then this structure of the absolutely continuous 
spectrum is preserved. At the same time in each spectral band precisely two critical points appear. 
At these points ``instable'' embedded eigenvalues may exist. We obtain locations of the critical points and 
discuss for each of them the possibility of an embedded eigenvalue to appear. 
We also show that the spectrum in gaps remains pure point.
\end{abstract}


\maketitle


\section{Introduction}

In the classical paper~\cite{vNW29} von~Neumann and Wigner studied the
one-dimensional Schr\"odinger operator with the potential of the form $\frac{c\sin(2\omega x)}x$ 
and discovered that such an operator may have an eigenvalue at the point of the continuous 
spectrum $\lambda=\omega^2$. Since then such potentials permanently attracted the interest of different authors \cite{Al72,Ma73, RS78, Be91,Be94,HKS91,KN07,N07,L10,JS10}. Potentials of the type $\frac{c\sin(2\omega x)}{x^{\gamma}}$ with $\gamma\in(0,1)$ also possess such a property and they are often called Wigner-von Neumann potentials.

Consider the operator $H_{\varkappa}$ on $\dR_+$ that formally corresponds to the differential expression
\begin{equation*}
	-\frac{d^2}{dx^2} + \sum_{n\in\dN}\alpha_0\delta_{nd}\langle\delta_{nd},\cdot\rangle
\end{equation*}
with the boundary condition
\begin{equation}\label{bc}
    \psi(0)\cos\varkappa-\psi'(0)\sin\varkappa = 0,
\end{equation}
at the origin, where $\alpha_0\in\dR,d >0,\varkappa\in[0,\pi)$ and $\delta_x$ 
is the delta-distribution supported on the point $x\in\dR_+$ (for the mathematically rigorous definition of the operator see Section~\ref{sec:def}, cf. \cite[Chapter III]{AGHH05}). The operator $H_{\varkappa}$ is self-adjoint in $L^2(\dR_+)$. It describes the behavior of a free non-relativistic charged quantum particle interacting with
the lattice $d\dN$. The constant $\alpha_0$ characterizes the strength of interaction between the free charged quantum particle and each interaction center in the lattice.  The spectrum of the operator $H_{\varkappa}$ 
has a band-gap structure: it consists of infinitely many bands of the purely 
absolutely continuous spectrum and outside these bands the spectrum of  $H_{\varkappa}$ is discrete. The investigation of such operators was initiated in the classical paper~\cite{KP31} by 
Kronig and Penney.

%It is a tradition to consider an analogue of the Kronig-Penney model with 
%$\delta$-interactions replaced by $\delta'$-interactions. 
%The operator $H_{\varkappa}'$ formally corresponds to the expression
%\begin{equation*}
%	-\frac{d^2}{dx^2} + \sum_{n\in\dN}\alpha_0\delta'_{nd}\langle\delta'_{nd},\cdot\rangle
%\end{equation*}
%with the same boundary condition at the origin. 
%The spectrum of $H'_{\varkappa}$ has a band-gap structure as well.

In the present paper we study what happens with
the spectrum of  the Kronig-Penney model  in the case
of perturbation of strengths or positions of interactions 
by a slowly decaying oscillating sequence resembling the Wigner-von Neumann potential.                                                                                                                                                                                                                                         
Let  constants $d,\alpha_0,c,\omega,\gamma$ and a 
real-valued sequence $\{q_n\}_{n\in\mathbb N}$ be such that
\begin{equation}\label{conditions on constants}
     d>0, \alpha_0\in\dR, \omega\in\left(0,\frac{\pi}2\right),\gamma\in\left(\frac12,1\right],\{q_n\}_{n\in\dN}\in \ell^1(\dN).
\end{equation}

\emph{Model I: Wigner-von Neumann amplitude perturbation.}
We add a discrete Wigner-von Neumann potential to the constant sequence of interaction strengths. 
Let the discrete set $\wt X = \{\wt x_n\colon n\in\dN\}$ supporting the interaction centers be defined as
\begin{equation}\label{wt X}
    \wt x_n := nd, \quad n\in\dN,
\end{equation}
and let the sequence of interaction strengths $\wt \alpha = \{\wt\alpha_n\}_{n\in\dN}$ be as below:
\begin{equation}\label{wt alpha}
    \wt\alpha_n:=\alpha_0+\frac{c\sin(2\omega n)}{n^{\gamma}}+q_n,\quad n\in\dN.
\end{equation}
We study the self-adjoint operator $H_{\varkappa,\wt X,\wt \alpha}$
which formally corresponds to the differential expression
\begin{equation*}
	-\frac{d^2}{dx^2} + \sum_{n\in\dN}\wt\alpha_n\delta_{\wt x_n}\langle\delta_{\wt x_n},\cdot\rangle
\end{equation*}
with the boundary condition \eqref{bc}. This operator reflects an amplitude  perturbation of the Kronig-Penney model.

\emph{Model II: Wigner-von Neumann positional perturbation.}
We change the distances between interaction centers in a ``Wigner-von Neumann''  way, i.e. 
we add a sequence of the form of Wigner-von Neumann potential to the coordinates 
of interaction centers leaving the strengths constant. 
Let the discrete set $\wh X = \{\wh x_n\colon n\in\dN\}$ be as below:
\begin{equation}\label{wh X}
    \wh x_n :=nd+\frac{c\sin(2\omega n)}{n^{\gamma}}+q_n, \quad n\in\dN,
\end{equation}
and the sequence of strengths be defined as
\begin{equation}\label{wh alpha}
    \wh\alpha_n := \alpha_0, \quad n\in\dN.
\end{equation}
We study the operator $H_{\varkappa,\wh X,\wh \alpha}$ 
which formally corresponds to the differential expression
\begin{equation*}
	-\frac{d^2}{dx^2} + \sum_{n\in\dN}\wh\alpha_n\delta_{\wh x_n}\langle\delta_{\wh x_n},\cdot\rangle
\end{equation*}
with the boundary condition \eqref{bc}. This operator reflects a positional perturbation of the Kronig-Penney model
and describes properties of one-dimensional crystals with global defects. 
%--------------------------
We also mention that local defects in the Kronig-Penney are discussed in \cite[\S III.2.6]{AGHH05};
situations of random perturbations of positions  were recently considered 
in~\cite{HIT10}.

The essential spectrum of the operator $H_\varkappa$ has a band-gap structure
similar to the case of Schr\"odinger operator with regular periodic potential:
%\marginpar{!!! $\alpha = 0$ is possible and no gaps in  that case.}
\begin{equation*}
\begin{split}
	&\sigma_{\rm ess}(H_\varkappa) =
    \sigma_{\rm ac}(H_\varkappa) =
    \bigcup_{n=1}^\infty
    \Bigl(\bigl[\lambda_{2n-1}^+,	
	\lambda_{2n-1}^-\bigr]\cup \bigl[\lambda_{2n}^-,\lambda_{2n}^+\bigr]\Bigr),\\
	&\text{where } \lambda_1^+ < \lambda_1^- \le \lambda_2^- < \lambda_2^+ \le \dots \le \lambda_{2n-1}^+ 	
	<\lambda_{2n-1}^- \le \lambda_{2n}^-<\lambda_{2n}^+  \le \dots
\end{split}
\end{equation*}
The locations of boundary points of the spectral bands are determined 
by the parameters $\alpha_0$ and $d$. Namely, the values $\lambda_n^\pm$ are the $n$-th roots of 
the corresponding Kronig-Penney equations
\begin{equation*}
    L_\delta\left(\sqrt{\lambda}\right) = \pm 1,
\end{equation*}
where
\begin{equation}\label{Lyapunov function for delta}
    L_\delta(k) = \cos(kd)+\alpha_0 \frac{\sin(kd)}{2k}.
\end{equation}
%$\lambda_{2n-1}^- = \bigl(\frac{\pi(2n-1)}{d}\bigr)^2$, $\lambda_{2n}^+ = \bigl(\frac{2\pi n}{d}\bigr)^2$.
For the details the reader is referred to the monograph~\cite[Chapter III]{AGHH05}.

%One has:
%\begin{equation*}
%	0<\mu_1^+<\mu_1^-<\mu_2^-<\mu_2^+<\dots<
%        \mu_{2n-1}^+<\mu_{2n-1}^-<\mu_{2n}^-<\mu_{2n}^+<\dots
%\end{equation*}
%and
%\begin{equation*}
%\begin{split}
%	&\sigma_{\rm ess}(H'_\varkappa) =
%   \sigma_{\rm ac}(H'_\varkappa) =
%    \bigcup_{n=1}^\infty
%   \Bigl(\bigl[\mu_{2n-1}^+,\mu_{2n-1}^-\bigr]
%    \cup
%    \bigl[\mu_{2n}^-,\mu_{2n}^+\bigr]\Bigr).
%\end{split}
%\end{equation*}
%The values  $\mu_n^\pm$ are the $n$-th roots  of the corresponding equations
%\begin{equation}
%\label{eq:delta'}
%   L_{\delta'}\left(\sqrt{\mu}\right)=\pm 1,
%\end{equation}
%where
%\begin{equation}\label{Lyapunov function for delta'}
%    L_{\delta'}(k)=\cos(kd)-\alpha_0\frac{k\sin(kd)}2,
%\end{equation}
%and, in particular, 
%$\mu_{2n-1}^- = \bigl(\frac{\pi(2n-1)}{d}\bigr)^2$ and 
%$\mu_{2n}^+ = \bigl(\frac{2\pi n}{d}\bigr)^2$.




In the present paper we show that the absolutely continuous spectra of the operators $H_{\varkappa,\wt X,\wt \alpha}$ and
$H_{\varkappa,\wh X,\wh \alpha}$ coincide with the absolutely continuous spectra of the non-perturbed operator $H_{\varkappa}$.  However the spectrum in bands may not remain purely absolutely continuous. Namely,
at certain (critical) points embedded eigenvalues may appear.  In each band there are two such points. 
The critical points $\lambda_{n,\rm cr}^\pm$ in the $n$-th spectral band
 are the  $n$-th roots of the equations
\begin{equation*}
    L_\delta\left(\sqrt{\lambda}\right) = \pm\cos\omega.
\end{equation*}
The illustration is given on the Figure I.   
\input{Fig1}
For the operators considered we give exact conditions which ensure that a given critical point  is 
indeed  an embedded eigenvalue for some $\varkappa\in[0,\pi)$.  This can occur 
only for one value of $\varkappa\in[0,\pi)$. The possibility of the appearance of an embedded 
eigenvalue at certain critical point depends on the rate of the decay of the subordinate generalized 
eigenvector. We calculate the asymptotics of  generalized
eigenvectors for all values of the spectral parameter $\lambda$ including the critical
points, except the endpoints of the bands. We also show that the spectrum in gaps remains pure point.

%In the case of $\delta'$-type potentials the critical points in the $n$-th spectral band $\mu_{n,\rm cr}^\pm$ are the $n$-th %roots of the equations
%\begin{equation*}
%   L_{\delta'}\left(\sqrt\mu\right)=\pm\cos\omega,
%\end{equation*}
%The illustration is given on Figure~2.  
%\input{Fig2}

Our results are close to the results for one-dimensional Schr\"od\-inger operator 
with the Wigner-von Neumann potential and a periodic background potential. Such  operators 
were considered very recently in~\cite{KN07, NS11, KS11}.
%Here we obtain asymptotics of the subordinate solutions explicitly in terms of the parameters. 
%In this sense our model has a remarkable property to be exactly  solvable.

To study spectra in bands we make a discretization 
of the spectral equations and further we perform an asymptotic integration of obtained discrete linear system
using Benzaid-Lutz-type theorems \cite{BL87}. As the next step we apply a modification of 
Gilbert-Pearson subordinacy theory \cite{GP87}.  To study spectra in gaps 
we apply  compact perturbation argument in Weyl function method, see, e.g.,
\cite{DM95}, \cite{BGP08}.  

The reader can trace some analogies of our case with Jacobi matrices. The coefficient matrix of the
discrete linear system that appears in our analysis has a form similar to the transfer-matrix 
for some Jacobi matrix and the Weyl function that appears in our analysis takes Jacobi matrices
as its values.

The body of the paper contains two parts: the preliminary part --- which consists of mostly known
material --- and the main part, where we obtain  new results. 
In the preliminary part we give a
rigorous definition of one-dimensional Schr\"odinger operators with $\delta$-interactions (Section~\ref{sec:def}),
show how to reduce the spectral equations for these operators
to  discrete linear systems in $\dC^2$ (Section~\ref{sec:system}), provide a formulation of the subordinacy theory 
analogue for the operators considered (Section~\ref{sec:subord}). Further we  
formulate few results from asymptotic integration theory for discrete linear
systems (Section~\ref{sec:BL}) and provide a characterization of the essential spectra
of considered operators using the Weyl function associated with certain ordinary boundary triple (Section~\ref{sec:Weyl}).
In the main part, in Section~\ref{sec:class} we study a special class of discrete linear systems in $\dC^2$ and find asymptotics of solutions of these systems. After certain technical preliminary calculations in Section~\ref{sec:decomp}, 
we proceed to Section~\ref{sec:main}, where we obtain asymptotics of generalized eigenvectors
for Schr\"odinger operators with point interactions subject to Model I and Model II. Further we pass to the conclusions about  the spectra in bands putting an emphasis on critical points. 
Section~\ref{sec:gaps} is devoted to the analysis of spectra in gaps by means of the Weyl function.

