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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 26, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/26\hfil Inverse spectral and inverse nodal problems]
{Inverse spectral and inverse nodal problems for energy-dependent Sturm-Liouville
equations with $\delta $-interaction}

\author[M. Dzh. Manafov, A. Kablan \hfil EJDE-2015/26\hfilneg]
{Manaf Dzh. Manafov, Abdullah Kablan}

\dedicatory{In memory of M. G. Gasymov}

\address{Manaf Dzh. Manafov \newline
Faculty of Arts and Sciences,
Department of Mathematics,
Adiyaman University, \newline
Adiyaman 02040, Turkey}
\email{mmanafov@adiyaman.edu.tr}

\address{Abdullah Kablan \newline
Faculty of Arts and Sciences,
Department of Mathematics,
Gaziantep University, \newline
Gaziantep 27310, Turkey}
\email{kablan@gantep.edu.tr}

\thanks{Submitted March 19, 2014. Published January 28, 2015.}
\subjclass[2000]{34A55, 34B24, 34L05, 47E05}
\keywords{Energy-dependent Sturm-Liouville equations; \hfill\break\indent 
inverse spectral and inverse nodal problems; point $\delta$-interaction}

\begin{abstract}
 In this article, we study the inverse spectral and inverse nodal problems
 for energy-dependent Sturm-Liouville equations with $\delta$-interaction.
 We obtain uniqueness, reconstruction and stability using the nodal
 set of eigenfunctions for the given problem.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

We consider the boundary value problem (BVP) generated by the
differential equation
\begin{equation}
-y''+q(x)y=\lambda ^2y, \quad  x\in ( 0,\frac{\pi }{2}
) \cup ( \frac{\pi }{2},\pi )  \label{a1}
\end{equation}
with the boundary conditions
\begin{equation}
U(y):=y(0)=0,\quad V(y):=y'(\pi )=0  \label{a2}
\end{equation}
and at the point $x=\frac{\pi }{2}$ satisfying
\begin{equation}
\begin{gathered}
y(\frac{\pi }{2}+0)=y(\frac{\pi }{2}-0)=y(\frac{\pi }{2}), \\
y'(\frac{\pi }{2}+0)-y'(\frac{\pi }{2}-0)=2\alpha \lambda
y(\frac{\pi }{2})
\end{gathered}
 \label{a3}
\end{equation}
where $q(x)$ is a nonnegative real valued function in $L_2(0,\pi )$,
$\alpha \neq \pm 1$ is real number and $\lambda $ is spectral parameter.
Without loss of generality we assume that
\begin{equation}
\int_0^\pi q(x)dx=0.  \label{a4}
\end{equation}
We denote the BVP \eqref{a1}, \eqref{a2} and \eqref{a3} by $L=L(q,\alpha )$.

Notice that, we can understand problem \eqref{a1} and \eqref{a3} as studying
the equation
\begin{equation}
y''+(\lambda ^2-2\lambda p(x)-q(x))y=0,\quad x\in (0,\pi)  \label{a5}
\end{equation}
when $p(x)=\alpha \delta (x-\frac{\pi }{2})$, where $\delta (x)$ is the
Dirac function (see \cite{Albeverio}).

We consider the inverse problems of recovering $q(x)$ and $\alpha $ from
the given spectral and nodal characteristics. Such problems play an
important role in mathematics and have many applications in natural sciences
(see, for example, monographs
 \cite{Freiling 1,Levitan,Marchenko,Poschel}). 
Inverse nodal problems consist in constructing
operators from the given nodes (zeros) of eigenfunctions 
(see \cite{Cheng,Hold,Law,McLaughlin,Shen}). Discontinuous
inverse problems (in various formulations) have been considered in 
\cite{Amirov,Freiling 2,Hryniv,Savchuk,Shepelsky, Shieh,Yang}.

Sturm-Liouville spectral problems with potentials depending on the spectral
parameter arise in various models quantum and classical mechanics. 
There $\lambda ^2$ is related to the energy of the system, this explaining the
term ``energy-dependent'' in \eqref{a5}. 
The non-linear dependence of equation \eqref{a5} on the spectral parameter
 $\lambda $ should be regarded as a spectral problem for a quadratic operator pencil. 
The inverse spectral and nodal problems for energy-dependent 
Schr\"{o}dinger operators 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{Buterin}, \cite{Gsymov}. Such problems for separated
and nonseparated boundary conditions were considered 
(see \cite{Akhtyamov,Geseinov,Yurko2} and the references therein). The inverse
scattering problem for equation \eqref{a5} with eigenparameter-dependent
boundary condition on the half line solved in \cite{Manafov}.

