\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 11, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/11\hfil Inverse spectral problems]
{Inverse spectral problems for energy-dependent Sturm-Liouville
equations with finitely many point $\delta $-interactions}

\author[M. Dzh. Manafov \hfil EJDE-2016/11\hfilneg]
{Manaf Dzh. Manafov}

\dedicatory{In memory of  M. G. Gasymov (1939-2008)}

\address{Manaf Dzh. Manafov \newline
Ad\i{y}aman University, Faculty of Science and Arts,
Department of Mathematics, \newline Ad\i{y}aman, Turkey}
\email{mmanafov@adiyaman.edu.tr}

\thanks{Submitted August 20, 2015. Published January 6, 2016.}
\subjclass[2010]{34A55, 34B24, 34L05}
\keywords{Energy-dependent Sturm-Liouville equation;
\hfill\break\indent inverse spectral problems; point delta-interactions}

\begin{abstract}
 In this study, inverse spectral problems for a energy-dependent
 Sturm-Liouville equations with finitely many point $\delta $-interactions.
 The uniqueness theorems for the inverse problems of reconstruction of the
 boundary value problem from the Weyl function, from the spectral data and
 from two spectra are proved and a constructive procedure for finding its
 solution are obtained.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{algorithm}[theorem]{Algorithm}
\allowdisplaybreaks

\section{Introduction}

We consider inverse problems for the boundary value problem L (BVP L)
generated by the differential equation
\begin{equation}
-y''+q(x)y=\lambda ^2y,\quad
x\in \cup _{s=0}^{m}( a_s,a_{s+1})\quad (0=a_0,\;a_{m+1}=\pi )
\label{1.1}
\end{equation}
with the Robin boundary conditions
\begin{equation}
\begin{gathered}
U(y):=y'(0)-hy(0)=0,\\
V(y):=y'(\pi )+Hy(\pi )=0,
\end{gathered} \label{1.2}
\end{equation}
and the conditions at the points $x=a_s$:
\begin{equation}
I(y):=\{
\begin{gathered}
y(a_s+0)=y(a_s-0)=y(a_s) \\
y'(a_s+0)-y'(a_s-0)=2\alpha _s\lambda y(a_s),
\end{gathered}  \label{1.3}
\end{equation}
$q(x)$ is a complex-valued function in $L_2(0,\pi )$;
$h,H$ and $\alpha _s$ are complex numbers, $s= \overline{1,m}:= 1,2,\dots m$;
and $\lambda $ is a spectral parameter.

Note that, we can understand problem \eqref{1.1} and \eqref{1.3} as
studying the equation
\begin{equation}
y''+(\lambda ^2-2\lambda p(x)-q(x)) y=0,\quad
x\in (0,\pi ) ,  \label{1.5}
\end{equation}
when $p(x)=\sum_{s=1}^{m}\alpha _s\delta (x-a_s) $, where
$\delta (x) $ is the Dirac function (see \cite{2}).

Sturm-Liouville spectral problems with potentials depending on the spectral
parameter arise in various models of quantum and classical mechanics. For
instance, the evolution equations that are used to model interactions
between colliding relativistic spineless particles can be reduced to the
form \eqref{1.1}. Then $\lambda ^2$ is associated with the energy of the
system (see \cite{15,17}).

Spectral problems of differential operators are studied in two main
branches, namely, direct and inverse problems. Direct problems of spectral
analysis consist in investigating the spectral properties of an operator. On
the other hand, inverse problems aim at recovering operators from their
spectral characteristics. One takes for the main spectral data, for
instance, one, two, or more spectra, the spectral function, the spectrum,
and the normalized constants, the Weyl function. Direct and inverse problems
for the classical Sturm-Liouville operators have been extensively studied
(see \cite{8,16,18} and the references therein). Some classes of direct and
inverse problems for discontinuous $BVPs$ (special case $p(x)\equiv 0$) in
various statements have been considered in \cite{3,11,13,21,23} and for
further discussion see the references therein. Notice that, inverse spectral
problems for non-selfadjoint Sturm-Liouville operators on a finite interval
with possibly multiple spectra have been investigated in \cite{B1, B2}.

Non-linear dependence of  \eqref{1.5} on the spectral parameter
$\lambda $ should be regarded as a spectral problem for a quadratic operator
pensil. The problem with $p(x)\in W_2^{1}(0,1)$ and $q(x)\in L_2(0,1)$
and with Robin boundary conditions was discussed in \cite{10}. Such problems
for separated and non-separated boundary conditions were considered
(see \cite{9,12,24} and the references therein). In this aspect, various inverse
spectral problems for the equation \eqref{1.5} have been investigated in
\cite{19,20} for the case $m=1$.

In this article we establish various uniqueness results for inverse spectral
problems of energy-dependent Sturm-Liouville equations with finitely many
point $\delta $-interactions, and obtain a  process for finding its
solution.

\section{Properties of the spectral characteristics of  BVP L}

In this section, we provide the spectral characteristics of the BVP L
and present the relationship among these spectral characteristics. Moreover,
we formulate the inverse problem of the reconstruction of BVP L:
from the Weyl function, from the spectral data, and from two spectra. The
technique employed is similar to those used in \cite{8}.

Let $y(x)$ and $z(x)$ be continuously differentiable functions on the
intervals $(0,a_1) $, $(a_1,a_2) ,\dots,(a_{m-1},a_m) $ and $(a_m,\pi ) $. Denote $\langle y,z\rangle :=yz'-y'z$. If $y(x)$ and $z(x)$ satisfy
the conditions \eqref{1.3}, then
\begin{equation}
\langle y,z\rangle _{x=a_s-0}=\langle y,z\rangle
_{x=a_s+0},\quad s=\overline{1,m},  \label{2.1}
\end{equation}
i.e. the function $\langle y,z\rangle $ is continuous on $(0,\pi ) $.

Let $\varphi (x,\lambda ) $, $\psi (x,\lambda ) $ be
solutions of \eqref{1.1} under the conditions
\begin{equation}
\begin{gathered}
\varphi (0,\lambda ) =\psi (\pi ,\lambda ) =1 \\
\varphi '(0,\lambda ) =h,\quad \psi '(\pi ,\lambda ) =-H,
\end{gathered}  \label{2.2}
\end{equation}
and under  condition \eqref{1.3}.

Denote $\Delta (\lambda ) :=\langle \varphi (x,\lambda ) ,\psi (x,\lambda )
\rangle $. By  \eqref{2.1} and the Ostrogradskii-Liouville theorem 
(see \cite[p.83]{5}),
$\Delta (\lambda ) $ does not depend on $x$. The function 
$\Delta (\lambda ) $ is called the characteristic function of $L$.
Clearly,
\begin{equation}
\Delta (\lambda ) =-V(\varphi ) =U(\psi) .  \label{2.3}
\end{equation}
Obviously, the function $\Delta (\lambda ) $ is entire in 
$\lambda $ and it has at most a countable set of zeros 
$\{ \lambda_n\} $.