\subsection*{Notations}
By small letters with integer subindices, e.g. $\xi_n$, we denote sequences of 
complex numbers. By small letters with integer subindices and arrows above, e.g. $\overrightarrow u_n$, we denote sequences of  $\dC^2$-vectors.
By capital letters with integer indices, e.g. $R_n$, we denote sequences of 
$2\times 2$ matrices with complex entries. We use notations 
$\ell^p(\dN)$, $\ell^p(\dN,\dC^2)$ and $\ell^p(\dN,\dC^{2\times 2})$  for spaces of  summable ($p=1$), square-summable ($p=2$) and bounded ($p=\infty$) sequences of  complex values, complex two-dimensional vectors and complex $2\times 2$ matrices, respectively.
We denote for a self-adjoint operator its pure point, absolutely continuous, singular continuous, essential and discrete spectra by
$\sigma_{\rm pp}$, $\sigma_{\rm ac}$, $\sigma_{\rm sc}$,  $ \sigma_{\rm ess}$ and 
$\sigma_{\rm d}$, respectively. We write $T\in\sS_\infty$ in the case $T$ is a compact operator. 
Other notations are assumed to be clear for the reader.
\section{Preliminaries}
\label{sec:prelim}

%In Section~\ref{sec:def} we define Schr\"odinger operators with point interactions rigorously,
%clarify what is spectral equations for such operators and formulate subordinacy principle 
%for the spectral analysis of such operators in terms of asymptotical properties of solutions 
%for spectral equations. Section~\ref{sec:system} contains a reduction to discrete linear systems
% of spectral equations introduced in Section~\ref{sec:def}. In Section~\ref{sec:BL} we formulate
%certain simplifications of Benzaid-Lutz theorems suitable for our purposes. 
%Finally, in Section~\ref{sec:Weyl} we give another approach to the spectral analysis 
%of operators with point interactions based on Weyl functions associated with ordinary boundary triples.
 
\subsection{Definition of operators with point interactions}
\label{sec:def}
In this section we give a rigorous definition of operators with $\delta$-interactions, see, e.g.,  
\cite{GK85, Ko89}. 
Let $\alpha=\{\alpha_n\}_{n\in\dN}$ be a sequence of real numbers and let $X = \{x_n\colon n\in\dN\}$ be a discrete set on $\dR_+$ ordered as $0<x_1<x_2< \dots$. Assume that the set $X$ satisfies
\begin{equation}\label{condition on X}
	\inf_{n\in\dN}\big|x_{n+1}-x_n\big|>0\quad\text{and}\quad \sup_{n\in\dN}\big|x_{n+1}-x_n\big|<\infty.
\end{equation}
Denote also $x_0:=0$. In order to define the operator corresponding to the expression
\begin{equation*}
    \tau_{X,\alpha}:=-\frac{d^2}{dx^2}+\sum_{n\in\mathbb N}\alpha_n\delta_{x_n}
	\langle\delta_{x_n},\cdot\rangle
\end{equation*}
and the boundary condition
\begin{equation*}
    \psi(0)\cos\varkappa-\psi'(0)\sin\varkappa = 0,
\end{equation*}
consider the following set of functions:
\begin{equation*}
	\cS_{X,\alpha} \defeq
	\Biggl\{
 	\psi:\ \psi,\psi^{\prime} \in AC_{\rm loc}(\dR_+\setminus X),
	\begin{smallmatrix}
	 	\psi(x_n+) = \psi(x_n-)=\psi(x_n)
        \\
		\psi^\prime(x_n+) - \psi^{\prime}(x_n-) = \alpha_n\psi(x_n)
	\end{smallmatrix},
    ~n\in\dN\Biggr\}
\end{equation*}
and let the operator $H_{\varkappa,X,\alpha}$ be defined by its action
\begin{equation*}
	H_{\varkappa, X, \alpha}\psi:=-\psi''
\end{equation*}
on the domain
\begin{equation*}
	\dom H_{\varkappa, X, \alpha}
                    :=
	\left\{
 	\psi \in L^2(\dR_+)\cap\cS_{X,\alpha}\colon\psi''\in L^2(\dR_+),\psi(0)\cos\varkappa = \psi^\prime(0)\sin\varkappa
	\right\}.
\end{equation*}
The  spectral equation $\tau_{X,\alpha}\psi=\lambda\psi$ is understood as the equation $-\psi''(x)=\lambda\psi(x)$ for $\psi\in\cS_{X,\alpha}$. The latter is equivalent to the following system:
\begin{equation}\label{spectral equation for delta}
\begin{split}
    &-\psi''(x)=\lambda\psi(x),\quad x\in\dR_+\setminus X,
    \\[0.5ex]
    &\psi(x_n+)=\psi(x_n-),\ \psi'(x_n+)=\psi'(x_n-)+\alpha_n\psi(x_n-),\quad n\in\dN.
\end{split}
\end{equation}
The equation \eqref{spectral equation for delta} has two linear independent 
solutions which are called generalized eigenvectors.
If $\psi\in L^2(\dR_+)$ satisfies~\eqref{spectral equation for delta} and the boundary condition, 
then $\psi$ is an eigenfunction of $H_{\varkappa,X,\alpha}$. 

%One can define the operator $H'_{\varkappa,X,\alpha}$ corresponding to the expression
%\begin{equation*}
%	\tau'_{X,\alpha}:=-\frac{d^2}{dx^2}+\sum_{n\in\mathbb N}\alpha_n\delta'_{x_n}\langle\delta_{x_n}',\cdot\rangle
%\end{equation*}
%with the boundary condition
%\begin{equation*}
%    \psi(0)\cos\varkappa-\psi'(0)\sin\varkappa = 0,
%\end{equation*}
%in an analogous way, if the set $X = \{x_n\}_{n\in\dN}$ and the sequence $\alpha = \{\alpha_n\}_{n\in\dN}$ are given and the condition \eqref{condition on X} holds. Let
%\begin{equation*}
%    \cS'_{X,\alpha} \defeq
%	\Biggl\{
% 	\psi:\ \psi,\psi^{\prime} \in AC_{\rm loc}(\dR_+\setminus X),
%	\begin{smallmatrix}
%		\psi^\prime(x_n+) = \psi^\prime(x_n-)=\psi^\prime(x_n)\\
%		 \psi(x_n+) - \psi(x_n-) = \alpha_n\psi^\prime(x_n)
%	\end{smallmatrix},~n\in\dN
%	\Biggr\}
%\end{equation*}
%and define
%\begin{equation*}
%    H'_{\varkappa, X,\alpha}\psi:=-\psi^{\prime\prime}
%\end{equation*}
%on the domain
%\begin{equation*}
%	\dom H'_{\varkappa, X,\alpha} :=
%	\left\{
% 	\psi \in L^2(\dR_+)\cap\cS'_{X,\alpha}\colon\psi''\in L^2(\dR_+),
%    \psi(0)\cos\varkappa = \psi^\prime(0)\sin\varkappa
%	\right\}.
%\end{equation*}
%The spectral equation for the differential expression $\tau'_{X,\alpha}$ has the form
%\begin{equation}\label{spectral equation for delta'}
%\begin{split}
%&-\psi''(x)=\lambda\psi(x),\quad x\in\dR_+\setminus X, \\[0.5ex]
%&\psi(x_n+)=\psi(x_n-)+\alpha_n\psi'(x_n-),\ \psi'(x_n+)=\psi'(x_n-), \quad n\in\dN.
%    \end{split}
%\end{equation}


\subsection{Reduction of the eigenfunction equation to a discrete linear system}
\label{sec:system}
In this subsection we recall rather well-known way of reduction of the spectral equation 
\eqref{spectral equation for delta} to a discrete linear system.  
Let the discrete set $X$ and the sequence of strengths $\alpha$ be as in the previous section. To make our formulas more compact we introduce the following notations
\begin{equation*}
    s_n(k):=\sin\bigl(k(x_{n+1}-x_n)\bigr)\quad\text{and}\quad c_n(k):=\cos\bigl(k(x_{n+1}-x_n)\bigr), \qquad n\in\dN_0.
\end{equation*}
%\marginpar{!!! $\psi$ must be a solution}
For a solution $\psi$ of \eqref{spectral equation for delta} corresponding to $\lambda = k^2$
we introduce the sequence below
\begin{equation*}
%\label{xi}
    \xi_n:=\psi(x_n),\quad n\in\dN_0.
\end{equation*}
If the condition \eqref{condition on X} is satisfied, then by \cite[Chapter III, Theorem 2.1.5]{AGHH05} for $k\in\mathbb C_+\cup \mathbb R$ and $n\in\dN$ such that $s_{n-1}(k),s_n(k)\neq 0$ it holds that
\begin{equation}\label{equation for xi}
	-k\Biggl(\frac{\xi_{n-1}}{s_{n-1}(k)}+\frac{\xi_{n+1}}{s_n(k)}\Biggr)
    +\Biggl(\alpha_n
    +k\Bigr(\frac{c_{n-1}(k)}{s_{n-1}(k)}+ \frac{c_n(k)}{s_n(k)}\Bigr)\Biggr)\xi_n=0.
\end{equation}
Inversely, solutions of the eigenfunction equation on each of the intervals $(x_n,x_{n+1})$ can be recovered from their values at the endpoints $x_n$ and $x_{n+1}$:
\begin{equation}\label{psi from xi}
     \psi(x)=\frac{\xi_n\sin(k(x_{n+1}-x))+\xi_{n+1}\sin(k(x-x_n))}{\sin(k(x_{n+1}-x_n))},
    \quad
x\in[x_n;x_{n+1}].
\end{equation}
%Analogously, for a function $\psi\in\cS'_{X,\alpha}$ we introduce the following sequence
%\begin{equation}\label{eta}
%    \eta_n:= \psi'(x_n),\quad n\in\dN_0.
%\end{equation}
%By \cite[Chapter III, Theorem 3.3]{AGHH05} for $k\in\mathbb C_+\cup \mathbb R$ such that for every $n\in\mathbb N_0$ $s_n(k)\neq 0$ it holds that
%\begin{equation}\label{equation for eta}
%	-\frac{1}{k}\Biggl(\frac{\eta_{n-1}}{s_{n-1}(k)}+\frac{\eta_{n+1}}{s_n(k)}\Biggr)
%	+\Biggl(-\alpha_n
%    +\frac{1}{k}\Bigr(\frac{c_{n-1}(k)}{s_{n-1}(k)}+\frac{c_{n}(k)}{s_{n}(k)}\Bigr)\Biggr)\eta_n=0,\ n\in\mathbb N.
%\end{equation}
%Solutions of the eigenfunction equation on each of the intervals $(x_n,x_{n+1})$ can be recovered from the values of their derivatives at the endpoints $x_n$ and $x_{n+1}$:
%\begin{equation}\label{psi from eta}
%    \psi(x)=\frac{\eta_n\cos(k(x_{n+1}-x))-\eta_{n+1}\cos(k(x-x_n))}{k\sin(k(x_{n+1}-x_n))},
%\quad x\in[x_n,x_{n+1}],\ n\in\mathbb N.
%\end{equation}
The reader may confer with \cite{E97}, where a more general case of a quantum graph is considered. 

Instead of working with recurrence relation \eqref{equation for xi}
 we will consider a discrete linear system in $\dC^2$. Define
\begin{equation*}\label{vectors u and v}
    \overrightarrow u_n := \begin{pmatrix}\xi_{n-1} \\ \xi_n \end{pmatrix}.
%\quad\text{and}\quad
%    \overrightarrow v_n := \begin{pmatrix}\eta_{n-1} \\ \eta_n \end{pmatrix}.
\end{equation*}
Then \eqref{equation for xi} is equivalent to
\begin{equation}\label{system for delta}
    \overrightarrow u_{n+1}=T_n(k)\overrightarrow u_n
\end{equation}
with
\begin{equation}\label{T-n-k}
    T_n(k):=\left(
    \begin{array}{cc}
    0 & 1 \\
    -\frac{s_n(k)}{s_{n-1}(k)}
	&
    \frac{\sin(k(x_{n+1}-x_{n-1}))}{s_{n-1}(k)}+\frac{\alpha_ns_n(k)}{k}   \\
    \end{array}
    \right)
\end{equation}
%and \eqref{equation for eta} is equivalent to
%\begin{equation}\label{system for delta'}
%    \overrightarrow v_{n+1}
%    =
%    \left(
%    \begin{array}{cc}
%    0 & 1 \\
%    -\frac{s_n(k)}{s_{n-1}(k)}
%	&
%    \frac{\sin(k(x_{n+1}-x_{n-1})}{s_{n-1}(k)}-\alpha_nks_n(k)   \\
%    \end{array}
%    \right)
%    \vec v_n.
%\end{equation}
The coefficient matrix of this system $T_n(k)$ is called the transfer-matrix.
% and $M_n'(k)$, respectively.