In this article we obtain some results on inverse spectral and inverse
nodal problems and establish connections between them.

\section{Inverse spectral problems}

In this section we study so-called incomplete inverse problem of recovering
the potential $q(x)$ from a part of the spectrum BVP $L$. The
technique employed is similar to those used in 
\cite{Hochstadt,Ramm}. Similar problems for the Sturm-Liouville and 
Dirac operators were
formulated and studied in \cite{Mochizuki 1,Mochizuki 2}.

Let $y(x)$ and $z(x)$ be continuously differentiable functions on the intervals
 $(0,\pi/2)$ and $(\pi/2,\pi )$. Denote
 $\langle y,z\rangle :=yz'-y'z$. If $y(x)$ and $z(x)$ satisfy the matching
conditions \eqref{a3}, then
\begin{equation}
\langle y,z\rangle _{x=\frac{\pi }{2}-0}=\langle y,z\rangle 
_{x=\frac{\pi }{2}+0}  \label{b1}
\end{equation}
i.e. the function $\langle y,z\rangle $ is continuous on $( 0,\pi ) $.

Let $\varphi (x,\lambda )$ be solution of equation \eqref{a1} satisfying the
initial conditions $\varphi (0,\lambda )=0$, $\varphi '(0,\lambda)=1$ and 
the matching condition \eqref{a3}. Then $U(\varphi )=0$. Denote
\begin{equation}
\Delta (\lambda ):=-V(\varphi )=-\varphi '(\pi ,\lambda ).
\label{b2}
\end{equation}
By  \eqref{b1} and the Liouville's formula (see \cite[p.83]{Coddington}), 
$\Delta (\lambda )$ does not depend on $x$. The function 
$\Delta (\lambda )$ is called characteristic function on $L$.

\begin{lemma} \label{lem1}
The eigenvalues of the BVP $L$ are real, nonzero and simple.
\end{lemma}

\begin{proof}
Suppose that $\lambda $ is an eigenvalue BVP $L$ and that $y(x,\lambda )$ is
a corresponding eigenfunction such that 
$\int_0^\pi | y(x,\lambda )| ^2dx=1$. Multiplying both sides of
 \eqref{a1} by $\overline{y(x,\lambda )}$ and integrate the result with
respect to $x$ from $0$ to $\pi$:
\begin{equation}
-\int_0^\pi y''(x,\lambda )\overline{
y(x,\lambda )}dx+\int_0^\pi q(x)|
y(x,\lambda )| ^2dx=\lambda ^2\int_0^\pi | y(x,\lambda )| ^2dx  \label{b3}
\end{equation}
Using the formula of integration by parts and the conditions \eqref{a2} and 
\eqref{a3} we obtain
\[
\int_0^\pi y''(x,\lambda )\overline{y(x,\lambda )}dx
=-2\alpha \lambda | y(0,\lambda )| ^2-\int_0^\pi | y'(x,\lambda)| ^2dx.
\]
It follows from this and \eqref{b3} that
\begin{equation}
\lambda ^2+B(\lambda )\lambda +C(\lambda )=0,  \label{b4}
\end{equation}
where
\begin{gather*}
B(\lambda )=-2\alpha .| y(0,\lambda )| ^2,\\
C(\lambda )=-\int_0^\pi q(x)| y(x,\lambda
)| ^2dx-\int_0^\pi | y'(x,\lambda )| ^2dx.
\end{gather*}
Thus the eigenvalue $\lambda $ of the BVP $L$ is a root of the quadratic
equation \eqref{b4}. Therefore, $B^2(\lambda )-4C(\lambda )>0$.
Consequently, the equation \eqref{b4} has only real roots.