We confine ourselves, for simplicity, to the case when all zeros of the
characteristic function $\Delta (\lambda ) $ are simple.

\begin{lemma} \label{lem2.1}
The eigenvalues $\{ \lambda _n^2\} _{n\geq0} $ of the BVP L coincide
with the zeros of the characteristic function.
The functions $\varphi (x,\lambda _n) $ and $\psi (x,\lambda _n) $ 
are eigenfunctions, and there exists a sequence $\{ \beta _n\} $ such that
\begin{equation*}
\psi (x,\lambda _n) =\beta _n\varphi (x,\lambda_n), \quad \beta _n\neq 0.
\end{equation*}
\end{lemma}

\begin{proof}
Let $\Delta (\lambda _0) =0$. Then by $\langle
\varphi (x,\lambda _0) ,\psi (x,\lambda _0)\rangle =0$, we have 
$\psi (x,\lambda _0) =C\varphi(x,\lambda _0) $ for some constant $C$. 
Hence $\lambda _0$ is an eigenvalue and $\varphi (x,\lambda _0) $, 
$\psi (x,\lambda _0) $ are eigenfunctions related to $\lambda _0$.

 Conversely, let $\lambda _0$ be an eigenvalue of $L$. We shall show
that $\Delta (\lambda _0) =0$. Assuming the converse, suppose
that $\Delta (\lambda _0) \neq 0$. In this case the functions $
\varphi (x,\lambda _0) $ and $\psi (x,\lambda_0) $ are linearly independent. 
Then $y(x,\lambda _0)=c_1\varphi (x,\lambda _0) +c_2\psi (x,\lambda _0) $ 
is a general solution of the problem $L$. If $c_1\neq 0$, we can write
\begin{equation*}
\varphi (x,\lambda _0) 
=\frac{1}{c_1}y(x,\lambda _0)-\frac{c_2}{c_1}\psi (x,\lambda _0) .
\end{equation*}
Then we have
\begin{equation*}
\langle \varphi (x,\lambda _0) ,\psi (x,\lambda_0) \rangle 
=-\frac{1}{c_1}[ y'(\pi,\lambda _0) +Hy(\pi ,\lambda _0) ] =0,
\end{equation*}
which is a contradiction.

 Note that for each eigenvalue there exists only one eigenfunction.
Therefore there exists sequence $\beta _n$ such that $\psi (
x,\lambda _n) =\beta _n\varphi (x,\lambda _n) $.
\end{proof}

Denote
\begin{equation*}
\gamma _n=\int_0^{\pi }\varphi ^2(x,\lambda _n) dx
-\frac{1}{\lambda _n}\sum_{s=1}^{m}\alpha _s\varphi ^2(a_s,\lambda_n) .
\end{equation*}
The set $\{ \lambda _n,\gamma _n\} _{n=0,\pm 1,\pm 2,\dots}$ is
called the \textit{spectral data} of $L$.

\begin{lemma}\label{lem2.2}
The equality
\begin{equation}
\dot{\Delta }(\lambda _n) =-2\lambda _n\beta _n\gamma _n  \label{2.4}
\end{equation}
holds. Here $\dot{\Delta }(\lambda ) =\frac{d}{d\lambda }\Delta (\lambda ) $.
\end{lemma}

\begin{proof}
Since
\begin{equation*}
-\varphi ^{''}(x,\lambda _n) +q(x)\varphi
(x,\lambda _n) =\lambda _n^2\varphi (x,\lambda _n) ,\quad
-\psi ^{''}(x,\lambda )+q(x)\psi (x,\lambda ) 
=\lambda^2 \psi (x,\lambda ) ,
\end{equation*}
we obtain
\begin{equation*}
\frac{d}{dx}\langle \varphi (x,\lambda _n) ,\psi (
x,\lambda ) \rangle =(\lambda ^2-\lambda _n^2)
\varphi (x,\lambda _n) \psi (x,\lambda ) .
\end{equation*}
Integrating from $0$ to $\pi $ and using the conditions \eqref{1.2}, 
\eqref{1.3}, we obtain
\begin{equation*}
\int_0^{\pi }\varphi (x,\lambda _n) \psi (x,\lambda) dx
 -\frac{2}{\lambda _n+\lambda }\sum_{s=1}^{m}\alpha _s\varphi
(a_s,\lambda _n) \psi (a_s,\lambda ) 
=\frac{\Delta (\lambda _n) -\Delta (\lambda ) }{\lambda ^2-\lambda _n^2}.
\end{equation*}
Because $\lambda \to \lambda _n$, by Lemma \ref{lem2.1}, this yields
\begin{equation*}
\dot{\Delta }(\lambda _n) =-2\lambda _n\beta_n\gamma _n.
\end{equation*}
\end{proof}

\begin{corollary}
For BVP L, $\lambda _n\neq 0$, $\gamma _n\neq 0$ and $\lambda
_n\neq \lambda _{k}$ $(n\neq k)$ hold.
\end{corollary}

Now, consider the solutions $\varphi (x,\lambda ) $ and 
$\psi (x,\lambda ) $. Let $\tau :=\operatorname{Im}\lambda $. 
The following asymptotic estimates hold uniformly with respect to 
$x\in ( a_s,a_{s+1}) $, $0\leq s\leq m$, as $| \lambda |
\to \infty$:
\begin{equation}  \label{2.7} 
\begin{aligned}
&\varphi (x,\lambda ) \\
&=\alpha _1'\alpha _2'\dots\alpha _s'\cos (\lambda x-\theta _1-\theta
_2-\dots-\theta _s)   \\
&\quad +\sum_{1\leq i\leq s}\alpha _1'\dots\alpha _{i}\dots\alpha
_s'\sin [ \lambda (x-2a_{i}) +\theta
_1+\dots+\theta _{i-1}-\theta _{i+1}-\dots-\theta _s]   \\
&\quad +\sum_{1\leq i<j\leq s}\alpha _1'\dots\alpha _{i}\dots\alpha
_{j}\dots\alpha _s'  
 \cos \big[ \lambda (x+2a_{i}-2a_{j}) +\theta_1+\dots \\
&\quad +\theta _{i-1}-\theta _{i+1}-\dots-\theta _{j-1}+\theta
_{j+1}+\dots+\theta _s\big]   \\
&\quad +\dots+\alpha _1\alpha _2\dots\alpha _s\{
\begin{cases}
\cos & \text{if $s$ is even} \\
\sin & \text{if $s$ is odd}
\end{cases}\\
&\quad \big[ \lambda (x+2(-1) ^{s}a_1+2(-1)
^{s-1}a_2+\dots-2\alpha _s) \big] +O\Big(\frac{1}{|
\lambda | }\exp (| \tau | x)\Big) ,
\end{aligned}
\end{equation}