\subsection{Subordinacy}
 \label{sec:subord}
The subordinacy theory as suggested in \cite{GP87} by D.~Gilbert and D.~Pearson 
produced a strong influence on the spectral theory of one-dimensional Schr\"odinger operators. 
Later on the subordinacy theory was translated to difference equations \cite{KP92}. 
For Schr\"odinger operators with $\delta$-interactions there exists a modification of the subordinacy theory, 
see, e.g., \cite{SCS94} which  relates the spectral properties of the operator $H_{\varkappa,X,\alpha}$ 
with the asymptotic behavior of the solutions of the spectral equation~\eqref{spectral equation for delta}. 
%A similar type relation holds for the operator $H_{\varkappa, X,\alpha}'$ and the spectral equation \eqref{spectral equation for delta'}. 
Analogously, to the classical definition of the subordinacy \cite{GP87} 
we say that a solution $\psi_1$ of the equation $\tau_{X,\alpha}\psi =\lambda\psi$ 
%with $\tau = \tau_{X,\alpha}$ or $\tau = \tau_{X,\alpha}'$ 
is subordinate if and only  if for any other solution $\psi_2$ of the same equation not proportional to 
$\psi_1$  the following limit property holds:
\begin{equation*}
    \lim_{x\rightarrow+\infty}\frac{\int_0^x |\psi_1(t)|^2dt}{\int_0^x |\psi_2(t)|^2dt} = 0.
\end{equation*}
We will use the following propositions to find location of the absolutely continuous spectrum.
\begin{prop}\cite[Proposition 7]{SCS94}
\label{prop:stolz}
Let  $H_{\varkappa, X,\alpha}$  be the self-adjoint operator
corresponding to the discrete set $X$ and the sequence of strengths $\alpha$ as in Section~\ref{sec:def}. 
Assume that for all $\lambda\subset (a,b)$ there is no subordinate solution for 
the spectral equation $\tau_{X,\alpha}\psi = \lambda\psi$. 
Then $[a,b] \subset \sigma(H_{\varkappa, X,\alpha})$ and $\sigma(H_{\varkappa, X,\alpha})$ 
is purely absolutely continuous in $(a,b)$. 
%The same is true for the operator $H'_{\varkappa, X,\alpha}$ and the expression $\tau'_{X,\alpha}$.
\end{prop}

\begin{prop}
\label{prop:Stolz2}
Let the discrete set $X = \{x_n\colon n\in\dN\}$ on $\dR_+$ be ordered as $0<x_1<x_2< \dots$. Assume that it satisfies conditions 
$\Delta x_* := \inf_{n\in\dN} | x_{n+1} - x_n | > 0$ and $\Delta x^* := \sup_{n\in\dN} | x_{n+1} - x_n | < \infty$. Let
$\alpha = \{\alpha_n\}_{n\in\dN}$ be a sequence of real numbers and let $\lambda\in\dR$.
Assume that $\liminf_{n\rightarrow\infty} \big| s_n\big(\sqrt{\lambda}\big) \big| > 0$.
If every solution of the equation $\tau_{X,\alpha}\psi = \lambda\psi$ (see \eqref{spectral equation for delta})
is bounded, then for such $\lambda$ there exists no subordinate solution.
\end{prop}
\begin{proof}
Let $\psi_1$ be an arbitrary solution of $\tau_{X, \alpha}\psi = \lambda\psi$. Differentiating \eqref{psi from xi} one gets 
in the case $s_n(k) \ne 0$
\[
|\psi_1'(x)|\le\frac{|k|(|\xi_n|+|\xi_{n+1}|)}{|s_n(k)|}\quad\text{for}~x\in[x_n,x_{n+1}].
\]
There exists $N(k)$ such that $\inf_{n\ge N(k)}|s_n(k)|>0$. Since the sequence $\{\xi_n\}_{n\in\dN}$ is bounded, one has that $\psi_1'(x)$ is also bounded for $x\ge x_{N(k)}$. Obviously for $x
\le x_{N(k)}$ it is bounded too, since it is piecewise continuous with finite jumps at the points of discontinuity.
Let $\psi_2$ be any other solution of $\tau_{X, \alpha}\psi = \lambda\psi$, which is linear independent with $\psi_1$. It follows that there exists a constant $C > 0$ such that 
\begin{equation}
\label{C}
\|\psi_1\|_\infty, \|\psi_2\|_\infty, \|\psi_1'\|_\infty, \|\psi_2'\|_\infty\le C.
\end{equation}
The Wronskian of the solutions $\psi_1$ and $\psi_2$ is independent of $x$:
\begin{equation}
\label{Wronskian}
W\{\psi_1,\psi_2\}:= \psi_1(x)\psi_2'(x) - \psi_1'(x)\psi_2(x),\quad\text{for all}~ x\in\dR_+\setminus X.
\end{equation}
This is easy to check: it is constant on every interval $(x_n,x_{n+1})$ and at the points $\{x_n\}_{n\in\dN}$ one has
$W\{\psi_1,\psi_2\}(x_n+)=W\{\psi_1,\psi_2\}(x_n-)$ from \eqref{spectral equation for delta}. The Wronskian is non-zero since the solutions are linear independent. From \eqref{Wronskian} one has:
\[
|W\{\psi_1,\psi_2\}|\le C(|\psi_1(x)| + |\psi_1'(x)|),\quad \text{for all}~ x\in\dR_+\setminus X,
\]
and therefore there exist constants $C_*$ and $C^*$ such that
\begin{equation}
\label{Stolz estimate}
0<C_* \le |\psi_1(x)| + |\psi_1'(x)| \le C^*,\quad\text{for all} ~ x\in\dR_+\setminus X.
\end{equation}

Now we apply the trick used in the proof of \cite[Lemma 4]{S92}. We consider for an arbitrary
$n\in\dN$ the interval $[x_n, x_{n+1}]$. Since the function $\psi_1$ is continuous on $[x_n, x_{n+1}]$, 
the formula 
\begin{equation}
\label{id1}
\psi_1(x_{n+1}) - \psi_1(x_n) = \int_{x_n}^{x_{n+1}}\psi_1'(t)dt
\end{equation}
holds. Set 
\[
p_* := \frac{\Delta x_*}{\Delta x_* + 2}.
\] 
Next we show that there exists a point 
$x_n^*\in[x_n,x_{n+1}]$ such that $|\psi_1(x_n^*)| \ge p_*C_*$. 
Let us suppose that such a point does not exist, i.e. $|\psi_1(x)| < p_*C_*$ for all $x\in[x_n,x_{n+1}]$. 
We get from \eqref{Stolz estimate} that 
$|\psi_1'(x)| > (1 - p_*)C_*$ for every $x\in [x_n,x_{n+1}]$. 
In particular $\psi_1'$ is sign-definite in $[x_n, x_{n+1}]$,
so using \eqref{id1} and $\Delta x_n \ge \Delta x_*$ we get a contradiction
\begin{equation*}
\begin{split}
&2p_*C_* > 
|\psi_1(x_{n+1})| + |\psi_1(x_n)| \ge 
|\psi_1(x_{n+1}) - \psi_1(x_n)| = \\[0.5ex]
&\qquad\qquad\qquad =\int_{x_n}^{x_{n+1}}|\psi_1'(t)|dt > \Delta x_n (1 - p_*)C_* \ge \Delta x_* (1 - p_*)C_*=2p_*C_*.
\end{split}
\end{equation*}
Thus the point $x_n^*$ with required properties exists.

Since $|\psi_1'(x)|\le C$, for every $x\in[x_n,x_{n+1}]$ such that 
$| x - x_n^*| \le \frac{p_*C_*}{2C}$ one has $|\psi_1(x)| \ge \frac{p_*C_*}{2}$.
We have shown that every interval $[x_n,x_{n+1}]$ contains a subinterval of length 
$l:=\min\big(\Delta x_*,\tfrac{p_*C_*}{2C}\big)$, where $|\psi_1(x)| \ge \frac{p_*C_*}{2}$. Therefore
\[
\int_0^{x_n} |\psi_1(t)|^2dt \ge \frac{lp_*^2C_*^2}{4}n.
\]
On the other hand,
\[
\int_0^{x_n} |\psi_1(t)|^2dt \le \Delta x^*C^2n.
\]
Summing up, for every solution $\psi$ the integral $\int_0^{x_n}|\psi(t)|^2dt$ has two-sided linear estimate.
Thus no subordinate solution exists.
\end{proof}

%\begin{lem}
%Let $X = \{x_n\}_{n\in\dN}$ and $\alpha = \{\alpha_n\}$ be as in \eqref{} and \eqref{}, respectively.
%Assume $\Delta x_* := \inf_{n\in\dN} | x_{n+1} - x_n | > 0$.
%Let the spectral equations $\tau_{X,\alpha}'\psi = \lambda\psi$ be as in \eqref{spectral equation for delta}. 
%If for some $\lambda\in\dR$ every solution of the equation $\tau_{X,\alpha}'\psi = \lambda\psi$
%is bounded, then for such $\lambda$ there exists no subordinate solution.
%\end{lem}
%\begin{proof}
%The proof is almost the same. One has to check that for any two solutions $\psi$ and $\psi_2$ of 
%$\tau_{X,\alpha}'\psi = \lambda\psi$ the Wronskian $W_{12}'(x)$ is independent on $x$. This is clear
%on the intervals $(x_n, x_{n+1})$ for $n\in\dN$. Let us check that the Wronskian is stable when passing
%through interior $\delta'$-discontinuity.
%\begin{equation}
%\label{wronskian2}
%\begin{split}
%W_{12}'(x_n+) &= \psi_1(x_n+)\psi_2'(x_n+) - \psi_1'(x_n+)\psi_2(x_n+) \\
%&=(\psi_1(x_n-)+\alpha_n\psi'_1(x_n)\big)\psi_2'(x_n-) - \psi_1'(x_n-)\big(\psi_2(x_n-) + \alpha_n\psi_2'(x_n-)\big) \\
%& =\psi_1(x_n-)\psi_2'(x_n-) - \psi_1'(x_n-)\psi_2(x_n-) = W_{12}'(x_n-).
%\end{split}
%\end{equation}
%\end{proof}



%Analogously, to the classical definition in \cite{KP92} 
%we say that a solution $\xi^{(1)} =  \{\xi_k^{(1)}\}_{k\in\dN}$ of the difference equation 
%\eqref{equation for xi} is subordinate if and only if for any other solution $\xi^{(2)} = \{\xi_k^{(2)}\}_{k\in\dN}$ 
%of the same equation not proportional to $\xi^{(1)} $ 
%the following limit property holds:
%\begin{equation*}
%    \lim_{n\rightarrow+\infty}
%\frac{\sum_{k=0}^n \big|\xi^{(1)}_k\big|^2}{\sum_{k=0}^n\big|\xi_k^{(2)}\big|^2} = 0.
%\end{equation*}
%Similar definition is applicable to the equation \eqref{equation for eta}.
%
%\begin{prop}
%\label{prop:equivalence}
%Let \eqref{condition on X} holds.
%The spectral equations \eqref{spectral equation for delta} and \eqref{spectral equation for delta'}
% with $\lambda = k^2$ have no subordinate solutions if and only if 
%the difference equations \eqref{equation for xi} and \eqref{equation for eta} have no subordinate solutions, respectively.
%\end{prop}
%\begin{proof}
%\marginpar{Do we need a proof of this proposition?}
%\end{proof}
%
%
%Non-existence of the subordinate solution does not follow immediately from the fact of the existence 
%of a basis of complex conjugate solutions or solutions with complex conjugate asymptotics.
%
%\marginpar{You can explain me this lemma during my visit.}
%    \begin{lem}\label{lem non-subordinacy}
%    Let
%    \begin{equation*}
%        \begin{split}
%        u_n^+=g_n(1+o(1)),\quad\text{as}~n\rightarrow\infty,\\[0.5ex]
%        u_n^-=\overline{g_n}(1+o(1)),\quad \text{as}~n\rightarrow\infty.
%        \end{split}
%    \end{equation*}
%If the following
%\[\sum\limits_{n=1}^{\infty}|g_n|^2=\infty\quad\text{and}\quad\sum\limits_{k=1}^ng_k^2=o\left(\sum\limits_{k=1}^n|g_k|^2\right),
%\]
%holds, then every non-trivial linear combination of $u_+$ and $u_-$ is non-subordinate.
%\end{lem}
%
%
%\begin{proof}
%Firstly, if $g_n\sim h_n$ as $n\rightarrow\infty$ and $\sum\limits_{n=1}^{\infty}|g_n|=\infty$ or $\sum\limits_{n=1}^{\infty}|h_n|=\infty$, then $\sum\limits_{k=1}^ng_k\sim\sum\limits_{k=1}^nh_k$ as $n\rightarrow\infty$ (we use the standard notation for the equivalence: $g_n\sim h_n$, iff $\frac{g_n}{h_n}\rightarrow1$ as $n\rightarrow\infty$). Hence it is enough to see that
%\begin{multline*}
%    \frac{\sum\limits_{k=1}^n|c_+u_k^++c_-u_k^-|^2}{\sum\limits_{k=1}^n|h_k|^2}
%    \sim
%    \frac{(|c_+|^2+|c_-|^2)\sum\limits_{k=1}^n|h_k|^2+2\Re
%    \left[c_+\overline{c_-}\sum\limits_{k=1}^nh_k^2\right]}
%    {\sum\limits_{k=1}^n|h_k|^2}
%    \\
%    =|c_+|^2+|c_-|^2+o(1)\text{ as }n\rightarrow\infty.
%\end{multline*}
%\end{proof}




\subsection{Benzaid-Lutz theorems for discrete linear systems in $\dC^2$}
\label{sec:BL}