Let us show that $\lambda _0$ is a simple eigenvalue. Assume that this is
not true. Suppose that $y_1(x)$ and $y_2(x)$ are linearly independent
eigenfunctions corresponding to the eigenvalue $\lambda _0$. Then for a
given value of $\lambda _0$, each solution $y_0(x)$ of \eqref{a5} will
be given as linear combination of solutions $y_1(x)$ and $y_2(x)$.
Moreover it will satisfy boundary conditions \eqref{a2} and conditions 
\eqref{a3} at the point $x=\pi/2$. However it is impossible.
\end{proof}

\begin{lemma} \label{lem2}
The BVP $L$ has a countable set of eigenvalues $\{\lambda _n\}_{n\geq 1}$.
Moreover, as $n\to \infty $,
\begin{equation}
\lambda _n:=n-\frac{\theta }{\pi }+\frac{1}{2(\pi n-\theta )}
(w_0+(-1)^{n-1}w_1)+o( \frac{1}{n}) ,  \label{b5}
\end{equation}
where
\begin{equation}
\tan \theta =\frac{1}{\alpha },\quad 
w_0=\int_0^\pi q(t)dt,\quad 
w_1=\frac{\alpha }{\sqrt{1+\alpha ^2}}
\Big( \int_0^{\pi/2} q(t)dt-\int_{\pi/2}^{\pi} q(t)dt\Big) .  \label{b55}
\end{equation}
\end{lemma}

\begin{proof}
Let $\tau :=\operatorname{Im}\lambda $. For 
$| \lambda |\to \infty $ uniformly in $x$ one has (see \cite[Chapter 1]{Yurko1})
\begin{gather}
\varphi (x,\lambda )=\frac{\sin \lambda x}{\lambda }-\frac{\cos \lambda x}{
2\lambda ^2}\int_0^x q(t)dt+o\Big( \frac{1}{
\lambda ^2}\exp (| \tau | x)\Big) ,\quad x<\frac{\pi }{2},  \label{b6} \\
\label{b7}
\begin{aligned}
&\varphi (x,\lambda )\\ 
&= \frac{1}{\lambda }\Big( \sqrt{1+\alpha ^2}\cos
(\lambda x+\theta )+\alpha \cos \lambda (\pi -x)\Big) 
+\sqrt{1+\alpha ^2} \frac{\sin (\lambda x+\theta )}{2\lambda ^2}
\int_0^x  q(t)dt \\
&\quad +\alpha \frac{\sin \lambda (\pi -x)}{2\lambda ^2}
\Big(\int_0^{\pi/2} q(t)dt
-\int_{\pi/2}^x  q(t)dt\Big) +o\Big( \frac{1}{\lambda ^2}\exp (| \tau
| x)\Big) ,\quad x>\frac{\pi }{2}
\end{aligned}
\\
\varphi '(x,\lambda )
=\cos \lambda x+\frac{\sin \lambda x}{2\lambda
}\int_0^x q(t)dt+o\Big( \frac{1}{\lambda }\exp
(| \tau | x)\Big) ,\quad x<\frac{\pi }{2}  \label{b8}\\
\label{b9}
\begin{aligned}
&\varphi '(x,\lambda )\\
&=-\sqrt{1+\alpha ^2}\sin (\lambda x+\theta)+\alpha \sin \lambda (\pi -x)
+\sqrt{1+\alpha ^2}\frac{\cos (\lambda x+\theta )}{2\lambda }
 \int_0^x q(t)dt \\
&\quad -\alpha \frac{\cos \lambda (\pi -x)}{2\lambda }
\Big(\int_0^{\pi/2} q(t)dt
-\int_{\pi /2}^x  q(t)dt\Big) +o\Big( \frac{1}{\lambda }\exp (| \tau |
x)\Big) ,\quad x>\frac{\pi }{2}
\end{aligned}
\end{gather}
It follows from \eqref{b9} that as $| \lambda | \to \infty $
\begin{equation} \label{b10}
\begin{aligned}
\Delta (\lambda )
&=\sqrt{1+\alpha ^2}\sin (\lambda \pi +\theta )-\sqrt{
1+\alpha ^2}\frac{\cos (\lambda \pi +\theta )}{2\lambda }
\int_0^{\pi} q(t)dt   \\
&\quad +\frac{\alpha }{2\lambda }\Big( \int_0^{\pi /2}q(t)dt
-\int_{\pi /2}^{\pi} q(t)dt\Big) +o\Big( \frac{1
}{\lambda }\exp (| \tau | x)\Big) .
\end{aligned}
\end{equation}
Using \eqref{b10} and Rouch\'e's theorem, by the well-known method (see
\cite{Freiling 1}) one has that as $n\to \infty $,
\[
\lambda _n:=n-\frac{\theta }{\pi }+\frac{1}{2(\pi n-\theta )}
(w_0+(-1)^{n-1}w_1)+o( \frac{1}{n}) .
\]
\end{proof}

