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\begin{document} \vspace*{3mm}
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\title{Inverse Problem for Interior Spectral Data
of the Sturm-Liouville Operator}
\author{\sc Kiyoshi MOCHIZUKI\footnote{Department of Mathematics, Tokyo
Metropolitan University, Hachioji, Tokyo 192-0397, Japan.E-mail:
mochizuk@comp.metro-u.ac.jp}
AND Igor TROOSHIN\footnote{Precision Mechanics and Control Institute
of Russian Academy of Sciences, 24, Rabochaya, Saratov,410028,
Russia}}
%\begin{center}
%{Department of Mathematics, Tokyo Metropolitan University\\
%Hachioji, Tokyo 192-0397, Japan}
%\end{center}

\maketitle


We  consider the Sturm-Liouville operator $L$ generated by the
differential expression $l(y)= -y^{\prime\prime} + q(x)y \; \;
(0 \le  x \le 1) $ subject to the boundary conditions $y(0)
\cos \alpha +y^{\prime } (0) \sin \alpha =0, \; 0 \le
\alpha < \pi, $ and $y(1) \cos \beta  +y^{\prime } (1) \sin
\beta =0, \; 0 \le \beta < \pi,$ where $q(x)  \in L (0,1) $
is a real-valued  function.

As is well known, the operator $L$ has a discrete spectrum
consisting of simple eigenvalues $\lambda_n, \; n=0,1,\ldots$.
We denote by $y_n (x),  \; n=0,1,\ldots$ the corresponding
eigenfunctions. The problem of reconstructing a potential $q(x)$
from the spectra of its boundary value problem is known as the
inverse Sturm-Liouville problem. It was first posed by Ambarzumian
(1929).  Later Borg (1946) showed that one spectra does not define
the potential uniquely ,
 to prove the uniqueness of the potential $q(x)$, one needs
  additional
  spectral data. Borg proved
  that the
  spectra of two boundary value problems for an operator with different
  boundary conditions at one end (and identical conditions repeated at
  the other end) uniquely determined the potential $q(x)$,
  i.e. he  took the second spectrum as an additional spectral
  data. Later
  Marchenko (1950)
  substituted the second spectrum by  normalising
  coefficients $\alpha_n =\int_0^1 y_n^2 (x)dx,  n=0,1,2,\ldots$ and
  proved that the eigenvalues $\lambda_n, n=0,1,2,\ldots$ and normalising
  coefficients
  $\alpha_n ,  n=0,1,2,\ldots$
  %$\alpha_n =\int_0^1 y_n^2 (x)dx,  n=0,1,2,\ldots$
  uniquely
  determined the potential $q(x)$.

  The aim of our present work is to investigate a possibility to recover
  potential from the known eigenvalues and some information
  on eigenfunctions in the internal point $b \in (0,1)$. We proved, for
example, the following theorem.

  Let us introduce a second   Sturm-Liouville operator $\tilde{L}$
generated

by differential expression with a real-valued  potential
$\tilde{q} (x)  \in L (0,1) $: $\tilde{l}(y)=
-y^{\prime\prime} + \tilde{q} (x)y \; \; (0 \le  x \le 1) $
subject to the same  boundary conditions . We denote by $\tilde{
\lambda}_n, \; n=0,1,\ldots$ and $\tilde{y}_n (x),  \;
n=0,1,\ldots$ eigenvalues and corresponding eigenfunctions of
operator $\tilde{L}$.



{\bf Theorem }
%\begin{thm}
  {\it If for any} $n=0,1,\ldots,
\lambda_n = \tilde{\lambda}_n, \;  \frac {y^{\prime}_n
  (\frac{1}{2})}{y_n(\frac{1}{2})}
=\frac {\tilde{y}^{\prime}_n
(\frac{1}{2})}{\tilde{y}_n(\frac{1}{2})},
$
{\it then $q(x)=\tilde{q} (x)$ a.e. on the $[0,1]$.}
%\end{thm}\par
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In the case $b \neq 1/2$, the uniqueness of $q(x)$ can be proven
if we require the knowledge of part of the second spectrum.

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\end{document}