The results of \cite{BL87} translate classical theorems due to  
N.~Levinson \cite{L48} and W.~Harris and D. Lutz  \cite{HL75} on the asymptotic 
integration of ordinary differential linear systems to the case of discrete linear systems. 
The major advance of these methods that they allow to reduce under certain assumptions
the asymptotic integration of general discrete linear systems to almost trivial asymptotic integration 
of diagonal discrete linear systems. For our applications it is sufficient to formulate Benzaid-Lutz theorems only 
for discrete linear systems in $\dC^2$. 
The first lemma of this subsection is a direct consequence of ~\cite[Theorem 3.3]{BL87}.
 \begin{lem}\label{lemma Benzaid-Lutz different modules}
Let $\mu_{\pm}\in\dC\setminus\{0\}$ be such that $|\mu_+|\neq|\mu_-|$ 
and let $\{V_n\}_{n\in\dN}\in \ell^2(\dN,\dC^{2\times 2})$. 
If the coefficient matrix of the discrete linear system
    \begin{equation*}
        \overrightarrow u_{n+1}=
        \left[
        \left(
          \begin{array}{cc}
            \mu_+ & 0 \\
            0 & \mu_- \\
          \end{array}
        \right)
        +V_n
        \right]
        \overrightarrow u_n
    \end{equation*}
is non-degenerate for every $n\in\dN$, 
then this system has a basis of solutions 
$\overrightarrow u_n^{\pm}$ with the following asymptotics:
        \begin{equation*}
          \begin{split}
            &\overrightarrow u_n^+=
            \left[
            \begin{pmatrix}    1 \\
                0 \\
              \end{pmatrix}
                      +o(1)
            \right]
            \prod\limits_{k=1}^n(\mu_++(V_k)_{11})\quad \text{as}~n\rightarrow\infty,
            \\[0.5ex]
            &\overrightarrow u_n^-=
            \left[
\begin{pmatrix}
               0 \\
                1 \\
              \end{pmatrix}
            +o(1)
            \right]
            \prod\limits_{k=1}^n(\mu_-+(V_k)_{22})\quad\text{as}~n\rightarrow \infty,
\end{split}
        \end{equation*}
 where by $(V_k)_{11}$ and $(V_k)_{22}$ we denote the diagonal entries of the matrices $V_k$, and the factors $(\mu_+ + (V_k)_{11})$ and $(\mu_-+(V_k)_{22})$ should be replaced by 1 for those values of the index $k$ for which they vanish. 
   \end{lem}
The following lemma is a simplification of \cite[Theorem 3.2]{BL87}.
  \begin{lem}\label{lemma Benzaid-Lutz same modules}
   Let  $\{t_n\}_{n\in\mathbb N}\in \ell^2(\dN)$ and $\{V_n\}_{n\in\mathbb N}\in \ell^2(\dN,\dC^{2\times 2})$ satisfy the following conditions.
\begin{itemize}
 \item [\rm (i)] \emph{Levinson condition:}  there exists $M_-$ such that for every 
        $n_1\ge n_2$
    \begin{equation*}
        \prod_{n=n_1}^{n_2}|1-t_n|<M_-
    \end{equation*}
    and either there exists $M_+$ such that for every $n_1\ge n_2$
    \begin{equation*}
        \prod_{n=n_1}^{n_2}|1+t_n|>M_+
    \end{equation*}
    or
    \begin{equation*}
        \prod_{k=1}^n|1+t_k|\rightarrow\infty\quad\text{as}~n\rightarrow\infty.
    \end{equation*}
\item[\rm (ii)]  The sum $\sum\limits_{n=1}^{\infty}V_n$ is conditionally convergent and
    \begin{equation*}
        \left\{\sum_{k=n}^{\infty}V_k\right\}_{n\in\dN}\in\ell^2(\dN,\dC^{2\times 2}).
   \end{equation*}
\end{itemize} 
If for every $n\in\dN$
\[
 \det
        \left[
        \begin{pmatrix}
            1+t_n & 0 \\
            0 & 1-t_n \\
    \end{pmatrix}
         +V_n
        \right]
        \neq0,
\]  
    then the discrete linear system
    \begin{equation*}
        \overrightarrow u_{n+1}=
         \left[
        \begin{pmatrix}
            1+t_n & 0 \\
            0 & 1-t_n \\
    \end{pmatrix} +V_n
        \right]
        \overrightarrow u_n
    \end{equation*}
    has a basis of solutions $\overrightarrow u_n^{\pm}$ with the following asymptotics:
    \begin{equation*}
\begin{split}
      &\overrightarrow u_n^+=
        \left[
        \begin{pmatrix}
            1 \\
            0 \\
          \end{pmatrix}
        +o(1)
        \right]
        \prod\limits_{k=1}^n(1+t_k)\quad \text{as}~n\rightarrow \infty,
        \\[0.5ex]
        &\overrightarrow u_n^-=
        \left[
        \begin{pmatrix}
              0 \\
            1 \\
       \end{pmatrix}
        +o(1)
        \right]
        \prod\limits_{k=1}^n(1-t_k) \quad\text{as}~n\rightarrow\infty,
  \end{split}  
  \end{equation*}
 where the factor $(1-t_k)$ should be replaced by 1 for those values of the index $k$ for which it vanishes.
    \end{lem}

\subsection{Characterization of spectra for operators with point interactions via Weyl function}
\label{sec:Weyl}
In this section we view the operator $H_{\varkappa,X,\alpha}$ as a self-adjoint extension of a
certain symmetric operator and we parametrize this self-adjoint extension via suitable ordinary 
boundary triple \cite{Ko75,Br76}. 
As a benefit of this parametrization we reduce
the spectral analysis of the self-adjoint operator $H_{\varkappa,X,\alpha}$ 
to the spectral analysis of an operator-valued function in the auxiliary Hilbert space. 
The reader is referred to \cite{DM95, BGP08} for an extensive discussion of ordinary boundary triples and their applications. The Weyl function method was applied to Schr\"odinger operators with $\delta$-interactions by many authors
\cite{Ko89, M95, KM10, GO10, AKM10}.

Let $X = \{x_n\colon n\in\dN\}$ be a discrete set satisfying 
the condition \eqref{condition on X} and let $\varkappa$ be a parameter in the interval $[0,\pi)$.  
Define the minimal symmetric operator via its action and domain
\begin{equation*}
 \begin{split}
&H_{\varkappa,X} \psi = -\psi'',\\
&\dom H_{\varkappa, X} =
    \left\{
    \psi\colon \psi,\psi'' \in L^2(\dR_+), \psi, \psi' \in AC_{\rm loc}(\dR_+\setminus X), 
\begin{smallmatrix} 
\cos(\varkappa)\psi(0) = \sin(\varkappa)\psi'(0)\\
\psi(x_n-) =\psi(x_n+) = 0, ~n\in\dN,\\ 
\psi'(x_n-) = \psi'(x_n+), ~n\in\dN.
\end{smallmatrix}
 \right\}.
\end{split}
\end{equation*}
It can be easily verified that  $H_{\varkappa, X}$ is a closed symmetric densely defined operator with the adjoint (the maximal operator)
%\marginpar{Earlier wrong domain for the adjoint!!}
\begin{equation*}
    \begin{split}
    &H_{\varkappa, X}^* \psi = -\psi'',\\
    &\dom H_{\varkappa, X}^* =
    \left\{
    \psi\colon \psi,\psi'' \in L^2(\dR_+), 
\psi, \psi' \in AC_{\rm loc}(\dR_+\setminus X),~ 
\begin{smallmatrix} 
\cos(\varkappa)\psi(0) = \sin(\varkappa)\psi'(0)\\ 
\psi(x_n+) = \psi(x_n-), ~n\in\dN. \end{smallmatrix} 
\right\}.
    \end{split}
\end{equation*}

\begin{dfn}
\label{def:triple1}
For the discrete set $X$ and the parameter $\varkappa$ as above we define
$\Gamma_{0,\varkappa, X}$ and $\Gamma_{1,\varkappa,X}$ as
\begin{equation*}
    \begin{split}
    &\Gamma_{0,\varkappa, X} \colon \dom H_{\varkappa, X}^* \rightarrow \ell^2(\dN),\quad \Gamma_{0,\varkappa, X}\psi :=  \bigl\{-\psi(x_n)\bigr\}_{n\in\dN},\\
    &\Gamma_{1,\varkappa,X} \colon \dom H_{\varkappa, X}^* \rightarrow \ell^2(\dN),\quad \Gamma_{1,\varkappa,X}\psi :=  \bigl\{\psi'(x_n+)-\psi'(x_n-)\bigr\}_{n\in\dN}.
    \end{split}
\end{equation*}
\end{dfn}
By \cite[Proposition 3.17]{BGP08} the triple $\Pi_{\varkappa, X} := \big\{\ell^2(\dN), \Gamma_{0,\varkappa, X}, \Gamma_{1,\varkappa, X}\big\} $ is an ordinary boundary triple for $H_{\varkappa, X}^*$ in the following sense:
\begin{itemize}\setlength{\itemsep}{1.2ex}
    \item[\rm (i)] $\ran \Big(\begin{smallmatrix}
 \Gamma_0\\ \Gamma_1\end{smallmatrix}\Big) = \ell^2(\dN)\times\ell^2(\dN)$;
    \item[\rm (ii)] for all $f,g\in\dom H_{\varkappa, X}^*$
    \begin{equation*}
    \bigl(H_{\varkappa, X}^*f,g\bigr)_{L^2}-\bigl(f,H_{\varkappa, X}^*g\bigr)_{L^2} = \big(\Gamma_{1,\varkappa,X}f,\Gamma_{0,\varkappa,X}g\big)_{\ell^2} - 
\big(\Gamma_{0, \varkappa, X}f,\Gamma_{1,\varkappa,X}g\big)_{\ell^2}.
   \end{equation*}
\end{itemize}

%For the purpose of studying $\delta'$-interactions we define another minimal operator
%via its action and domain
%\begin{equation}
%    \begin{split}
%    &H_{\varkappa,X}' \psi = -\psi'',\\
%    &\dom H_{\varkappa, X}' =
%    \left\{
%    \psi\colon \psi,\psi'' \in L^2(\dR_+), \psi, \psi' \in AC_{\rm loc}(\dR_+\setminus X), 
%\begin{smallmatrix} 
%\cos(\varkappa)\psi(0) = \sin(\varkappa)\psi'(0),\\ 
%\psi(x_n-) =\psi(x_n+), ~n\in\dN,\\
%\psi'(x_n-) = \psi'(x_n+)=0, ~n\in\dN.
%\end{smallmatrix},
%\right\}.
% \end{split}
%\end{equation}
%It can be easily verified that  $H_{\varkappa, X}'$ is also a closed symmetric densely defined operator with the adjoint (the maximal operator)
%\begin{equation}
%    \begin{split}
%    &(H_{\varkappa, X}')^* \psi = -\psi'',\\
%    &\dom \big((H_{\varkappa, X}')^*\big) =
%    \left\{
%    \psi\colon \psi,\psi'' \in L^2(\dR_+), \psi, \psi' \in AC_{\rm loc}(\dR_+\setminus X), \begin{smallmatrix}\psi'(x_n+) = \psi'(x_n-),~n\in\dN\end{smallmatrix} \right\}.
%    \end{split}
%\end{equation}
%\begin{dfn}[$\delta'$-triple]
%\label{def:triple2}
%Let the discrete set $X$ and the parameter $\varkappa$ be as above.
%The triple $\Pi_{\varkappa, X}'= \{\ell^2(\dN),\Gamma_{0,\varkappa, X}',\Gamma_{1,\varkappa,X}'\}$ with the boundary mappings
%\begin{equation*}
%    \begin{split}
%    &\Gamma_{0,\varkappa, X}'\colon \dom( (H_{\varkappa, X}')^*) \rightarrow \ell^2(\dN),\quad \Gamma_{0,\varkappa, X}'\psi =  \bigl\{\psi'(x_n)\bigr\}_{n\in\dN},\\[0.5ex]
%    &\Gamma_{1,\varkappa,X}' \colon \dom( (H_{\varkappa, X}')^*) \rightarrow \ell^2(\dN),\quad \Gamma_{1,\varkappa,X}'\psi =  \bigl\{\psi(x_n+)-\psi(x_n-)\bigr\}_{n\in\dN}.
%    \end{split}
%\end{equation*}
%is further called the $\delta'$-triple.
%\end{dfn}
%It can be shown in a way similar to Proposition 3.17 in \cite{BGP08} that this triple is an ordinary boundary triple for $(H_{\varkappa, X}')^*$ in the same sense as $\delta$-triple with $H_{\varkappa,X}^*$ replaced by $(H_{\varkappa, X}')^*$.

\begin{rem}
The assumption $\inf\limits_{n\in\dN} |x_{n+1} - x_n| >0$ is essential. 
In a recent work~\cite{KM10} an ordinary boundary triple was constructed for a class of sets $X$  not satisfying this condition.
\end{rem}

For further purposes we define the set
$\sigma_{\varkappa, X}: = \sigma(H_{\varkappa, X}^*\upharpoonright\ker\Gamma_{0,\varkappa, X})$.
We associate the Weyl function with the triple $\Pi_{\varkappa, X}$
\begin{equation}
\label{weyl}
M_{\varkappa, X}(\lambda) := 
\Gamma_{1,\varkappa, X}\Big(\Gamma_{0,\varkappa, X}\upharpoonright\ker(H_{\varkappa, X}^* - \lambda I)\Big)^{-1}
%M_{\varkappa, X}'(\lambda) := \Gamma_{1,\varkappa, X}'\Big(\Gamma_{0,\varkappa, X}'\upharpoonright\ker((H_{\varkappa, X}')^* - \lambda I)\Big)^{-1} 
\end{equation}
which is defined for all $\lambda\in\dC\setminus\sigma_{\varkappa, X}$.