Together with $L$ we consider a BVP $\tilde{L}=\tilde{L}(\tilde{q},\alpha )$
of the same form but with different coefficient $\tilde{q}$. The following
theorem has been proved in \cite{Horvarth} for the Sturm-Liouville equation.
We show it also holds for \eqref{a1}-\eqref{a3}.

\begin{theorem} \label{thm1}
If for any $n\in\mathbb{N}\cup \{0\}$,
\[
\lambda _n=\tilde{\lambda}_n,\quad \langle y_n,\tilde{y}_n\rangle_{x=\frac{\pi }{2}
-0}=0,
\]
then $q(x)=\tilde{q}(x)$ almost everywhere (a.e) on $(0,\pi )$.
\end{theorem}

\begin{proof}
Since
\begin{gather*}
-y''(x,\lambda )+q(x)y(x,\lambda )=\lambda ^2y(x,\lambda ), \quad
 -\tilde{y}''(x,\lambda )+\tilde{q}(x)\tilde{y}(x,\lambda
)=\lambda ^2\tilde{y}(x,\lambda ), \\
y(0,\lambda )=0,\quad  y'(0,\lambda )=1, \quad 
 \tilde{y}(0,\lambda )=0,\quad  \tilde{y}'(0,\lambda )=1,
\end{gather*}
it follows from \eqref{b1} that
\begin{equation}
\int_0^{\pi/2} r(x)y(x,\lambda )\tilde{y}(x,\lambda)dx
=\langle y,\tilde{y}\rangle_{x=\frac{\pi }{2}-0}  \label{b11}
\end{equation}
where $r(x)=q(x)-\tilde{q}(x)$. Since 
$\langle y_n,\tilde{y}_n\rangle_{x=\frac{\pi }{2}-0}=0$ for 
$n\in \mathbb{N}\cup \{0\}$, it follows from \eqref{b11} that
\begin{equation}
\int_0^{\pi/2} r(x)y(x,\lambda _n)\tilde{y} (x,\lambda _n)dx=0,\quad n\in
\mathbb{N}\cup \{0\}.  \label{b12}
\end{equation}
For $x\leq \pi /2$ the following representation holds
 (see \cite{Levitan,Marchenko});
\[
y(x,\lambda )=\frac{\sin \lambda x}{\lambda }+\int_0^x K(x,t)
\frac{\sin \lambda x}{\lambda }dt,
\]
where $K(x,t)$ is a continuous function which does not depend on $\lambda $.
Hence \begin{equation}
2\lambda ^2y(x,\lambda )\tilde{y}(x,\lambda )
=1-\cos 2\lambda x-\int_0^x V(x,t)\cos 2\lambda tdt,  \label{b13}
\end{equation}
where $V(x,t)$ is a continuous function which does not depend on $\lambda $.
Substituting \eqref{b13} into \eqref{b12} and taking the relation \eqref{a4}
into account, we calculate
\[
\int_0^{\pi/2} \Big( r(x)+\int_x^{\pi/2} V(t,x)r(x)dt\Big) \cos 2\lambda _nxdx=0,
\quad n\in \mathbb{N}\cup \{0\},
\]
which implies from the completeness of the function cosine, that
\[
r(x)+\int_x^{\pi/2} V(t,x)r(x)dt=0\quad \text{a.e. on }
[ 0,\frac{\pi }{2}] .
\]
But this equation is a homogeneous Volterra integral equation and has only
the zero solution, it follows that $r(x)=0$ a.e. on $[0,\frac{\pi }{2}]$. To
prove that $q(x)=\tilde{q}(x)$ a.e. on $[\pi/2,\pi ]$ we will
consider the supplementary problem $\hat{L}$;
\begin{gather*}
-y''(x,\lambda )+q_1(x)y(x,\lambda )=\lambda
^2y(x,\lambda ),\quad q_1(x)=q(\pi -x),\quad 0<x<\frac{\pi }{2}, \\
U(y):=y(0,\lambda )=0, \\
y( \frac{\pi }{2}+0,\lambda ) =y( \frac{\pi }{2}-0,\lambda) ,\quad
 y'( \frac{\pi }{2}+0,\lambda ) -y'( \frac{\pi }{2}-0) 
=2\alpha \lambda y( \frac{\pi }{2} -0,\lambda ) .
\end{gather*}
It follows from \eqref{b1} that 
$\langle y_n,\tilde{y}_n\rangle_{x=\frac{\pi }{2} +0}=0 $. 
A direct calculation implies that $\tilde{y}_n(x):=y_n(\pi -x)$
is the solution to the supplementary problem $\hat{L}$, the $\hat{L}$ and 
$\tilde{y}_n(\frac{\pi }{2}-0)=y_n(\frac{\pi }{2}+0)$. Thus for the
supplementary problem $\hat{L}$ the assumption conditions in Theorem \ref{thm1} are
still satisfied. If we repeat the above arguments then yields $r(\pi -x)=0$
and $0 <x <\pi/2$, that is $q(x)=\tilde{q}(x)$ a.e. on
 $[\pi/2,\pi]$.
\end{proof}