\begin{equation} \label{2.8} 
\begin{aligned}
&\varphi '(x,\lambda ) \\
&= \lambda \Big[ -\alpha
_1'\alpha _2'\dots\alpha _s'\sin (
\lambda x-\theta _1-\theta _2-\dots-\theta _s)    \\
&\quad +\sum_{1\leq i\leq s}\alpha _1'\dots\alpha _{i}\dots\alpha
_s'\cos \big[ \lambda (x-2a_{i}) +\theta
_1+\dots+\theta _{i-1}-\theta _{i+1}-\dots-\theta _s\big]   \\
&\quad -\sum_{1\leq i<j\leq s}\alpha _1'\dots\alpha _{i}\dots\alpha
_{j}\dots\alpha _s'  \sin \big[ \lambda (x+2a_{i}-2a_{j}) +\theta
_1+\dots+\theta _{i-1} \\
&\quad -\theta _{i+1}-\dots-\theta _{j-1}+\theta
_{j+1}+\dots+\theta _s\big]  
+\dots+\alpha _1\alpha _2\dots\alpha _s
\begin{cases}
-\sin  & \text{if $s$ is even} \\
\cos   & \text{if $s$ is odd}
\end{cases}  \\
&\quad \big[ \lambda \big(x+2(-1) ^{s}a_1+2(-1)
^{s-1}a_2+\dots-2\alpha _s\big) \big] +O\Big(\exp (|
\tau | x) \Big) ,
\end{aligned}
\end{equation}
and
\begin{equation}  \label{2.9}
\begin{aligned}
&\psi (x,\lambda ) \\
&= \alpha _{s+1}'\alpha_{s+2}'\dots\alpha _m'\cos [ \lambda (\pi
-x) -\theta _{s+1}-\theta _{s+2}-\theta _m]   \\
&\quad -\sum_{s+1\leq i\leq m}\alpha _{s+1}'\dots\alpha _{i}\dots\alpha
_m'\sin [ \lambda (\pi +x-2a_m) -\theta
_{s+1}-\theta _{s+2}-\theta _{m-1}]   \\
&\quad +\sum_{s+1\leq i<j\leq m}\alpha _{s+1}'\dots\alpha _{i}\dots\alpha
_{j}\dots\alpha _m'   
 \cos \big[ \lambda (\pi +x+2a_{i}-2a_{j}) +\theta
_{s+1}+\\
&\quad \dots+\theta _{i-1} -\theta _{i+1}-\dots-\theta _{j-1}+\theta_{j+1}+\dots
 +\theta _m\big] +\dots\\
&\quad +\alpha _{s+1}\alpha _{s+2}\dots\alpha _m
\begin{cases}
\cos &\text{if $s$ is even} \\
\sin &\text{if $s$ is odd}
\end{cases}   \\
&\quad [ \lambda (\pi +x-2a_{s+1}+\dots+2(-1)
^{s-1}a_{m-1}+2(-1) ^{s}a_m) ]\\
&\quad +O\Big(\frac{1}{ | \lambda | }\exp (| \tau |(\pi -x) ) \Big) ,
\end{aligned}
\end{equation}

\begin{equation} \label{2.10}
\begin{aligned}
&\psi '(x,\lambda ) \\
&=\lambda [ \alpha _{s+1}'\alpha _{s+2}'\dots\alpha _m'\sin
[ \lambda (\pi -x) -\theta _{s+1}-\theta _{s+2}-\theta _m]
  \\
&\quad -\sum_{s+1\leq i\leq m}\alpha _{s+1}'\dots\alpha _{i}\dots\alpha
_m'\cos [ \lambda (\pi +x-2a_m) -\theta
_{s+1}-\theta _{s+2}-\theta _{m-1}]   \\
&\quad +\sum_{s+1\leq i<j\leq m}\alpha _{s+1}'\dots\alpha _{i}\dots\alpha
_{j}\dots\alpha _m'   \sin \big[ \lambda (\pi +x+2a_{i}-2a_{j}) +\theta
_{s+1}\\
&\quad +\dots+\theta _{i-1}
 -\theta _{i+1}-\dots-\theta _{j-1}+\theta
_{j+1}+\dots+\theta _m\big]   
+\dots\\
&\quad +\alpha _{s+1}\alpha _{s+2}\dots\alpha _m
\begin{cases}
-\sin &\text{ if $s$ is even} \\
\cos &\text{ if $s$ is odd}
\end{cases}   \\
&\quad \big[ \lambda (\pi +x-2a_{s+1}+\dots+2(-1)
^{s-1}a_{m-1}+2(-1) ^{s}a_m) \big] \\
&\quad +O\Big(\exp (| \tau | (\pi -x) ) \Big) ,
\end{aligned}
\end{equation}
where $\alpha _{i}'=\sqrt{1+\alpha _{i}^2}$, $\theta _{i}=\tan
^{-1}\alpha _{i}$, $i=\overline{1,m}$.

It is easy to have the following asymptotic formulae for
$\varphi (x,\lambda ) $ and $\psi (x,\lambda )$,
\begin{gather}
| \varphi ^{(\nu ) }(x,\lambda )| =O\Big(| \lambda | ^{\nu }e^{|
\tau | x}\Big) ,\quad  0\leq x\leq \pi ,\; \nu =0,1, \label{2.11} \\
| \psi ^{(\nu ) }(x,\lambda ) |
=O(| \lambda | ^{\nu }e^{| \tau| (\pi -x)}) .  \label{2.12}
\end{gather}
It follows from \eqref{2.3}, \eqref{2.7} and \eqref{2.8} that
\begin{equation}
\Delta (\lambda ) =\Delta _0(\lambda ) +O\Big(\exp (| \tau | \pi ) \Big) ,  
\label{2.13}
\end{equation}
where
\begin{equation}
\begin{aligned}
&\Delta _0(\lambda ) \\
&= \lambda \Big[ -\alpha _1'\alpha _2'\dots\alpha _s'\sin (\lambda \pi
-\theta _1-\theta _2-\dots-\theta _s)    \\
&\quad +\sum_{1\leq i\leq s}\alpha _1'\dots\alpha _{i}\dots\alpha
_s'\cos \big[ \lambda (\pi -2a_{i}) +\theta
_1+\dots+\theta _{i-1}-\theta _{i+1}-\dots-\theta _s\big]   \\
&\quad -\sum_{1\leq i<j\leq s}\alpha _1'\dots\alpha _{i}\dots\alpha
_{j}\dots\alpha _s'  
 \sin \big[ \lambda (\pi +2a_{i}-2a_{j}) +\theta
_1+\dots+\theta _{i-1}\\
&\quad -\theta _{i+1}-\dots-\theta _{j-1}+\theta
_{j+1}+\dots+\theta _s\big]  
+\dots+\alpha _1\alpha _2\dots\alpha _s
\begin{cases}
-\sin &\text{if $s$ is even} \\
\cos &\text{if $s$ is odd}
\end{cases}
   \\
&\quad \big[ \lambda (\pi +2(-1) ^{s}a_1+2(-1)
^{s-1}a_2+\dots-2\alpha _s) \big] +O\Big(\exp (|\tau | \pi ) \Big) .
\end{aligned}
\end{equation}
Let $\{ \lambda _n^{0}\} $ be zeros of $\Delta _0(\lambda ) $. 
Using \eqref{2.13}, by the well known methods (see, for
example, \cite{4}) one can obtain the following properties of the
characteristic function $\Delta (\lambda ) $ of the BVP L:
\begin{enumerate}
\item For $| \lambda | \to \infty $, $\Delta
(\lambda ) =O(| \lambda | \exp (| \tau | \pi ) ) $.