Recall that $s_n(k)=\sin(k(x_{n+1} -x_n))$ and $c_n(k)=\cos(k(x_{n+1}-x_n))$. Given a vector $\xi\in\ell^2(\mathbb N)$ we can find a solution $\psi\in\ker(H_{\varkappa, X}^* - \lambda I)$ such that $\psi(x_n)=-\xi_n$. The operator $M_{\varkappa, X}(\lambda)$ maps $\xi$ to the vector $\{\psi'(x_n+0)-\psi'(x_n-0)\}_{n\in\mathbb N}\in\ell^2(\mathbb N)$. A straightforward calculation 
based on the formula \eqref{psi from xi} gives us %that for $\lambda\in\mathbb R\setminus\sigma_{\varkappa , X}$ 
the values of all entries of the matrix $M_{\varkappa,X}(\lambda)$ except the first row. To find the values of the first row entries one needs to know $\psi(x)$ on the interval $[0,x_1]$, i.e., to solve the non-homogenious Sturm-Liouville problem $-\psi''(x)=\lambda\psi(x),\,\psi(0)\cos\varkappa-\psi'(0)\sin\varkappa=0,\psi(x_1)=-\xi_1$. The result of calculations is the following:
\begin{equation*}
\label{Weyl1}
    M_{\varkappa,X}(k^2)=
    \begin{pmatrix}
    b_1(k)    &   a_1(k)  &   0    & \dots\\
    a_1(k)    &   b_2(k)  &   a_2(k)  & \dots\\
    0      &   a_2(k)  &   b_3(k)  & \dots\\
    \vdots & \vdots & \vdots & \ddots\\
    \end{pmatrix},
\end{equation*}
with
\begin{equation*}
\begin{split}
   & b_1(k):=k\frac{c_0(k)\cos\varkappa-ks_0(k)\sin\varkappa}
    {s_0(k)\cos\varkappa+kc_0(k)\sin\varkappa}
    +k\frac{c_1(k)}{s_1(k)},\\[0.5ex]
    &b_n(k):=k\Bigg(\frac{c_n(k)}{s_n(k)} + \frac{c_{n-1}(k)}{s_{n-1}(k)}\Bigg),\quad n\ge 2,\\[0.5ex] 
    &a_n(k):=-\frac {k}{s_n(k)}.
\end{split}
\end{equation*}

\begin{rem}\label{remark sigma varkappa X}
Note that the set $\sigma_{\varkappa, X}$ consists of squares of all zeroes of denominators in the formulas above:
\begin{equation}\label{sigma-varkappa-X}
\sigma_{\varkappa, X}=\{k^2\in\dR: s_n(k)=0\text{ for some } n\in \mathbb N\text{ or }s_0(k)\cos\varkappa+kc_0(k)\sin\varkappa=0\}.
\end{equation}
\end{rem}

%The matrix $M'_{\varkappa, X}(\lambda)$ can be obtained in a similar way using the Definition~\ref{def:triple2}. One has: the operator $M_{\varkappa, X}(\lambda)$ maps the vector $\eta\in\ell^2(\dN)$ to the vector $\{\psi(x_n+0)-\psi(x_n-0)\}_{n\in\mathbb N}\in\ell^2(\mathbb N)$, where $\psi(x)$ is given by \eqref{psi from eta} on $(x_1,\infty)$ and is a solution non-homogenious Sturm-Liouville problem on $[0,x_1)$: $-\psi''(x)=\lambda\psi(x),\,\psi(0)\cos\varkappa-\psi'(0)\sin\varkappa=0,\psi'(x_1)=\eta_1$. Calculations yield:
%\begin{equation}
%\label{Weyl2}
%    M_{\varkappa,X}'(k^2)=
%    \begin{pmatrix}
%    b_1'(k)    &   a_1'(k)  &   0    & \dots\\
%    a_1'(k)    &   b_2'(k)  &   a_2'(k)  & \dots\\
%    0      &   a_2'(k)  &   b_3'(k)  & \dots\\
%    \vdots & \vdots & \vdots & \ddots\\
%    \end{pmatrix},
%\end{equation}
%with
%\marginpar{Did not check!}
%\begin{equation*}
%\begin{split}
%   & b_1(k):=-\frac{1}{k}\frac{s_0(k)\cos\varkappa+kc_0(k)\sin\varkappa}
%    {c_0(k)\cos\varkappa-ks_0(k)\sin\varkappa}
%    +\frac1k\frac{c_1(k)}{s_1(k)},\\[0.5ex]
%    &b_n(k):=\frac{1}{k}\Bigg(\frac{c_n(k)}{s_n(k)} + \frac{c_{n-1}(k)}{s_{n-1}(k)}\Bigg),\quad n\ge 2,\\[0.5ex] 
%    &a_n(k) :=-\frac {k}{s_n(k)}.
%\end{split}
%\end{equation*}
%
%\begin{rem}\label{remark sigma varkappa X prime}
%The set $\sigma_{\varkappa, X}'$ consists of squares of all zeroes of denominators in the formulas above:
%\begin{equation}\label{sigma-varkappa-X-prime}
%\sigma_{\varkappa, X}'=\{k^2: s_n(k)=0\text{ for some } n\in \mathbb N\text{ or }c_0(k)\cos\varkappa-ks_0(k)\sin\varkappa=0\}.
%\end{equation}
%\end{rem}

Let $ \alpha = \{\alpha_n\}_{n\in\dN}$ be a bounded sequence of real numbers.
Define an operator in $\ell^2(\dN)$
\begin{equation}
\label{Balpha}
B_\alpha \xi = \{-\alpha_n\xi_n\}_{n\in\dN}, \quad \xi \in\ell^2(\dN),
\end{equation}
which is bounded and self-adjoint.
The operator $H_{\varkappa, X,\alpha}$ is a restriction of the operator $H_{\varkappa, X}^*$  
corresponding  to $B_{\alpha}$ in the following sense
\[
H_{\varkappa, X,\alpha} = H_{\varkappa,X}^* \upharpoonright \ker(\Gamma_{1,\varkappa, X} -B_{\alpha}\Gamma_{0,\varkappa, X}). 
\]
%Analogously, the operator $H_{\varkappa, X,\alpha}'$ is a self-adjoint extension of the symmetric operator $H_{\varkappa, X}'$ corresponding in view of the ordinary boundary triple $\Pi_{\varkappa, X}'$ to the self-adjoint operator $B_\alpha$ in the same sense 
%\[
%H'_{\varkappa, X,\alpha} = (H_{\varkappa,X}')^* \upharpoonright \ker(\Gamma'_{1,\varkappa, X} - B_\alpha\Gamma'_{0,\varkappa, X}). 
%\]

\begin{prop}\cite[Theorem 3.3]{BGP08}
\label{prop:spec}
Let the discrete set $X$, the sequence of strengths $\alpha$ and the self-adjoint operator $H_{\varkappa, X,\alpha}$ be defined as in Section~\ref{sec:def}. 
Let $\sigma_{\varkappa, X}$ be the set defined in \eqref{sigma-varkappa-X} and $B_{\alpha}$ be given by \eqref{Balpha}.
Let $M_{\varkappa, X}$ be the Weyl function \eqref{weyl} associated with the ordinary boundary triple $\Pi_{\varkappa,X}$.
For $\lambda\in\dR\setminus\sigma_{\varkappa, X}$ the following holds:
\[
\lambda\in \sigma_{\rm ess}(H_{\varkappa,X,\alpha})\quad\text{if and only if}\quad 0\in\sigma_{\rm ess}\big(B_{\alpha} - M_{\varkappa, X}(\lambda)\big).
\]
\end{prop}


\section{Spectral and asymptotic analysis}

\subsection{Asymptotic analysis of a special class of discrete linear systems}
\label{sec:class}
In this section we study a special class of discrete linear systems 
that encapsulates system \eqref{system for delta} corresponding to $X = \{x_n\colon n\in\dN\}$ and $\alpha = \{\alpha_n\}_{n\in\dN}$ as in Models I and II 
described in the introduction.  

Let the parameters $l$, $a$, $b$, $\gamma$ and $\omega$ satisfy conditions
\begin{equation}
\label{parameters}
l \in \dR_+,\quad a,b\in\dC, \quad\gamma\in\left(\frac12,1\right]\quad
\text{and}\quad\omega\in\left(0,\frac{\pi}2\right).
\end{equation}

For further purposes we define
\begin{equation}
\label{mu}
\mu_\pm  :=  l \pm \sqrt{l^2-1},
\end{equation}
\begin{equation}
\label{betaphi}
z:=\frac{a e^{-i\omega} + be^{-2i\omega}}{2i\sin\omega},\qquad
\beta :=|z|,\qquad
\varphi :=\arg z,
\end{equation}
and
\begin{equation}
\label{fn}
f^{\pm}_n(\beta) :=
        \left\{
        \begin{array}{ll}
        \exp\left(\pm\frac{\beta n^{1-\gamma}}{1-\gamma}\right),&\text{ if }\gamma<1,
        \\
        n^{\pm\beta},&\text{ if }\gamma=1.
        \end{array}
        \right.
\end{equation}
The following has technical nature and helps to simplify the analysis of cases (Model I and Model II).
%----------Main Lemma------------------------------------------------------------------------
\begin{lem}\label{lemma general}
Let the parameters $l$, $a,b$, $\gamma$ and $\omega$ be as in \eqref{parameters}. 
Let $\mu_\pm$, $\beta, \varphi$ and $f_n^\pm(\cdot)$ be as in \eqref{mu}, \eqref{betaphi} 
and \eqref{fn}, respectively. Let the sequence of matrices $\{R_n\}_{n\in\dN}\in\ell^1(\dN,\dC^{2\times 2}\}$.
If the coefficient matrix of the discrete linear system
 \begin{equation}\label{system on u}
        \overrightarrow u_{n+1}=
        \left[
        \begin{pmatrix}
             0 & 1 \\
            -1 & 2l \\
          \end{pmatrix}
        +
       \begin{pmatrix} 
           0 & 0 \\
            a & b \\
        \end{pmatrix}
        \frac{e^{2i\omega n}}{n^{\gamma}}
        +
\begin{pmatrix}
           0 & 0 \\
            \ov a & \ov b \\
\end{pmatrix}
        \frac{e^{-2i\omega n}}{n^{\gamma}}
        +R_n
        \right]
        \overrightarrow u_n
    \end{equation}
is non-degenerate for every $n\in\dN$, 
then this system has a basis of solutions $\overrightarrow u_n^{\pm}$ with the following asymptotics.
\begin{itemize}
 \item [\rm (i)] If $l\in \dR_+\setminus \{\cos\omega, 1\}$, then
    \begin{equation*} 
    \overrightarrow u_n^{\pm}=
        \left[
        \left(
          \begin{array}{c}
            1 \\
            \mu_{\pm} \\
          \end{array}
        \right)
        +o(1)
        \right]
        \mu_{\pm}^n\quad\text{as}~ n\rightarrow\infty.
    \end{equation*}
  \item[\rm (ii)]  If $l=\cos\omega$, then
    \begin{equation*}
        \overrightarrow u_n^+=
        \left[
        \left(
          \begin{array}{c}
            \cos(\omega n+\varphi/2) \\
            \cos(\omega(n+1)+\varphi/2) \\
          \end{array}
        \right)
        +o(1)
        \right]
        f^+_n(\beta) \quad \text {as}~n\rightarrow \infty,
    \end{equation*}
    and
    \begin{equation*}
        \overrightarrow u_n^-=
        \left[
        \left(
          \begin{array}{c}
            \sin(\omega n+\varphi/2) \\
            \sin(\omega(n+1)+\varphi/2) \\
          \end{array}
        \right)
        +o(1)
        \right]
        f^-_n(\beta)\quad\text{as}~n\rightarrow \infty.
    \end{equation*}
\end{itemize}
 \end{lem}
Before providing the proof we make several remarks.

\begin{rem}
\label{rem1}
We do not consider the (double-root) case $l  = 1$ . 
The analysis in this special case is technically involved
and we refer the reader to \cite{J06, JNSh07}.
\end{rem}

\begin{rem}
\label{rem2}
The case of negative values of the parameter $l$ 
can be treated by the same lemma if one makes the substitution
\begin{equation*}
    \overrightarrow u_n\mapsto
    (-1)^n
    \left(
      \begin{array}{cc}
        1 & 0 \\
        0 & -1 \\
      \end{array}
    \right)
    \overrightarrow u_n.
\end{equation*}
This leads to the same answer in the case $l\in\dR_-\setminus\{-1,-\cos\omega\}$
and to the following answer in the case $l=-\cos\omega$:
\begin{equation*}
    \overrightarrow u_n^+=
    (-1)^n
    \left[
    \begin{pmatrix}
    \cos(\omega n+\varphi_1/2) \\
        -\cos(\omega(n+1)+\varphi_1/2) \\
      \end{pmatrix}
    +o(1)
    \right]
    f^+_n\big(\beta_1\big) \quad   \text{as} ~n\rightarrow\infty,
\end{equation*}
and
\begin{equation*}
    \overrightarrow u_n^-=
    (-1)^n
    \left[
    \begin{pmatrix}
        \sin(\omega n+\varphi_1/2) \\
        -\sin(\omega(n+1)+\varphi_1/2) \\
      \end{pmatrix}
    +o(1)
    \right]
    f^-_n\big(\beta_1\big) \quad   \text{as} ~n\rightarrow\infty,
\end{equation*}
where 
\begin{equation}
\label{betaphi2}
z_1:=\frac{a e^{-i\omega} - be^{-2i\omega}}{2i\sin\omega},\qquad
\beta_1:= |z_1|,\qquad
\varphi _1:= \arg z_1.
\end{equation}
The (double-root) case $l =-1$ is out of our considerations.
\end{rem}

\begin{proof}[Proof of Lemma~\ref{lemma general}]
Since $l \in \dR_+\setminus  \{1\}$,  the constant term in the coefficient matrix in~\eqref{system on u} 
can be diagonalized as follows 
\begin{equation*}
\begin{pmatrix}    
   1 & 1 \\
  \mu_+ & \mu_- \\
    \end{pmatrix}^{-1}
\begin{pmatrix}    
    0 & 1 \\
      -1 & 2l \\
     \end{pmatrix}
     \begin{pmatrix}    
    1 & 1 \\
        \mu_+ & \mu_- \\
      \end{pmatrix}
    =
   \begin{pmatrix}
        \mu_+ & 0 \\
        0 & \mu_- \\
\end{pmatrix}.
\end{equation*}
In view of the identity
\[
\begin{pmatrix}    
    1 & 1 \\
        \mu_+ & \mu_- \\
   \end{pmatrix}^{-1} 
=
\frac1{\mu_--\mu_+}
\begin{pmatrix}
         \mu_- & -1 \\
        -\mu_+ & 1 \\
      \end{pmatrix},
\]
the substitution
\begin{equation}
\label{substitution1}
    \overrightarrow u_n=
    \left(
      \begin{array}{cc}
        1 & 1 \\
        \mu_+ & \mu_- \\
      \end{array}
    \right)
    \overrightarrow{v}_n
\end{equation}
transforms the system \eqref{system on u} on $\overrightarrow u_n$ 
into the system on $\overrightarrow v_n$ given below
\begin{equation}
\label{system on v}
\begin{split}
 \overrightarrow{v}_{n+1}=&
 \Bigg[
 \begin{pmatrix}    
 \mu_+ & 0 \\
   0        & \mu_- \\
  \end{pmatrix}
 +
 \frac{1}{\mu_- -\mu_+}
\begin{pmatrix}    
-(a+\mu_+b) & -(a+\mu_-b) \\
(a+\mu_+b) & (a+\mu_-b) \\
\end{pmatrix}
\frac{e^{2i\omega n}}{n^{\gamma}} \\[0.5ex]
&\qquad + 
\frac{1}{\mu_- -\mu_+}
\begin{pmatrix}    
-(\ov a+\mu_+\ov b) & -( \ov a+\mu_-  \ov b) \\
(\ov a+\mu_+ \ov b) & (\ov a+\mu_- \ov b) \\
\end{pmatrix}
\frac{e^{-2i\omega n}}{n^{\gamma}}
    +R_n \Bigg]   \overrightarrow{ v}_n.
\end{split}
\end{equation}

(i) The case $l \in \dR_+\setminus\{\cos\omega, 1\}$ splits into two subcases:
 $ l > 1$ and $\l\in [0,1)\setminus \{\cos\omega\}$.