\section{Inverse nodal problems}

In this section, we obtain uniqueness theorems and a procedure of recovering
the potential $q(x)$ on the whole interval $(0,\pi )$ from a dense subset of
nodal points.

The eigenfunctions of the BVP $L$ have the form 
$y_n(x)=\varphi (x,\lambda _n)$. We note that $y_n(x)$ are real-valued functions. 
Substituting \eqref{b5} into \eqref{b6} and \eqref{b7} we obtain the 
following asymptotic formulae for $n\to \infty $ uniformly in $x$:
\begin{gather} \label{c1}
\begin{aligned}
\lambda _ny_n(x)
&=\sin (n-\frac{\theta }{\pi })x+\frac{1}{2(\pi n-\theta )
}\Big( -\pi \int_0^x q(t)dt+(w_0+(-1)^{n-1}w_1)x\Big)  \\
&\times \cos (n-\frac{\theta }{\pi })x+o( \frac{1}{n}) ,\quad x<\frac{\pi }{2}
\end{aligned}\\
\label{c2}
\begin{aligned}
&\lambda _ny_n(x) \\
&= \cos ((n-\frac{\theta }{\pi })x+\theta )[ \sqrt{
1+\alpha ^2}+(-1)^{n}\alpha ] \\
&\quad +\frac{1}{2(\pi n-\theta )}
\Big[\pi \sqrt{1+\alpha ^2}\int_0^x q(t)dt
+(-1)^{n-1}\alpha \pi \Big(\int_0^{\pi/2} q(t)dt
-\int_{\pi /2}^x q(t)dt\Big) \\
&\quad -(\sqrt{1+\alpha ^2}x+(-1)^{n-1}\alpha (\pi -x))(w_0+(-1)^{n-1}w_1)\Big]
\\
&\quad \times \sin ((n-\frac{\theta }{\pi })x+\theta )+o( \frac{1}{n})
,\quad x>\frac{\pi }{2}.
\end{aligned}
\end{gather}
For the BVP $L$ an analog of Sturm's oscillation theorem is true. 
More precisely, the eigenfunction $y_n(x)$ has exactly $(n-1)$ (simple) zeros
inside the interval $(0,\pi ):$ 
$0<x_n^{1}<x_n^2<\dots <x_n^{n-1}<\pi $. 
The set $X_L:=\{x_n^{j}\}_{n\geq 2,~j=\overline{1,n-1}}$ is called the
set of nodal points of the BVP $L$. 
Denote $X_L^k:=\{x_{2m-k}^{j}\}_{m\geq 1,j=1,2m-k-1}$, $k=0,1$. Clearly, 
$X_L^{0}\cup X_L^{1}=X_L$. Denote $\mu _n^{0}:=0$, $\mu _n^{n}:=1$,
$\mu _n^{j}:=\frac{j}{\pi n-\theta }\pi ^2$, 
$\gamma _n^{j}:=\mu _n^{j}-\frac{\pi ^2+2\theta \pi }{2(\pi n-\theta )}$, 
$j=\overline{1,n-1}$.