\item Denote $G_{\delta }:=\{ \lambda :| \lambda -\lambda
_n^{0}| \geq \delta \} $. Then exist $C_{\delta }>0$ such
that
\begin{equation}
| \Delta (\lambda ) | \geq C_{\delta
}| \lambda | \exp (| \tau | \pi
) \quad \text{for all }\lambda \in G_{\delta }\quad (\delta >0).
\label{2.14}
\end{equation}

\item For sufficiently large values of $n$, one has
\begin{gather*}
| \Delta (\lambda ) -\Delta _0(\lambda )
| <\frac{C_{\delta }}{2}\exp (| \tau |\pi ) ,\\
\lambda \in \Gamma _n:=\{ \lambda :|
\lambda | =| \lambda _n^{0}| +\frac{1}{2}
\inf_{n\neq m}| \lambda _n^{0}-\lambda _m^{0}|\} .
\end{gather*}
Then
\begin{equation}
\lambda _n=\lambda _n^{0}+o(1),\text{ }n\to \infty .\label{2.15}
\end{equation}
Substituting \eqref{2.7}, \eqref{2.8} and \eqref{2.14} into \eqref{2.3} we obtain
\begin{equation}
\gamma _n=\gamma _n^{0}+o(1),\text{ }n\to \infty ,  \label{2.16}
\end{equation}
where
\begin{equation*}
\gamma _n^{0}=\int_0^{\pi }\varphi ^2(x,\lambda _n^{0})
dx-\frac{1}{\lambda _n^{0}}\sum_{s=1}^{m}\alpha _s\varphi ^2(
a_s,\lambda _n^{0}) .
\end{equation*}
\end{enumerate}

\section{Formulation of the inverse problem. Uniqueness theorems}

In this section, we study three inverse problems of recovering L from its
spectral characteristics, namely
\begin{itemize}
\item[i] from the Weyl function,

\item[ii] from the so-called spectral data,

\item[iii] from two spectra.
\end{itemize}
For each class of inverse problems we prove the corresponding uniqueness
theorems and show the connection between the different spectral
characteristics.

Let $\Phi (x,\lambda ) $ be the solution of \eqref{1.5} under
the conditions $U(\Phi )=1$ and $V(\Phi )=0$. We set
 $M(\lambda ):=\Phi (\lambda )$. The functions $\Phi (x,\lambda ) $ and 
$M(\lambda )$ are called the Weyl solution and the Weyl function for BVP L,
respectively. Clearly
\begin{gather}
\Phi (x,\lambda ) =\frac{\psi (x,\lambda ) }{\Delta
(\lambda ) }=S(x,\lambda ) +M(\lambda )\varphi
(x,\lambda ),  \label{4.1} \\
M(\lambda )=\frac{\psi (0,\lambda ) }{\Delta (\lambda
) },  \label{4.2}
\end{gather}
where $\psi (0,\lambda ) $ is the characteristic function of 
BVP  L$_1$ which is equation \eqref{1.5} with the boundary conditions 
$U(y)=0$, $y(\pi )=0$, and $S(x,\lambda ) $ is defined from the
equality
\begin{equation*}
\psi (x,\lambda ) =\psi (0,\lambda ) \varphi (
x,\lambda ) +\Delta (\lambda ) S(x,\lambda ) .
\end{equation*}
Note that, by equalities $\langle \varphi (x,\lambda
) ,S(x,\lambda ) \rangle \equiv 1$ and \eqref{4.1},
one has
\begin{equation}
\langle \Phi (x,\lambda ) ,\varphi (x,\lambda )
\rangle \equiv 1.  \label{4.3}
\end{equation}
Let $\{ \mu _n^2\} _{n\geq 0\text{ }}$ be zeros of $\psi
(0,\lambda ) $, i.e. the eigenvalues of $L_1$.

First, let us prove the uniqueness theorems for the solutions of the
problems (i)-(iii). For this purpose we agree that together with L we
consider a BVP $\widetilde{L}$ of the same form but with different
coefficients $\widetilde{q}(x)$, $\widetilde{h}$, $\widetilde{H}$,
$\widetilde{a_s}$ and $\widetilde{\alpha _s}$, $s=\overline{1,m}$. 
Everywhere below if a certain symbol $e$ denotes an object related to $L$, then the
corresponding symbol $\widetilde{e}$ with tilde denotes the analogous object
related to $\widetilde{L}$.

\begin{theorem}\label{theorem 4.1} 
If $M(\lambda )=\widetilde{M}(\lambda )$, then 
$L=\widetilde{L}$. Thus, the specification of the Weyl function $M(\lambda )$
uniquely determines $L$.
\end{theorem}

\begin{proof}
Let us define the matrix $P(x,\lambda )=[ P_{jk}(x,\lambda )]_{j,k=1,2}$ 
by the formula
\begin{equation}
P(x,\lambda )\begin{bmatrix}
\widetilde{\varphi }(x,\lambda ) &   \widetilde{\Phi }(
x,\lambda ) \\
\widetilde{\varphi }'(x,\lambda ) &   \widetilde{\Phi }
'(x,\lambda )
\end{bmatrix}
= \begin{bmatrix}
\varphi (x,\lambda ) &   \Phi (x,\lambda ) \\
\varphi '(x,\lambda ) &   \Phi '(x,\lambda )
\end{bmatrix}. \label{4.4}
\end{equation}
By  \eqref{4.1}, we calculate
\begin{equation}
\begin{gathered}
P_{j1}(x,\lambda )=\varphi ^{(j-1) }(x,\lambda )
\widetilde{\Phi }'(x,\lambda ) -\Phi ^{(
j-1) }(x,\lambda ) \widetilde{\varphi }'(x,\lambda ) , \\
P_{j2}(x,\lambda )=\Phi ^{(j-1) }(x,\lambda )
\widetilde{\varphi }(x,\lambda ) -\varphi ^{(j-1)
}(x,\lambda ) \widetilde{\Phi }(x,\lambda ) ,
\end{gathered}  \label{4.5}
\end{equation}
and
\begin{equation}
\begin{gathered}
\varphi (x,\lambda ) =P_{11}(x,\lambda )\widetilde{\varphi }
(x,\lambda ) +P_{12}(x,\lambda )\widetilde{\varphi }'(x,\lambda ) , \\
\Phi (x,\lambda ) =P_{11}(x,\lambda )\widetilde{\Phi }(
x,\lambda ) +P_{12}(x,\lambda )\widetilde{\Phi }'(
x,\lambda ) .
\end{gathered} \label{4.6}
\end{equation}
Taking \eqref{2.7}, \eqref{2.8} and \eqref{2.14} into account, we infer that
\begin{equation}
| P_{11}(x,\lambda )-1| \leq C_{\delta }| \lambda | ^{-1},\quad
| P_{12}(x,\lambda )|\leq C_{\delta }| \lambda | ^{-1},\quad 
\lambda \in G_{\delta },  \label{4.7}
\end{equation}
where $G_{\delta }$ is defined in \eqref{2.14} and $C_{\delta }$ is a
constant.