The condition $l>1$ implies $0 < \mu_- < 1 <\mu_+$, thus Lemma~\ref{lemma Benzaid-Lutz different modules} 
is applicable to the system \eqref{system on v}  and it  gives us a basis. 
Reverting the substitution \eqref{substitution1} we get the statement. 

The condition $l\in[0,1)\setminus\{\cos\omega\}$ implies  $|\mu_+| = |\mu_-|=1$. 
We reduce this situation to Lemma \ref{lemma Benzaid-Lutz same modules} 
by the following substitution:
\begin{equation}
\label{substitution2}
\overrightarrow{v}_n=
    \begin{pmatrix}
        \mu_+^n & 0 \\
        0 & \mu_-^n \\
      \end{pmatrix}
    \overrightarrow w_n.
\end{equation}
The system on $\overrightarrow w_n$ has the following form
\begin{equation}
\label{system on w}
\begin{split}
 &\overrightarrow w_{n+1}= 
    \Bigg[
    I+\frac{1}{\mu_- -\mu_+}
      \begin{pmatrix}   
     -\mu_-( a+\mu_+ b) & -\mu_-( a+\mu_- b)\mu_-^{2n} \\
        \mu_+(a+\mu_+b)\mu_+^{2n} & \mu_+(a+\mu_- b) \\
      \end{pmatrix}
    \frac{e^{2i\omega n}}{n^{\gamma}}\\
    &\qquad +\frac{1}{\mu_--\mu_+}
    \begin{pmatrix}
        - \mu_-(\ov a+\mu_+\ov b) & -\mu_-(\ov a+\mu_-\ov b)\mu_-^{2n} \\
        \mu_+(\ov  a+\mu_+\ov b)\mu_+^{2n} & \mu_+(\ov a+\mu_- \ov b) \\
      \end{pmatrix}
    \frac{e^{-2i\omega n}}{n^{\gamma}}
    +R_n
    \Bigg]
    \overrightarrow w_n.
\end{split}
\end{equation}
Now Lemma \ref{lemma Benzaid-Lutz same modules} is applicable and 
reverting substitutions \eqref{substitution1} and \eqref{substitution2} we get the statement.

(ii) We start with the system \eqref{system on w}. In the case $l = \cos\omega$  
Lemma~\ref{lemma Benzaid-Lutz same modules} could not be applied immediately. 
In order to bring the system into the form in which Lemma~\ref{lemma Benzaid-Lutz same modules} 
is applicable we make further transformations. Since $l = \cos\omega$, we, clearly, have  $\mu_{\pm}=e^{\pm i\omega}$ 
and thus, grouping summands in a right way, we can rewrite our system in the form
\begin{equation*}
    \overrightarrow w_{n+1}=
    \left[
    I+\begin{pmatrix}
        0 & z \\
       \ov{z}& 0 \\
      \end{pmatrix}
    \frac1{n^{\gamma}}+V_n
    \right]
    \overrightarrow w_n,
\end{equation*}
with 
\[
z = \frac{a e^{-i\omega} + be^{-2i\omega}}{2i\sin\omega}
\] 
and with 
\begin{equation*}
\begin{split}
V_n :=& \frac{1}{(\mu_- -\mu_+)n^\gamma}\Bigg[
      \begin{pmatrix}   
     -\mu_-( a+\mu_+ b) & 0\\
        \mu_+(a+\mu_+b)\mu_+^{2n} & \mu_+(a+\mu_- b) \\
      \end{pmatrix}
    e^{2i\omega n}\\[0.6ex]
     &\qquad\qquad\qquad +
    \begin{pmatrix}
        - \mu_-(\ov a+\mu_+\ov b) & -\mu_-(\ov a+\mu_-\ov b)\mu_-^{2n} \\
        0 & \mu_+(\ov a+\mu_- \ov b) \\
      \end{pmatrix}
    e^{-2i\omega n}\Bigg]
    +R_n.
\end{split}
\end{equation*}
The sequence $\{V_n\}_{n\in\dN}$ satisfies the condition (ii) in Lemma~\ref{lemma Benzaid-Lutz same modules}. 
It can checked by  a straightforward calculation that 
$|z| = \beta$ and $\arg(z) = \varphi$ 
with $\beta$ and $\varphi$ defined in \eqref{betaphi}.

Since
\begin{equation*}
      \begin{pmatrix}
        e^{i\varphi} & -e^{i\varphi} \\
        1 & 1 \\
        \end{pmatrix}^{-1}
        \begin{pmatrix}
          0 & \beta e^{i\varphi} \\
          \beta e^{-i\varphi} & 0 \\
        \end{pmatrix}
      \begin{pmatrix}
         e^{i\varphi} & -e^{i\varphi} \\
        1 & 1 \\
      \end{pmatrix}
   = 
\begin{pmatrix}
     \beta & 0 \\        
       0 & -\beta \\
    \end{pmatrix},
   \end{equation*}
the substitution
\begin{equation}
\label{substitution3}
    \overrightarrow w_n=
\begin{pmatrix}    
    e^{i\varphi} & -e^{i\varphi} \\
        1 & 1 \\
      \end{pmatrix}
    \overrightarrow{x}_n
\end{equation}
leads to the system on $\overrightarrow{x}_n$ of the form
\begin{equation}
\label{system on x} 
    \overrightarrow{x}_{n+1}
    =
    \left[
    I+
    \left(
      \begin{array}{cc}
        \beta & 0 \\
        0 & -\beta \\
      \end{array}
    \right)\frac{1}{n^\gamma}
    +\widetilde V_n
    \right]
    \overrightarrow{x}_n,
\end{equation}
with $\{\wt V_n\}_{n\in\dN}$ satisfying the condition (ii) in Lemma \ref{lemma Benzaid-Lutz same modules}.

For sufficiently  large $n_0$
\begin{equation}
\label{product}
    \prod\limits_{k=n_0}^n\left(1\pm\frac \beta{k^{\gamma}}\right)
    \sim \text{const}\cdot f_n^\pm(\beta)\quad \text{as}~n\rightarrow\infty,
\end{equation}
with $f_n^\pm(\cdot)$ defined in \eqref{fn}.
Now we apply Lemma \ref{lemma Benzaid-Lutz same modules} 
to the system \eqref{system on x} and 
using \eqref{product} we get a basis of solutions of that system
of the form
\begin{equation}
 \begin{split}
\overrightarrow x_n^+ =& \Bigg[\begin{pmatrix} 1\\ 0\end{pmatrix} + o(1)\Bigg] f_n^+(\beta)\quad \text{as}~n\rightarrow\infty,\\[0.5ex]
\overrightarrow x_n^- =& \Bigg[\begin{pmatrix} 0\\1\end{pmatrix} + o(1)\Bigg] f_n^-(\beta)\quad \text{as}~n\rightarrow\infty.
 \end{split}
\end{equation}
The asymptotics above imply the statement immediately after reverting substitutions 
\eqref{substitution1}, \eqref{substitution2},  \eqref{substitution3} 
and coming back to the sequence $\overrightarrow u_n$.
\end{proof}

\subsection{Decomposition of the transfer matrices}
\label{sec:decomp}

Recall that $T_n(k)$ denotes the transfer matrix \eqref{T-n-k}. 
In this section we decompose this transfer matrix  subject
to Model I and Model II in the form
\begin{equation}
\label{Tnk}
     T_n(k) =   \begin{pmatrix}
             0 & 1 \\
            -1 & 2l \\
          \end{pmatrix}
        +
       \begin{pmatrix} 
           0 & 0 \\
            a & b \\
        \end{pmatrix}
        \frac{e^{2i\omega n}}{n^{\gamma}}
        +
\begin{pmatrix}
           0 & 0 \\
            \ov a & \ov b \\
\end{pmatrix}
        \frac{e^{-2i\omega n}}{n^{\gamma}}
        +R_n,
\end{equation}
where $R_n\in\ell^1(\dN;\dC^{2\times 2})$. This decomposition
allows to apply later Lemma~\ref{lemma general}.

 
{\em Model I: Amplitude perturbation.} 
The sequence of strengths $\wt\alpha = \{\wt \alpha_n\}_{n\in\dN}$ and the discrete set 
$\wt X = \{\wt x_n\colon n\in\dN\}$ are defined as follows
\begin{equation*}
\label{amplitude}
    \wt\alpha_n:=\alpha_0+\frac{c\sin(2\omega n)}{n^{\gamma}}+q_n,\quad    \wt x_n:=nd \quad\text{and}\quad     \wt x_{n+1}- \wt  x_n\equiv d.
\end{equation*}
Substitution of these expression into \eqref{T-n-k} and tedious calculation show that $T_n(k)$ for $k\notin\pi\dZ/d$
has the representation of the type \eqref{Tnk} with the following values of the parameters:
\begin{equation}
\label{ab1}
\begin{split}
& l = \cos(kd) + \frac{\alpha_0 \sin(kd)}{2k} = L_\delta(k),\\[0.5ex]
& a =0 ,\qquad  b =  \frac{c\sin(kd)}{2ik}. 
\end{split}
\end{equation}

{\em Model II: Positional perturbation.} 
The sequence of strengths $\wh\alpha = \{\wh \alpha_n\}_{n\in\dN}$ 
and the discrete set $\wh X = \{\wh x_n\colon n\in\dN\}$ are defined as follows
\begin{equation*}
\label{position}
\wh \alpha_n\equiv\alpha_0,\quad \wh x_n:= nd+\frac{c\sin(2\omega n)}{n^{\gamma}}+q_n.
\end{equation*}
In this case substitution into \eqref{T-n-k} and calculation give for $k\notin\pi\dZ/d$
the representation of the type \eqref{Tnk} with the following values of the parameters:
\begin{equation}
\label{ab2}
\begin{split}
 l  = & \cos(kd) + \frac{\alpha_0 \sin(kd)}{2k} = L_\delta(k),\\[0.5ex]
 a = & -2ikc\cot(kd) \sin^2(\omega),\\[0.5ex]
 b = & c\cos(kd)\sin(\omega)\Big[4k\cos(\omega)\cot(2kd) - 2k\cot(kd)e^{-i\omega} +\alpha_0e^{i\omega}\Big].
\end{split}
\end{equation}
%
%\rm III. {\em $\delta'$-Kronig-Penney model with a Wigner-von Neumann amplitude disorder.}
%It follows from \eqref{spectral equation for delta'} that $M_n'(k)$ 
%has the representation of the type \eqref{system on u} with 
%\begin{equation}
%\label{ab3}
%\begin{split}
%&l  =  \cos(kd) - \alpha_0 \frac{k \sin(kd)}{2} = L_{\delta'}(k),\\[0.5ex]
%&a = 0,\qquad b  =  -\frac{ck\cos(kd)}{2i}.
%\end{split}
%\end{equation}
%
%\rm IV. {\em $\delta'$-Kronig-Penney model with a Wigner-von Neumann position disorder.}
%Combining \eqref{spectral equation for delta'} and \eqref{position} we get after tedious calculations
%that $M_n'(k)$ has the representation of the type \eqref{system on u} with 
%\begin{equation}
%\label{ab4}
%\begin{split}
%l  = & \cos(kd) - \alpha_0 \frac{k \sin(kd)}{2} = L_{\delta'}(k),\\[0.5ex]
%a = & -2ikc\cot(kd) \sin^2(\omega),\\[0.5ex] 
%b = & c\cos(kd)\sin(\omega)\Big[4k\cos(\omega)\cot(2kd) - 2k\cot(kd)e^{-i\omega} - \alpha_0k^2e^{i\omega}\Big].
%\end{split}
%\end{equation}
%
\begin{rem}
We do not provide here long calculations for the decompositions of the transfer 
matrices. In these calculations we in fact expand the transfer matrix in powers of $\frac1n$ as $n\rightarrow\infty$ up to a summable term.
\end{rem}

\subsection{Asymptotics of generalized eigenvectors for the spectral bands}
\label{sec:main}
In this section we obtain 
using Lemma~\ref{lemma general} asymptotics of  solutions $\xi$ 
of the difference equation \eqref{equation for xi}
in the cases of Model I and Model II. 
Using the statements from Section~\ref{sec:subord}
we come from the asymptotics of these solutions to the conclusions 
about the spectral bands and critical points inside them.
 