Inverse nodal problems consist in recovering the problem $q(x)$ from the
given set $X_L$ of nodal points or from a certain part.

Taking \eqref{c1}-\eqref{c2} into account, we obtain the following
asymptotic formulae for nodal points as $n\to \infty $ uniformly in $j$:\\
for $x_n^{j}\in ( 0,\frac{\pi }{2})$:
\begin{equation}
x_n^{j}=\mu _n^{j}+\frac{\pi }{2(\pi n-\theta )^2}\Big( \pi \overset{
\mu _n^{j}}{\underset{0}{\int }}q(t)dt-(w_0+(-1)^{n}w_1)\mu
_n^{j}\Big) +o( \frac{1}{n^2}) ,  \label{c25}
\end{equation}
for $x_n^{j}\in ( \frac{\pi }{2},\pi )$:
\begin{equation}
x_n^{j}=\gamma _n^{j}+\frac{\pi }{2(\pi n-\theta )^2}
\Big[\pi \int_0^{\gamma _n^{j}} q(t)dt
-((w_0+(-1)^{n-1}w_1)\gamma _n^{j}+d_{k})\Big]+o( \frac{1}{n^2}) ,  \label{c3}
\end{equation}
where $k=0$ when $n$ is odd and $k=1$ when $n$ is even in $d_{k}$, and
\begin{equation}
d_{k}=( \sqrt{1+\alpha ^2}+(-1)^{n-1}\alpha ) 
\Big[ 2(-1)^{n-1}\alpha \pi \int_0^{\pi/2}
q(t)dt+(-1)^{n}\alpha \pi (w_0+(-1)^{n-1}w_1)\Big] .  \label{c4}
\end{equation}
Using these formulae we arrive at the following assertion.

\begin{theorem} \label{thm2}
Fix $k\in \{0,1\}$ and $x\in \lbrack 0,\pi ]$. 
Let $\{x_n^{j}\}\subset X_L^k$ be chosen such that $\lim_{n\to \infty}x_n^{j}=x$. 
Then there exists a finite limit
\begin{equation}
g_{k}(x):=\lim_{n\to \infty}\frac{2(\pi n-\theta )}{\pi }
\Big[ (\pi n-\theta )x_n^{j}-
\begin{cases}
j\pi ,& \text{if }x_n^{j}\in ( 0,\frac{\pi }{2}) \\
(j+\frac{1}{2}) \pi +\theta , &\text{if }x_n^{j}\in (\frac{\pi }{2},\pi )
\end{cases}
 \Big] ,  \label{c5}
\end{equation}
and
\begin{gather}
g_{k}(x)=\int_0^x q(t)dt-\frac{w_0+(-1)^{k-1}w_1}{
\pi }x,\quad x\leq \frac{\pi }{2},  \label{c6} \\
g_{k}(x)=\int_0^x q(t)dt-\frac{w_0+(-1)^{k-1}w_1}{
\pi }x+d_{k},\quad x\geq \frac{\pi }{2} \nonumber
\end{gather}
where $d_0$ and $d_1$ are defined by \eqref{c4}.
\end{theorem}

Let us now formulate a uniqueness theorem and provide a constructive
procedure for the solution of the inverse nodal problem.

\begin{theorem} \label{thm3}
Fix $k=0\vee 1$. Let $X\subset X_L^k$ be a subset of nodal points which
is dense on $(0,\pi )$. Let $X=\tilde{X}$. Then $q(x)=\tilde{q}(x)$ a.e. on 
$(0,\pi )$, $\alpha =\tilde{\alpha}$. Thus the specification of $X$ uniquely
determines the potential $q(x)$ on $(0,\pi )$ and the number $\alpha $. The
function $q(x)$ and the number $\alpha $\ can be constructed via the formulae
\begin{gather}
q(x)=g_{k}'(x)+\frac{1}{\pi }( g_{k}(\pi )-g_{k}(0)) , \label{c7} \\
\alpha =\Big[ \Big( \frac{2g_{k}(\pi )+4g_{k}(\frac{\pi }{2})-6g_{k}(0)}{
\pi (g_0'(x)-g_1'(x))}\Big) ^2-1\Big] ^{-2}
\label{c8}
\end{gather}
where $g_{k}(x)$ is calculated by \eqref{c6}.
\end{theorem}