On the other hand according to \eqref{4.2} and \eqref{4.5},
\begin{gather*}
P_{11}(x,\lambda )=\varphi (x,\lambda ) \widetilde{S}'(x,\lambda )
 -S(x,\lambda )\widetilde{\varphi }'(x,\lambda
) +(\widetilde{M}(\lambda )-M(\lambda ))\varphi (x,\lambda
) \widetilde{\varphi }'(x,\lambda ) ,
\\
P_{12}(x,\lambda )=S(x,\lambda )\widetilde{\varphi }(x,\lambda )
-\varphi (x,\lambda ) \widetilde{S}(x,\lambda )+(M(\lambda )-
\widetilde{M}(\lambda ))\varphi (x,\lambda ) \widetilde{\varphi }
(x,\lambda ) .
\end{gather*}
Since $M(\lambda )=\widetilde{M}(\lambda )$, it follows that for each fixed 
$x$ the functions $P_{1k}(x,\lambda )$, $k=1,2$ are entire in $\lambda $.
With the help of \eqref{4.7} and well-known Liouville's theorem, this yields
$P_{11}(x,\lambda )\equiv 1$, $P_{12}(x,\lambda )\equiv 0$. Substituting
into \eqref{4.6}, we obtain 
$\varphi (x,\lambda ) \equiv \widetilde{ \varphi }(x,\lambda ) $, 
$\Phi (x,\lambda ) = \widetilde{\Phi }(x,\lambda ) $ for all 
$x\in \cup _{s=0}^{m}(x_s,x_{s+1}) $ and $\lambda $. Taking this into
account, from \eqref{1.1} we obtain $q(x)=\widetilde{q}(x)$ a.e. on $(
0,\pi ) $, from \eqref{2.2} we obtain $h=\widetilde{h}$, $H=\widetilde{
H}$, and from \eqref{1.3} we conclude that $a_s=\widetilde{a_s}$, $
\alpha _s=\widetilde{\alpha _s}$ $(s=\overline{1,m}) $.
Consequently, $L=\widetilde{L}$.
\end{proof}

\begin{theorem}\label{thm4.2} 
If $\lambda _n=\widetilde{\lambda _n}$,
$\gamma _n= \widetilde{\gamma _n}$, $n=0,\pm 1,\pm 2,\dots$, then
 $L=\widetilde{L}$.
Thus, the specification of the spectral data $\{ \lambda _n,\gamma
_n\} _{n=0,\pm 1,\pm 2,\dots}$ uniquely determines the operator.
\end{theorem}

\begin{proof}
It follows from \eqref{4.2} that the Weyl function $M(\lambda )$ is
meromorphic with simple poles at points $\lambda _n^2$. 
Using \eqref{4.2}, Lemma \ref{lem2.1} and equality
 $\dot{\Delta }(\lambda_n)=-2\lambda _n\beta _n\gamma _n$, we have
\begin{equation}
\operatorname{Re}s_{\lambda-\lambda _n} M(\lambda )
=\frac{\psi (0,\lambda ) }{\dot{\Delta }(\lambda _n)}
=\frac{\beta _n}{\dot{ \Delta }(\lambda _n)}
=\frac{1}{-2\lambda _n\gamma _n}.  \label{4.8}
\end{equation}
Since the Weyl function $M(\lambda )$ is regular for
 $\lambda \in \Gamma _n$, applying the Rouche theorem \cite[p.112]{6}, 
we conclude that
\begin{equation*}
M(\lambda )=\frac{1}{2\pi i}\int_{\Gamma _n}\frac{M(\mu )}{\lambda -\mu }
d\mu ,\text{ }\lambda \in int\Gamma _n,
\end{equation*}
where the contour $\Gamma _n$ is assumed to have the counterclockwise
circuit.

Taking \eqref{2.14} and \eqref{4.2} into account, we arrive at 
$|M(\lambda )| \leq C_{\delta }| \lambda | ^{-1}$,
$\lambda \in G_{\delta }$. Hence, by the residue theorem, we have
\begin{equation}
M(\lambda )=\sum_{n=-\infty }^{\infty }\frac{1}{2\lambda _n\gamma
_n(\lambda -\lambda _n) }.  \label{4.9}
\end{equation}
Under the hypothesis of the theorem we obtain, in view of \eqref{4.9}, that 
$M(\lambda )=\widetilde{M}(\lambda )$ and consequently by Theorem 
\ref{theorem 4.1}, $L=\widetilde{L}$.
\end{proof}

\begin{theorem} \label{theorem 4.3} 
If $\lambda _n=\widetilde{\lambda _n}$ and 
$\mu _n= \widetilde{\mu _n}$, $n=0,\pm 1,\pm 2,\dots$, then 
$L=\widetilde{L}$. Thus the specification of two spectra 
$\{ \lambda _n,\mu _n\}_{n=0,\pm 1,\pm 2,\dots}$ uniquely determines $L$.
\end{theorem}

\begin{proof}
It is obvious that characteristic functions $\Delta (\lambda ) $
and $\psi (0,\lambda ) $ are uniquely determined by the
sequences $\{ \lambda _n^2\} $ and $\{ \mu_n^2\} $ $(n=0,\pm 1,\pm 2,\dots) $, 
respectively. 
If $\lambda _n=\widetilde{\lambda _n}$, $\mu _n=\widetilde{\mu _n}$, 
$n=0,\pm 1,\pm 2,\dots$, then 
$\Delta (\lambda ) \equiv \widetilde{\Delta }(\lambda ) $, 
$\psi (0,\lambda ) =\widetilde{\psi }(0,\lambda ) $. 
Together with \eqref{4.2} this yields $M(\lambda )=\widetilde{M}(\lambda )$.
 By Theorem \ref{theorem 4.1} we obtain $L=\widetilde{L}$.
\end{proof}

\begin{remark} \label{remark 4.4}  \rm
By \eqref{4.2}, the specification of two
spectra $\{ \lambda _n,\mu _n\} _{n=0,\pm 1,\pm 2,\dots}$ is
equivalent to the specification of the Weyl function $M(\lambda )$. On the
other hand, it follows from \eqref{4.9} that the specification of the Weyl
function $M(\lambda )$ is equivalent to the specification of the spectral
data $\{ \lambda _n,\gamma _n\} _{n=0,\pm 1,\pm 2,\dots}$.
Consequently, three statements of the inverse problem of reconstruction of
the problem $L$ from the Weyl function, from the spectral data and from two
spectra are equivalent.
\end{remark}

\section{Solution of the inverse problem}

In this section, we will solve the inverse problems of recovering the BVP 
L by the method of spectral mappings with the help of Cauchy's integral
formula and Residue theorem. Finally, we provide the algorithms for the
solution of the inverse problems by using the solution of the main equation.