In what follows we use the inverse Joukowsky mapping
$\Phi \colon \dC\mapsto\dC$ defined as 
\begin{equation}
\label{Phi}
\Phi(w) := w+\sqrt{w^2-1}, \quad w\in\dC,
\end{equation}
where the branch of the square root is chosen so that $|\Phi(w)|>1$ for $w\in\dC_+$.
In the table below we list certain functions playing a role in the main theorem of this section.
\begin{center}
\begin{table}[H]
\label{table}
\caption{Notation for Theorem~\ref{thm:asymp1}}
\begin{tabular}{|c| c | c |}\hline
\backslashbox{Funct.:}{Perturb.:}
&Amplitude&Positional\\
\hline
&&\\
$\beta(k) := $& $\Big|\frac{c\sin(kd)}{4k\sin\omega}\Big|$ & $\big|\frac{c\alpha_0}{2}\big|$\\
&&\\
\hline
&&\\
$\vartheta_\pm(k) := $ & $\frac{1}{2}\arg\big(\mp \frac{c\sin(kd)}{k}\big)$  & $\frac12\arg(ic\alpha_0)$\\
&&\\
\hline
\end{tabular}
\end{table}
\end{center}

%\begin{center}
%\begin{table}[H]
%\label{table}
%\caption{Notation for Theorems~\ref{thm:asymp1} and~\ref{thm:asymp2}}
%\begin{tabular}{| c | c | c | c | c |}
%    \hline &&&&\\
% Function   & $\delta$-amp. & $\delta$-pos. & $\delta'$-amp. & $\delta'$-pos. \\ 
%&&&&\\
%    \hline
%&&&&\\
%    $\beta(k) :=$ & $\frac{c|\sin(kd)|}{4k\sin\omega}$ & $\frac{c\alpha_0}2$
%    &
%    $\frac{ck|\sin(kd)|}{4\sin\omega}$ & $\frac{c\alpha_0k^2}2$\\
%&&&&\\
%    \hline
%&&&&\\
%    $\theta(k) :=$ & $-\frac{\arg(i\sin(kd))}2$  & $\frac{\pi}4$ & $\frac{\arg(i\sin(kd))}{2}$ & $-\frac{\pi}4$ \\
%&&&&\\    \hline
%\end{tabular}
%
%\end{table}
%\end{center}

%---------Asymptotics in the delta-case-------------------------------------------------------
\begin{thm}
\label{thm:asymp1}
Let the  parameters $d$, $\alpha_0$,  $c$, $\omega$,  $\gamma$ and $\{q_n\}_{n\in\dN}$ 
be as in \eqref{conditions on constants}.
Let the function $L_\delta$ be as in \eqref{Lyapunov function for delta}.
Assume either that $X =\wt X$,  $\alpha = \wt \alpha$ as in Model I or that $X = \wh X$, $\alpha = \wh \alpha$  as 
in Model II. Then for $\lambda\in\dR$ the finite difference equation \eqref{equation for xi} 
with $k = \sqrt{\lambda}$  has a basis of  solutions 
$\xi^{\pm}(k) = \{\xi_n^\pm(k)\}_{n\in\dN}$ with the following asymptotics:
\begin{itemize}\setlength{\itemsep}{1.2ex}

\item[\rm (i)] If $\big|L_{\delta}(k)\big|\in\dR_+\setminus\{\cos\omega,1\}$, then
\begin{equation*}
\xi_n^{\pm}(k)=\big(\Phi(L_{\delta}(k))\big)^{\pm n}\big(1+o(1)\big)\quad\text{as}~n\rightarrow\infty,
\end{equation*}
where $\Phi$ is defined in \eqref{Phi}. 

\item[\rm (ii)] 
If $L_{\delta}(k)=\cos\omega$, then
\begin{equation*}
\begin{split}    
\xi^+_n(k) =\big(\cos(\omega n+\vartheta_+(k))+o(1)\big)f^+_n(\beta(k)) \quad\text{as}~n\rightarrow\infty,\\[0.5ex]
\xi^-_n(k) = \big(\sin(\omega n+\vartheta_+(k))+o(1)\big)f^-_n(\beta(k)) \quad \text{as}~n\rightarrow\infty.
\end{split}
\end{equation*}
If $L_{\delta}(k)= - \cos\omega$, then 
\begin{equation*}
\begin{split}    
\xi^+_n(k) =(-1)^{n}\big(\cos(\omega n+\vartheta_-(k))+o(1)\big)f^+_n(\beta(k))\quad\text{as}~n\rightarrow\infty,\\[0.5ex]
\xi^-_n(k) = (-1)^{n}\big(\sin(\omega n+ \vartheta_-(k))+o(1)\big)f^-_n(\beta(k)) \quad \text{as}~n\rightarrow\infty.
\end{split}
\end{equation*}
Here $f_n^\pm(\cdot)$ are defined in \eqref{fn} and the functions  $\beta(\cdot)$ and $\vartheta_{\pm}(\cdot)$ are
given in the 1st column of the Table 1 in the case of Model I and 
in the 2nd column in the case of Model II, respectively.
\end{itemize}
\end{thm}
\begin{proof}
Let us fix $k = \sqrt{\lambda}$ such that $\big|L_\delta(k)\big| \ne 1$. For such $k$
the limit property $\liminf_{n\rightarrow +\infty}\big|s_n(k)\big| > 0$ holds, and 
the transfer matrix $T_n(k)$ of the discrete linear system \eqref{system for delta} 
decomposes into the form in \eqref{system on u} 
with the parameters $a$, $b$ and $l$ given in~\eqref{ab1} for the Model I 
and given in~\eqref{ab2} for the Model II, respectively.

In the case $\big|L_{\delta}(k)\big| \in\dR_+\setminus \{\cos\omega,1\}$ 
Lemma~\ref{lemma general}\,(i) could be applied and it automatically implies the item (i).
Assume now $L_\delta(k) = \pm \cos\omega$. In the case of Model I we plug into formulas 
\eqref{betaphi} and \eqref{betaphi2} the values $a$ and $b$ from \eqref{ab1} and we
easily get the following 
\[
 z(k)=-z_1(k)=-\frac{c\sin(kd)e^{-2i\omega}}{4k\sin\omega}.
\]
In the case of Model II we plug into formulas \eqref{betaphi} and \eqref{betaphi2} 
the values  $a$ and $b$  from \eqref{ab2} and using that $L_\delta(k) = \pm\cos\omega$ 
after a long and tedious calculation we get
\[
 z(k) = z_1(k) = \frac{ic\alpha_0e^{-2i\omega}}{2}.
\]
Now Lemma~\ref{lemma general}\,(ii) could be applied. 
It gives us the asymptotics for the sequences of $\dC^2$-vectors
\[
\overrightarrow u_n^\pm(k)= 
\begin{pmatrix}
\xi_{n-1}^\pm(k) \\   
\xi_n^\pm(k) 
\end{pmatrix}.
\] 
Extracting the second components from these asymptotics we obtain the statement.
\end{proof}


Given the asymptotics, we can now come to the conclusions about the structure 
of the purely absolutely continuous spectrum using the subordinacy theory, 
see Section~\ref{sec:subord}. 
Let the parameters $d$, $\alpha_0, c$, $\omega$ and  $\gamma$ be as in 
\eqref{conditions on constants}. Recall that the self-adjoint operator 
$H_{\varkappa}$ corresponds to the Kronig-Penney without perturbation, 
where the distance between interaction is the constant $d$
and the strength of interactions is the constant $\alpha_0$. 
The spectral properties of this operator are discussed in the introduction. 
The absolutely continuous spectrum of the operator $H_{\varkappa}$ is the set 
\[
\sigma_{\rm ac}(H_{\varkappa}) :=\Big\{\lambda\in\dR\colon L_{\delta}\left(\sqrt{\lambda}\right)\in[-1,1]\Big\}.
\]
Denote the set of all critical points by 
\[
\frs_{\rm cr} := 
\Big\{\lambda\in\dR\colon L_{\delta}\left(\sqrt{\lambda}\right) = \pm \cos\omega \Big\} 
\subset{\rm Int}\big(\sigma_{\rm ac}(H_{\varkappa})\big)
=\Big\{\lambda\in\dR\colon L_{\delta}\left(\sqrt{\lambda}\right)\in(-1,1)\Big\}.
\]
\begin{cor}
Let the parameters $d$, $\alpha_0, c$, $\omega$,  $\gamma$ and $\{q_n\}$ 
be as in \eqref{conditions on constants} and let
the sequences $\widetilde X, \widetilde\alpha$ and $\wh X, \wh\alpha$ 
be as in \eqref{wt X},\eqref{wt alpha} and in \eqref{wh X},\eqref{wh alpha}, respectively. Then
the following statements hold.
\begin{itemize}
\item [\rm (i)] 
The spectrum of the operators $H_{\varkappa,\wt X,\wt\alpha}$ 
and $H_{\varkappa,\wh X,\wh\alpha}$ 
is purely absolutely continuous on the set 
\[
{\rm Int}\big(\sigma_{\rm ac}(H_{\varkappa})\big)\setminus\frs_{\rm cr}.
\]
\item[\rm (ii)] 
Let $\lambda=k^2\in\frs_{\rm cr}$.
If either $\gamma < 1$ or
\[
\gamma=1\quad\text{and}\quad\Bigg|\frac{c\sin(kd)}{2k\sin\omega}\Bigg| > 1,
\]
then there exists a unique value $\varkappa$ such that $\lambda$  is an embedded eigenvalue 
of $H_{\varkappa, \wt X, \wt\alpha}$.
\item[\rm (iii)] 
Let $\lambda=k^2\in\frs_{\rm cr}$.
If either $\gamma < 1$ or
\[
\gamma=1\quad\text{and}\quad |c\alpha_0|>1,
\]
then there exists a unique value $\varkappa$ such that $\lambda$  is an embedded eigenvalue 
of $H_{\varkappa, \wh X, \wh\alpha}$.
\end{itemize}
\end{cor}
\begin{proof}
For $k^2\in{\rm Int}\big(\sigma_{ac}(H_{\varkappa})\big)$ one has that 
$\liminf_{n\rightarrow\infty}\big|s_n(k) \big |>0$. From the asymptotics of $\xi^\pm_n(k)$ and the formula \eqref{psi from xi} it follows that all the solutions of the spectral equations for both operators $H_{\varkappa,\wt X,\wt\alpha}$ 
and $H_{\varkappa,\wh X,\wh\alpha}$ are bounded. The assertion of the item (i) follows now from Proposition~\ref{prop:stolz}, Proposition~\ref{prop:Stolz2} and boundedness of all solutions. 

Conditions of the items (ii) and (iii) guarantee the existence of the square-integrable solution of the spectral equations at the point $\lambda$ for the operators $H_{\varkappa,\wt X,\wt\alpha}$ and $H_{\varkappa,\wh X,\wh\alpha}$, respectively. This fact means that there is a unique value of the boundary parameter $\varkappa$ such that this solution satisfies the boundary condition, and thus in this case $\lambda$ is an eigenvalue.

\end{proof}

\subsection{Spectral gaps}
\label{sec:gaps}
In this section we show that the essential spectra of perturbed operators 
coincide with the essential spectrum of the non-perturbed Kronig-Penney model.  
Let the sets $\sigma_{\varkappa, \wt X}$ and $\sigma_{\varkappa, \wh X}$ be defined by $\wt X$ and $\wh X$, respectively, by the formula \eqref{sigma-varkappa-X}.
Recall that the Weyl functions $M_{\varkappa, \wt X}$ and $M_{\varkappa, \wh X}$ are defined in \eqref{weyl} and operators $B_{\wt \alpha}$ and $B_{\wh\alpha}$ are defined in \eqref{Balpha}.


\begin{thm}\label{thm:essamp}
Let the parameters $d$, $\alpha_0, c$, $\omega$,  $\gamma$ and $\{q_n\}$ 
be as in \eqref{conditions on constants}. Let the sets $\wt X, \wh X$ and the sequences $\wt \alpha, \wh\alpha$
be as in \eqref{wt X},\eqref{wh X} and \eqref{wt alpha},\eqref{wh alpha}, respectively. Let the self-adjoint operator $H_\varkappa$ 
correspond to the non-perturbed Kronig-Penney model. Let the operators $H_{\varkappa, \wt X,\wt \alpha}$ and $H_{\varkappa, \wh X,\wh \alpha}$ correspond
to $\wt X,\wt \alpha$ and $\wh X,\wh \alpha$ as in Section~\ref{sec:def}, respectively. Then the spectrum of both operators $H_{\varkappa, \wt X,\wt \alpha}$ and $H_{\varkappa, \wh X,\wh \alpha}$ on the set $\dR\setminus\sigma_{\rm ess}(H_{\varkappa})$ is pure point.
\end{thm}
\begin{proof}
(i)
Define $B_{\alpha_0}:=-\alpha_0I$, a diagonal operator in $\ell^2(\dN)$. One has: $\wt\alpha_n\rightarrow\alpha_0$ as $n\rightarrow\infty$. Thus 
\[
B_{\wt\alpha} - B_{\alpha_0} \in \sS_\infty. 
\]
Consider $\lambda\in\dR\setminus\big(\sigma_{\rm ess}(H_{\varkappa})\cup\sigma_{\varkappa,\wt X}\big)$.
By Proposition \ref{prop:spec}, $0\notin\sess\big(B_{\alpha_0} - M_{\varkappa, \wt X}(\lambda)\big)$. Therefore by compact perturbation argument 
$0\notin\sess\big(B_{\wt\alpha} - M_{\varkappa, \wt X}(\lambda)\big)$ and again by Proposition \ref{prop:spec} 
$\lambda\notin\sigma_{\rm ess}\big(H_{\varkappa,\wt X,\wt\alpha}\big)$. Thus the spectrum of the operator $H_{\varkappa, \wt X,\wt \alpha}$ on the set 
$\dR\setminus\big(\sigma_{\rm ess}(H_{\varkappa})\cup\sigma_{\varkappa,\wt X}\big)$ is discrete. Since the set 
$\sigma_{\varkappa,\wt X}\cap\big(\dR\setminus\sigma_{\rm ess}(H_{\varkappa})\big)$ is at most countable, it cannot support absolutely continuous or singular continuous spectral measure.