\begin{proof}
Formulae \eqref{c7}, \eqref{c8} follow from \eqref{c6}, \eqref{a4} and 
\eqref{b55}. Note that by \eqref{c6}, we have
\begin{equation}
g_{k}'(x)=q(x)-\frac{w_0+(-1)^kw_1}{\pi },\quad 
x\in ( 0,\frac{\pi }{2}) \cup ( \frac{\pi }{2},\pi ) ,  \label{c9}
\end{equation}
hence
\begin{equation}
g_{k}(\pi )-g_{k}(0)=\int_0^\pi q(x)dx-(w_0+(-1)^{n-1}w_1),\quad 
w_1=\frac{\pi }{2}\left[ g_0'(x)-g_1'(x)\right] .  \label{c10}
\end{equation}
Then \eqref{c7} can be derived directly from \eqref{c9} and \eqref{c10}.
Similarly, we can derive \eqref{c8}. Note that if $X=\tilde{X}$, then 
\eqref{c5} yields $q_{k}(x)\equiv \tilde{q}_{k}(x)$, $x\in \lbrack 0,\pi ]$. By
 \eqref{c7} \eqref{c8}, we obtain $q_{k}(x)=\tilde{q}_{k}(x)$ a.e. on 
$(0,\pi )$, $\alpha =\tilde{\alpha}$.
\end{proof}

\section{Stability of inverse problem for operator L}

Finally, we also solve the stability problem. Stability is about a
continuity between two metric spaces. To show this continuity, we use a
homeomorphism between these two spaces. These type stability problems were
studied in \cite{Law,Marchenko2,McLaughlin2,Yang}.

\begin{definition} \rm
(i) Let $\mathbb{N}'=\mathbb{N}\backslash \{1\}$. We denote
\[
\Omega :=\big\{ q\in L_1( 0,\pi ) :\int_0^{\pi} q(x)dx=0\big\}, 
\]
$\Sigma :=$ the collection of all double sequences $X$, where
\[
X :=\big\{ x_n^{j}:j=\overline{1,n-1};n\in\mathbb{N}'\big\}
\]
such that $0<x_n^{1}<x_n^2<\dots <x_n^{k-1}
<x_n^k<\frac{\pi }{2}<x_n^{k+1}<\dots <x_n^{n-1}<\pi$ for each $n$.

We call $\Omega $ the space of discontinuous Sturm-Liouville operators and 
$\Sigma $ the space of all admissible sequences. Hence, when $\overline{X}$
is the nodal set associated with $( \overline{q},\alpha ) $ and $
\overline{X}$ is close to $X$ in $\Sigma $, then 
$( \overline{q},\alpha) $ is close to $( q,\alpha ) $.

(ii) Let $X\in \Sigma $ and define $x_n^{0}=0$, $x_n^{n}=1$, 
$L_n^{j}=x_n^{j+1}-x_n^{j}$ and $I_n^{j}=(x_n^{j},x_n^{j+1})$ for
$j=\overline{0,n-1}$. Note that, $L_n^{0}=x_n^{1}$ and 
$L_n^{n-1}=\pi-x_n^{n-1}$. We say $X$ is quasinodal to some $q\in \Omega $ 
if $X$ is an admissible sequence and satisfies the conditions:

(I) As $n\to \infty $ the limit of
\[
( \pi n-\theta ) \Big[ ( \pi n-\theta )
x_n^{j}-\begin{cases}
j\pi ,&\text{if } x_n^{j}\in ( 0,\frac{\pi }{2}) \\
( j+\frac{1}{2}) \pi +\theta ,&\text{if }
 x_n^{j}\in ( \frac{\pi }{2},\pi )
\end{cases}
 \Big] 
\]
exists in $ \mathbb{R}$ for all $j=\overline{1,n-1}$;