For definiteness we consider the inverse problem of recovering L from the
spectral data $\{ \lambda _n,\gamma _n\} _{n=0,\pm 1,\pm
2,\dots}$. Let BVPs  L and $\widetilde{L}$ be such that
\begin{equation}
a_s=\widetilde{a_s},\quad s=\overline{1,m},\quad 
\sum_{n=-\infty}^{\infty }\zeta _n| \lambda _n| <\infty ,
\label{5.1}
\end{equation}
where $\zeta _n:=| \lambda _n-\widetilde{\lambda }
_n| +| \gamma _n-\widetilde{\gamma }_n| $.
Denote
\begin{gather*}
\lambda _{n0}=\lambda _n,\text{ }\lambda _{n1}=\widetilde{\lambda }_n,
\quad \gamma _{n_0}=\gamma _n,\gamma _{n1}=\widetilde{\gamma }_n, 
\\
\varphi _{ni}(x)=\varphi (x,\lambda _{ni}) ,\quad 
\widetilde{ \varphi }_{ni}(x)=\widetilde{\varphi }(x,\lambda _{ni}) ,
\\
Q_{kj}(x,\lambda ) =\frac{\langle \varphi (x,\lambda
) ,\varphi _{kj}(x)\rangle }{2\lambda _{kj}\gamma _{kj}(
\lambda -\lambda _{kj}) }=\frac{1}{2\lambda _{kj}\gamma _{kj}}
\int_0^{x}\varphi (t,\lambda ) \varphi _{kj}(t)dt,
\\
Q_{ni,kj}(x) =Q_{kj}(x,\lambda _{ni})
\end{gather*}
for $i,j=0,1$ and $n,k=0,\pm 1,\pm 2,\dots$, where
 $\widetilde{\varphi }(x,\lambda ) $ is the solution of \eqref{1.5} 
with the potential $\widetilde{q}$ under the initial conditions 
$\widetilde{\varphi }(0,\lambda ) =1$, 
$\widetilde{\varphi }'(0,\lambda ) =\widetilde{h}$. Analogously, we can 
define $\widetilde{Q} _{kj}(x,\lambda ) $ by replacing $\varphi $ with 
$\widetilde{\varphi }$ in the above definition.

Using Schwarz's lemma \cite[p.130]{6} and \eqref{2.7}-\eqref{2.10}, 
\eqref{2.15} we obtain the following asymptotic estimates:
\begin{gather}
| \varphi _{nj}^{(\nu )}(x) | \leq C(| \lambda _n^{0}| +1) ^{\nu },  \label{5.2} \\
| Q_{ni,kj}(x) | \leq \frac{C}{| \lambda _n^{0}-\lambda _{k}^{0}| +1},\quad
|Q_{ni,kj}^{(\nu +1)}(x) | \leq C(|\lambda _n^{0}| +| \lambda _{k}^{0}|
+1) ^{(\nu )},  \label{5.3}
\end{gather}
where $n,k=0,\pm 1,\pm 2,\dots$,  $i,j,\nu =0,1$ and $C$ is a positive
constant. Analogous estimates are also valid for 
$\widetilde{\varphi }_{ni}(x)$, $\widetilde{Q}_{ni,kj}(x)$.

\begin{lemma}\label{lemma 5.1}
Let $\varphi _{nj}(x) $ and $Q_{ni,kj}(x) $ be defined as above. 
Then the following representations hold for $i,j=0,1$ and 
$n,k=0,\pm 1,\pm 2,\dots$:
\begin{equation}
\widetilde{\varphi }_{ni}(x)=\varphi _{n_{i}}(x)+\sum_{l=-\infty }^{\infty
}\Big(\widetilde{Q}_{ni,l0}(x)\varphi _{k0}(x)-\widetilde{Q}
_{ni,l1}(x)\varphi _{k1}(x)\Big) .  \label{5.4}
\end{equation}
Series \eqref{5.4} converge absolutely and uniformly with respect to $x\in
[ 0,\pi] \backslash \{ a_s\} _{s=1}^{m}$.
\end{lemma}

\begin{proof}
By \eqref{5.1} we have
$a_s=\widetilde{a_s}$, $\alpha _s=\widetilde{\alpha _s}$ for $s=\overline{1,m}$.
Then it follows from \eqref{2.7}-\eqref{2.8} that
\begin{equation}
| \varphi ^{(\nu )}(x,\lambda ) -\widetilde{\varphi }^{(\nu )}(x,\lambda ) | 
\leq C| \lambda | ^{\nu -1}\exp (| \tau | x) .
\label{5.5}
\end{equation}
Similarly,
\begin{equation}
| \psi ^{(\nu )}(x,\lambda ) -\widetilde{\psi} ^{(\nu )}(x,\lambda ) | 
\leq C| \lambda | ^{\nu -1}\exp (| \tau | (\pi -x) ) .
\label{5.6}
\end{equation}
Denote $G_{\delta }^{0}=G_{\delta }\cap \widetilde{G}_{\delta }$. In view of
\eqref{2.12}, \eqref{2.14}, \eqref{4.1} and \eqref{5.6} we obtain
\begin{equation}
| \Phi ^{(\nu )}(x,\lambda ) -\widetilde{\Phi} ^{(\nu )}(
x,\lambda ) | \leq C_{\delta }| \lambda |
^{\nu -2}\exp (-| \tau | x) ,\text{ }\lambda
\in G_{\delta }^{0}.  \label{5.7}
\end{equation}
Let $P(x,\lambda )$ be the matrix defined in Section 3. Since for each fixed
$x$, the functions $P_{1k}(x,\lambda )$ are meromorphic in $\lambda $ with
simple poles $\lambda _n$ and $\widetilde{\lambda }_n$, we obtain by
Cauchy's theorem \cite[p.84]{6}
\begin{equation}
P_{1k}(x,\lambda )-\delta _{1k}=\frac{1}{2\pi i}\int_{\Gamma _n}\frac{
P_{1k}(x,\zeta )-\delta _{1k}}{\lambda -\zeta },\text{ }k=1,2,  \label{5.9}
\end{equation}
where $\lambda \in int\Gamma _n$, and $\delta _{jk}$ is the Kronecker
delta.