(ii)
One has $B_{\wh\alpha}=B_{\alpha_0}$. Consider 
$\lambda\in\dR\setminus\big(\sigma_{\rm ess}(H_{\varkappa})\cup\sigma_{\varkappa,\wt X}\cup\sigma_{\varkappa,\wh X}\big)$. Let $\wh b_n(k)$ be diagonal and $\wh a_n(k)$ be anti-diagonal entries of the Jacobi matrix $M_{\varkappa,\wh X}(\lambda)$ (where $\lambda=k^2$). Clearly $\wh a_n(k)\rightarrow-\frac k{\sin(kd)}$ and $\wh b_n(k)\rightarrow2k\cot(kd)$ as $n\rightarrow\infty$. It follows that
\[
M_{\varkappa, \wt X}(k^2) -  M_{\varkappa, \wh X}(k^2)\in\sS_\infty.    
\]
Again, Proposition~\ref{prop:spec} implies that $\lambda\notin\sigma_{\rm ess}\big(H_{\varkappa,\wh X,\wh\alpha}\big)$. Hence the spectrum of $H_{\varkappa, \wh X,\wh \alpha}$ on $\dR\setminus\big(\sigma_{\rm ess}(H_{\varkappa})\cup\sigma_{\varkappa,\wt X}\cup\sigma_{\varkappa,\wh X}\big)$ is discrete. The set
$\big(\sigma_{\varkappa,\wt X}\cup\sigma_{\varkappa,\wh X}\big)\cap\big(\dR\setminus\sigma_{\rm ess}(H_{\varkappa})\big)$ is at most countable, so the spectrum of $H_{\varkappa, \wh X,\wh \alpha}$ on the set $\dR\setminus\sigma_{\rm ess}(H_{\varkappa})$ is pure point.
\end{proof}

\subsection*{Acknowledgments}
The authors wish to express their deep gratitude to Prof. Sergey Naboko 
for his constant attention to this work and many fruitful discussions of the subject 
and to Prof. G\"unter Stolz for important comments and valuable help. V.~L. was supported 
by the grant ''Scientific Staff of Innovative Russia'' (contracts P689 NK-526P, and    14.740.11.0879) 
and grant RFBR 11-08-00267-a. S. S. was supported by the Chebyshev Laboratory
 (Department of Mathematics and Mechanics, Saint-Petersburg State University) under the grant 11.G34.31.0026 of the Government of the Russian Federation, by grants RFBR-09-01-00515-a and 11-01-90402-Ukr\_f\_a and by the Erasmus Mundus Action 2 Programme of the European Union.

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%Elliptic operators on closed manifolds (Russian).
%Current problems in mathematics. Fundamental directions, Vol. 63 (Russian),  5--129,
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D.~Hinton, M.~Klaus and J.~Shaw,
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\textit{Math.\ Proc.\ Camb.\ Philos.\ Soc.} \textbf{142}  (2007), 161--183.

\bibitem[KS11]{KS11}
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1112.4717.extracted_bib
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%\textbf{or???} \\
%M.\,S.~Agranovich, \textit{Elliptic boundary problems}.
%Translated from the Russian by the author. Encyclopaedia Math. Sci., 79,
%Partial differential equations, IX,  1--144, 275--281, Springer, Berlin, 1997.
%
%\bibitem[Ag90]{Ag90} M.\,S.~Agranovich,
%Elliptic operators on closed manifolds (Russian).
%Current problems in mathematics. Fundamental directions, Vol. 63 (Russian),  5--129,
%\textit{Itogi Nauki i Tekhniki}, Akad. Nauk SSSR,
%Vsesoyuz.\ Inst.\ Nauchn.\ i Tekhn.\ Inform., Moscow, 1990.
%Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 63, 5-129 (1990).

\bibitem[AGHH05]{AGHH05}
S.~Albeverio, F.~Gesztesy, R.~Hoegh-Krohn and H.~Holden,
\textit{Solvable models in quantum mechanics. With an appendix by Pavel Exner.},
AMS Chelsea Publishing, 2005.

\bibitem[Al72]{Al72}
S.~Albeverio, 
On bound states in the continuum of $N$-body systems and the virial theorem,
\textit{Ann. Physics} \textbf{71} (1972), 167--276.

\bibitem[AKM10]{AKM10}	
S.~Albeverio, A.~Kostenko and M.~Malamud,
Spectral theory of semibounded Sturm-Liouville operators with local interactions on a discrete set,
\textit{J.\ Math.\ Phys.} \textbf{51}  (2010), 102102.

\bibitem[Be91]{Be91}
H.~Behncke, 
Absolute continuity of hamiltonians with von Neumann-Wigner Potentials I,
\textit{Proc. AMS} \textbf{111} (1991), 373--384.

\bibitem[Be94]{Be94}
H.~Behncke,
The $m$-function for Hamiltonians with Wigner-von Neumann potentials,
\textit{J.\ Math.\ Phys.} \textbf{35} (1994), 1445--1462.

\bibitem[BL87]{BL87}
Z.~Benzaid and D.~Lutz,
Asymptotic representation of solutions of perturbed systems of linear difference equations,
\textit{Stud.\ Appl.\ Math.} \textbf{77}  (1987), 195--221.

%\bibitem[BEKS94]{BEKS94}
%J.\,F.~Brasche, P.~Exner, Yu.\,A.~Kuperin, P.~Seba,
%Schr\"odinger operators with singular interactions,
%\textit{J.\ Math.\ Anal. \ Appl.} \textbf{184} (1994) 112--139.

\bibitem[Br76]{Br76} V.~M.~Bruk,
A certain class of boundary value problems with a spectral parameter in the
boundary condition,
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translated in: \textit{Math.\ USSR-Sb.} \textbf{29} (1976), 186--192.

\bibitem[BGP08]{BGP08}
J.~Br\"uning, V.~Geyler and K.~Pankrashkin,
Spectra of self-adjoint extensions and applications to solvable Schr\"odinger operators,
\textit{Rev.\ Math.\ Phys.} \textbf{20}  (2008), 1--70.

%\bibitem[DN07]{DN07}
%D.~Damanik and S.~Naboko,
%\newblock {Unbounded Jacobi matrices at critical coupling},
%\newblock {\em J. Approx. Theory}, \textbf{145 (2)} (2007), 221--236.

\bibitem[DM95]{DM95} V.\,A.~Derkach and M.\,M.~Malamud,
The extension theory of Hermitian operators and the moment problem,
\textit{J.\ Math.\ Sci.\ (New York)} \textbf{73} (1995), 141--242.

%\bibitem[EE87]{EE87} D.\,E.~Edmund and W.\,D.~Evans,
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%Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press,
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%Undergraduate Texts in Mathematics. New York, NY: Springer, 2005.

\bibitem[E97]{E97}
P.~Exner, A duality between Schr\"odinger operators on graphs and certain Jacobi matrices,
\textit{Ann.\ Inst.\ Henri\ Poincare,\ Phys.\ Theor.} \textbf{66}  (1997),  359--371.

\bibitem[GK85]{GK85}
F.~Gesztesy and W.~Kirsch,
One-dimensional Schr\"odinger operators with interactions singular on a discrete set,
\textit{J.\ Reine\ Angew.\ Math.} \textbf{362} (1985), 28--50.

\bibitem[GO10]{GO10}
N.~Goloschapova and L.~Oridoroga,
On the negative spectrum of one-dimensional Schr\"odinger operators with point interactions,
\textit{Integral\ Equations\ Operator\ Theory} \textbf{67} (2010), 1--14.

\bibitem[GP87]{GP87}
D.~Gilbert and D.~Pearson,
On subordinacy and analysis of the spectrum of one-dimensional Schr\"odinger operators,
\textit{J.\ Math.\ Anal.\ Appl.} \textbf{128} (1987), 30--56.

\bibitem[HIT10]{HIT10}
J.~Hernandez-Herrejon, F.~Izrailev and L.~Tessieri,
Anomalous localization in the aperiodic Kronig-Penney model,
\textit{J.\ Phys.\ A,\ Math.\ Theor.} \textbf{43} (2010), 425004.


%\bibitem[JM03]{JM03}
%J.~Janas and M.~Moszynski,
%Spectral properties of Jacobi matrices by asymptotic analysis,
%\textit{J.\ Approx.\ Theor} \textbf{120}  (2003), 309--336.

\bibitem[HL75]{HL75}
W.~A.~jun. Harris and D.~A.~Lutz,
Asymptotic integration of adiabatic oscillators,
\textit{J.\ Math.\ Anal.\ Appl.} \textbf{51} (1975), 76--93.

%\bibitem[HKS91-1]{HKS91-1}
%D.~B.~Hinton, M.~Klaus, J.~K.~Shaw
%Half-bound states and Levinson's theorem for discrete systems,
%\textit{SIAM \ J.\ Math. \ Anal.} \textbf{22} (3) (1991), 754--768.

\bibitem[HKS91]{HKS91}
D.~Hinton, M.~Klaus and J.~Shaw,
Embedded halfbound states for potentials of Wigner-von Neumann type,
\textit{Proc.\ Lond.\ Math.\ Soc.}, \textbf{6}  (1991), 607--646.

\bibitem[J06]{J06}
J.~Janas,
The asymptotic analysis of generalized eigenvectors of some Jacobi
  operators. Jordan box case,
\textit{ J.\ Difference\ Equ.\ Appl.}, \textbf{12} (2006), 597--618.

\bibitem[JNSh07]{JNSh07}
J.~Janas, S.~Naboko and E.~Sheronova,
Jacobi Matrices arising in the spectral phase transition phenomena:
  asymptotics of generalized eigenvectors in the "double root" case,
\textit{Centre for mathematical sciences, Lund Institute of Technology},
  (2007),  mp\_arc 07-41.

\bibitem[JS10]{JS10}
J.~Janas and S.~Simonov,
Weyl-Titchmarsh type formula for discrete Schroedinger operator with Wigner-von Neumann potential,
\textit{Studia Math.}. \textbf{201} (2010), 167--189. 
%arXiv:1003.3319.

%\bibitem[IKLMS09]{IKLMS09}
%F.\,M.~Izrailev, U.~Kuhl
%G.\,A.~Luna-Acosta, N.\,M.~Makarov,
%One dimensional Kronig-Penney model with positional disorder: Theory versus experiment,
%\textit{Phys.\, Rev. B} \textbf{80},  (2009), 115112, 8pp.

\bibitem[KP92]{KP92}
S.~Khan and D.~Pearson,
Subordinacy and spectral theory for infinite matrices,
\textit{Helv.\ Phys.\ Acta} \textbf{65} (1992), 505--527.

\bibitem[Ko75]{Ko75} A.~Kochubei,
Extensions of symmetric operators and symmetric binary relations,
\textit{Mat.\ Zametki} \textbf{17} (1975), 41--48 (in Russian);
translated in: \textit{Math.\ Notes} \textbf{17} (1975), 25--28.

\bibitem[Ko89]{Ko89}
A.~Kochubei,
One-dimensional point interactions,
\textit{Ukrain.\ Math.\ J.} \textbf{41} (1989), 1391--1395.

\bibitem[KM10]{KM10}
A.~Kostenko and M.~Malamud,
1-D Schr\"odinger operators with local point interactions on a discrete set,
\textit{J.\ Differential Equations} \textbf{249} (2010), 253--304.

\bibitem[KP31]{KP31}
R.~de L.~Kronig and W.\,G.~Penney,
Quantum mechanics of electrons in crystal lattices,
\textit{Proc.\ Roy.\ Soc.\ Lond.} \textbf{130} (A) (1931), 499--513.

\bibitem[KN07]{KN07} P.~Kurasov and S.~Naboko,
Wigner-von Neumann perturbations of a periodic potential: spectral singularities in bands,
\textit{Math.\ Proc.\ Camb.\ Philos.\ Soc.} \textbf{142}  (2007), 161--183.

\bibitem[KS11]{KS11}
P.~Kurasov and S.~Simonov,
Weyl-Titchmarsh type formula for periodic Schr\"odinger operator with Wigner-von Neumann potential,
\textit{submitted}, arXiv:1102.5213.

\bibitem[L48]{L48}
N.~Levinson, The asymptotic nature of solutions of linear systems of differential equations,
\textit{Duke Math.\ J.} \textbf{15} (1948), 111--126.

\bibitem[L10]{L10}
M.~Lukic, Orthogonal polynomials with recursion coefficients of generalized bounded variation,
\textit{Comm.\ Math.\ Phys.} \textbf{306} (2011),  485--509.

\bibitem[Ma73]{Ma73}
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