(II) $X$ has the following asymptotic uniformity for $j$ as $n\to \infty $,
\[
x_n^{j}=\begin{cases}
\mu _n^{j}+O( \frac{1}{n^2}) ,&\text{if } x_n^{j}\in ( 0,\frac{\pi }{2})\\[4pt]
\gamma _n^{j}+O( \frac{1}{n^2}) ,&\text{if } x_n^{j}\in ( \frac{\pi }{2},\pi)
\end{cases}
\]
for $j=\overline{1,n-1}$.
\end{definition}

\begin{definition} \rm
Suppose that $X$, $\overline{X}\in \Sigma $ with $L_{k}^{n}$ and 
$\overline{L}_{k}^{n}$ as their respective grid lengths. Let
\[
S_n(X,\overline{X})=( \pi n-\theta )
^2\sum_{k=1}^{n-1}| L_{k}^{n}-\overline{L}_{k}^{n}|
\]
and $d_0(X,\overline{X})=\limsup_{n\to \infty}
S_n(X,\overline{X})$ and 
$d_{\Sigma }(X,\overline{X})=\limsup_{n\to \infty } 
\frac{S_n(X,\overline{X})}{1+S_n(X,\overline{X})}$.
\end{definition}

Since the function $f(x)=\frac{x}{1+x}$ is monotonic, we have
\[
d_{\Sigma }(X,\overline{X})=\frac{d_0(X,\overline{X})}{1+d_0(X,\overline{
X})}\in [ 0,\pi] ,
\]
admitting that if $d_0(X,\overline{X})=\infty $, then 
$d_{\Sigma }(X,\overline{X})=1$. Conversely,
\[
d_0(X,\overline{X})=\frac{d_{\Sigma }(X,\overline{X})}{1-d_{\Sigma }(X,
\overline{X})}.
\]

After the following theorem, we can say that inverse nodal problem for operator 
$L$ is stable.

\begin{theorem} \label{thm4}
The matric spaces $(\Omega ,\|\cdot\| _1)$ and 
$(\Sigma/\sim ,d_{\Sigma })$ are homeomorphic to each other. 
Here, $\sim $ is the equivalence relation induced by $d_{\Sigma }$. Furthermore
\[
\| q-\overline{q}\| _1=\frac{2d_{\Sigma }(X,\overline{X})
}{1-d_{\Sigma }(X,\overline{X})},
\]
where $d_{\Sigma }(X,\overline{X})<1$.
\end{theorem}

\begin{proof}
According to  Theorem \ref{thm3}, using the definition of norm on $L_1$ for the
potential functions, we obtain
\begin{equation}  \label{d1}
\begin{aligned}
\| q-\overline{q}\| _1 
&\leq 2( n-\frac{\theta }{\pi }) ^{3}\int_0^\pi | L_n^{j}-
\overline{L}_n^{\overline{j}}| dx+o(1) \\
&\leq 2( n-\frac{\theta }{\pi }) ^{3}\overset{\pi }{\underset{0}{
\int }}| L_n^{j}-\overline{L}_n^{j}| dx+2( n-
\frac{\theta }{\pi }) ^{3}\int_0^\pi
| \overline{L}_n^{j}-\overline{L}_n^{\overline{j}}|
dx+o(1)
\end{aligned}
\end{equation}
Here, the integrals in the second and first terms can be written as
\[
\int_0^\pi | \overline{L}_n^{j}-\overline{
L}_n^{\overline{j}}| dx=o( \frac{1}{n^{3}})
\]
and
\[
\int_0^\pi | L_n^{j}-\overline{L}
_n^{j}| dx=\frac{1}{( \pi n-\theta ) }
\sum_{k=1}^{n-1} | L_{k}^{n}-\overline{L}_{k}^{n}| ,
\]
respectively. If we consider these equalities in \eqref{d1}, we obtain
\begin{equation}
\| q-\overline{q}\| _1\leq 2( \pi n-\theta )
^2\sum_{k=1}^{n-1}| L_{k}^{n}-\overline{L}_{k}^{n}|
+o(1)=2S_n(X,\overline{X})+o(1).  \label{d2}
\end{equation}
Similarly, we can easily obtain
\begin{equation}
\| q-\overline{q}\| _1\geq 2S_n(X,\overline{X})+o(1)
\label{d3}
\end{equation}
The proof is complete after by taking limits in \eqref{d2} and \eqref{d3} as 
$n\to \infty $.
\end{proof}

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\end{document}