Further, \eqref{4.3} and \eqref{4.5} imply
\begin{equation}
P_{11}(x,\lambda )=1+(\varphi (x,\lambda ) -\widetilde{
\varphi }(x,\lambda ) ) \widetilde{\Phi }'(
x,\lambda ) -(\Phi (x,\lambda ) -\widetilde{\Phi }
(x,\lambda ) ) \widetilde{\varphi }'(x,\lambda ) .  \label{5.10}
\end{equation}
Using \eqref{2.11}, \eqref{2.12}, \eqref{4.5}, \eqref{5.5}, \eqref{5.7} and 
\eqref{5.10} we infer
\begin{equation}
| P_{1k}(x,\lambda )-\delta _{1k}| 
\leq C_{\delta }| \lambda | ^{-1},\quad \lambda \in G_{\delta }^{0}.
\label{5.11}
\end{equation}
By \eqref{5.9} and \eqref{5.11}
\begin{equation*}
P_{1k}(x,\lambda )-\delta _{1k}=\lim_{n\to \infty }\frac{1}{2\pi i}
\int_{\Gamma _{n1}}\frac{P_{1k}(x,\zeta )}{\zeta -\lambda }d\zeta ,
\end{equation*}
where $\Gamma _{n1}=\{ \lambda :| \lambda |
=| \lambda _n^{0}| ,\text{ }n=0,\pm 1,\pm 2,\dots\}$. 
Substituting this into \eqref{4.6}, we obtain
\begin{equation*}
\varphi (x,\lambda ) =\widetilde{\varphi }(x,\lambda
) +\lim_{n\to \infty }\frac{1}{2\pi i}\int_{\Gamma _{n1}}\frac{
\widetilde{\varphi }(x,\lambda ) P_{11}(x,\zeta )+\widetilde{
\varphi }'(x,\lambda ) P_{12}(x,\zeta )}{\lambda
-\zeta }d\zeta .
\end{equation*}
Taking \eqref{4.1} and \eqref{4.5} into account we calculate
\begin{equation}
\widetilde{\varphi }(x,\lambda ) 
=\varphi (x,\lambda ) +\lim_{n\to \infty }\frac{1}{2\pi i}
\int_{\Gamma _{n1}}\frac{\langle \widetilde{\varphi }(x,\lambda ) ,
\widetilde{ \varphi }(x,\zeta ) \rangle }{\lambda -\zeta }\widetilde{M}
(\zeta ) \varphi (x,\zeta ) d\zeta .  \label{5.12}
\end{equation}
It follows from \eqref{4.8} that
\begin{equation*}
\operatorname{Re}s_{\zeta =\lambda _{kj}}\frac{\langle \widetilde{\varphi }
(x,\lambda ) ,\widetilde{\varphi }(x,\zeta )
\rangle }{\lambda -\zeta }\widetilde{M}(\zeta ) \varphi
(x,\zeta ) =\widetilde{Q}_{kj}(x,\lambda ) \varphi
_{kj}(x).
\end{equation*}
Consequently, calculating the integral in \eqref{5.12} by the residue
theorem \cite[p.112]{6} and then taking $\lambda =\lambda _{ni}$, we arrive
at \eqref{5.4} as $n\to \infty $. Furthermore, according to
asymptotic formulaes \eqref{5.2} and \eqref{5.3}, we derive for 
$x\in (a_s,a_{s+1}) $, $0\leq s\leq m$ that the series converges absolutely
and uniformly on $x\in [ 0,\pi] \backslash \{ a_s\}_{s=1}^{m}$.
\end{proof}

From the above arguments, it is seen that, for each fixed $x\in [ 0,\pi
] \backslash \{ a_s\} _{s=1}^{m}$, the relation \eqref{5.4}
 can be considered as a system of linear equations with respect to 
$\varphi _{ni}(x)$ for $n=0,\pm 1,\pm 2,\dots$ and $i=0,1$. 
But the series in \eqref{5.4}  converges only ``with brackets''. 
Therefore, it is not convenient to use \eqref{5.4} as a main equation 
of the inverse problem.

Let $V$ be a set of indices $u=(n,i) $, $n=0,\pm 1,\pm 2,\dots$
and $i=0,1$. For each fixed $x\in [ 0,\pi ] \backslash \{a_s\} _{s=1}^{m}$, 
we define the vectors
\begin{equation*}
\phi (x)=[ \phi _{u}(x)] _{u\in V}=[
\begin{bmatrix}
\phi _{n0}(x) \\
\phi _{n1}(x)
\end{bmatrix},\quad
\widetilde{\phi }(x)=[ \widetilde{\phi }_{u}(x)] _{u\in V}
=\begin{bmatrix}
\widetilde{\phi }_{n0}(x) \\
\widetilde{\phi }_{n1}(x)
\end{bmatrix} _{n=0,\pm 1,\pm 2,\dots}
\end{equation*}
by the formulas
\begin{equation}
\begin{bmatrix}
\phi _{n0}(x) \\
\phi _{n1}(x)
\end{bmatrix}
=\begin{bmatrix}
\zeta _n^{-1} & -\zeta _n^{-1} \\
0 & 1
\end{bmatrix}
\begin{bmatrix}
\varphi _{n0}(x) \\
\varphi _{n1}(x)
\end{bmatrix},\quad 
\begin{bmatrix}
\widetilde{\phi }_{n0}(x) \\
\widetilde{\phi }_{n1}(x)
\end{bmatrix}=
\begin{bmatrix}
\zeta _n^{-1} & -\zeta _n^{-1} \\
0 & 1
\end{bmatrix}
\begin{bmatrix}
\widetilde{\varphi }_{n0}(x) \\
\widetilde{\varphi }_{n1}(x)
\end{bmatrix}.  \label{5.13}
\end{equation}
If $\zeta _n=0$ for a certain $n$, then we put 
$\phi _{n0}(x)=\widetilde{\phi }_{n0}(x)=0$.

Further, we define the block matrix
\begin{equation*}
H(x)=[ H_{u,v}(x)] _{u,v\in V}=
\begin{bmatrix}
H_{n0,k0}(x) & H_{n0,k1}(x) \\
H_{n1,k0}(x) & H_{n1,k1}(x)
\end{bmatrix}_{n,k=0,\pm 1,\pm 2,\dots},
\end{equation*}
where $u=(n,i) $, $v=(k,j) $ and
\begin{equation*}
\begin{bmatrix}
H_{n0,k0}(x) & H_{n0,k1}(x) \\
H_{n1,k0}(x) & H_{n1,k1}(x)
\end{bmatrix}=
\begin{bmatrix}
\zeta _n^{-1} & -\zeta _n^{-1} \\
0 & 1
\end{bmatrix}
\begin{bmatrix}
Q_{n0,k0}(x) & Q_{n0,k1}(x) \\
Q_{n1,k0}(x) & Q_{n1,k1}(x)
\end{bmatrix}
\begin{bmatrix}
\zeta _{k} & \zeta _{k} \\
0 & -1
\end{bmatrix}.
\end{equation*}
Analogously we define  $\widetilde{\phi }(x)$,  $\widetilde{H}(x)$ by
replacing in the previous definitions $\varphi _{ni}(x)$, $Q_{ni,kj}(x)$ by 
$\widetilde{\varphi }_{ni}(x)$, $\widetilde{Q}_{ni,kj}(x)$, respectively. It
follows from \eqref{2.7}-\eqref{2.10}, \eqref{2.15}, \eqref{2.16}, 
\eqref{5.2}, \eqref{5.3} and Schwarz's lemma that
\begin{gather}
| \phi _{nj}^{(\nu )}(x) | ,\text{ }
| \widetilde{\phi }_{nj}^{(\nu )}(x) | \leq
C(| \lambda _n^{0}| +1) ^{\nu },\quad \nu =0,1, \nonumber \\
| H_{ni,kj}(x)| ,\text{ }| \widetilde{H}
_{ni,kj}(x)| \leq \frac{C\zeta _{k}}{| \lambda
_n^{0}-\lambda _{k}^{0}| +1},  \label{5.14} \\
| H_{ni,kj}^{(\nu +1) }(x)| ,\;
| \widetilde{H}_{ni,kj}^{(\nu +1) }(x)| \leq
C(| \lambda _n^{0}| +| \lambda _{k}^{0}| +1) ^{\nu }\zeta _{k},\quad \nu =0,1.
\label{5.15}
\end{gather}

Let us consider the Banach space $B$ of bounded sequences 
$\alpha =[\alpha _{u}] _{u\in V}$ with the norm 
$\| \alpha \|_{B}=\sup_{u\in V}| \alpha _{u}| $. It follows from 
\eqref{5.14}, \eqref{5.15} that for each fixed $x\in (a_s,a_{s+1}) $,
$0\leq s\leq m$, the operator $E+\widetilde{H}(x)$ and $E-H(x)$ (here $E$ is
the identity operator), acting from $B$ to $B$, are linear bounded operator,
and
\begin{equation*}
\| H(x)\| ,\| \widetilde{H}(x)\| \leq
C\sup_n\sum_{k=-\infty }^{\infty }\frac{\zeta _{k}}{| \lambda
_n^{0}-\lambda _{k}^{0}| +1}<\infty .
\end{equation*}
Here, we give the main result of the section.

\begin{theorem}\label{theorem 5.1}
For each fixed $x\in [ 0,\pi ] \backslash \{ a_s\} _{s=1}^{m}$, 
the vector $\phi (x)\in B$ satisfies the equation
\begin{equation}
\widetilde{\phi }(x)=(E+\widetilde{H}(x)) \phi (x)  \label{5.16}
\end{equation}
in the Banach space $B$. Moreover, the operator $E+\widetilde{H}(x)$ has a
bounded inverse operator, i.e. the equation \eqref{5.16} is uniquely
solvable.
\end{theorem}

\begin{proof}
Using the notation $\widetilde{\phi }(x)$ we rewrite \eqref{5.4} as
\begin{equation*}
\widetilde{\phi }_{ni}(x)=\phi _{ni}(x)
+\sum_{k,j}\widetilde{H} _{ni,kj}(x)\phi _{kj}(x),\quad (n,i) \in V,\quad 
(k,j) \in V,
\end{equation*}
which is equivalent to \eqref{5.4}. Similarly, using the notation
 $H(x)$:
\begin{equation*}
H_{ni,lj}(x)-\widetilde{H}_{ni,lj}(x)+\sum_{k,t}\widetilde{H}
_{ni,kt}(x)H_{kt,lj}(x)=0
\end{equation*}
for $(n,i) $, $(l,j) $, $(k,t) \in V$ or
in the other form
\begin{equation*}
(E+\widetilde{H}(x)) (E-H(x)) =E.
\end{equation*}
Interchanging places for $L$ and $\widetilde{L}$, we obtain analogously
\begin{equation*}
\phi (x)=(E-H(x)) \widetilde{\varphi }(x),\quad 
(E-H(x)) (E+\widetilde{H}(x)) =E.
\end{equation*}
Hence the operator $(E+\widetilde{H}(x)) ^{-1}$ exists, and it
is a linear bounded operator.
\end{proof}

Equation \eqref{5.16} is called the \textit{basic }equation of the inverse problem.
Solving \eqref{5.16} we find the vector $\phi (x)$, and consequently, the
functions $\varphi _{ni}(x)$. Thus, we obtain the following algorithms for
the solution of our inverse problems.

\begin{algorithm} \label{a1} \rm
Given the spectral data $\{ \lambda _n,\gamma _n\}_{n=0,\pm 1,\pm 2,\dots}$,
construct $q(x)$ and $h,H;a_s,\alpha _s$, $s=\overline{1,m}$.
\begin{enumerate}
\item Choose $\widetilde{L}$ and calculate $\widetilde{\phi }(x)$ and $
\widetilde{H}(x)$;

\item Find $\phi (x)$ by solving the equation \eqref{5.16} and calculate $
\varphi _{no}(x)$ via \eqref{5.13};

\item Choose some $n$ (e.g., $n=0$) and construct $q(x)$ and $
h,H;a_s,\alpha _s$, $s=\overline{1,m}$ by formulae
\begin{gather*}
q(x)=\frac{\varphi _{n0}''(x)}{\varphi _{n0}(x)}+\lambda _n, \quad
h=\varphi _{n0}'(0),\quad 
H=-\frac{\varphi _{n0}'(\pi )}{\varphi _{n0}(\pi )}; \\
\varphi _{n0}(a_s+0)=\varphi _{n0}(a_s-0); \quad
\alpha _s=\frac{\varphi _{n0}'(a_s+0)
-\varphi _{n0}'(a_s-0)}{2\lambda _n\varphi _{n0}(a_s-0)};\quad
s=\overline{1,m}.
\end{gather*}
\end{enumerate}
\end{algorithm}

\begin{algorithm} \label{a2} \rm
Given $M(\lambda )$, we construct $q(x)$ and $h,H;a_s,\alpha _s$,
$s=\overline{1,m}$.
\begin{enumerate}
\item According to \eqref{4.9} construct the spectral data $\{ \lambda
_n,\gamma _n\} _{n=0,\pm 1,\pm 2,\dots}$.

\item By Algorithm \ref{a1}, construct $q(x)$ and $h,H;a_s,\alpha _s$,
 $s=\overline{1,m}$.
\end{enumerate}
\end{algorithm}


\begin{algorithm} \label{a3} \rm
Given two spectra $\{ \lambda _n,\mu _n\} _{n=0,\pm
1,\pm 2,\dots}$, we construct $q(x)$ and $h,H;a_s,\alpha _s$,
 $s= \overline{1,m}$.
\begin{enumerate}
\item By \eqref{4.2} calculate $M(\lambda )$;

\item By considering Algorithm \ref{a2}, we construct $q(x)$ and 
$h,H;a_s,\alpha _s$, $s=\overline{1,m}$.
\end{enumerate}
\end{algorithm}


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\end{thebibliography}

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