Document Type : Research Paper


1 Department of Mathematics, Faculty of Science, University of Maragheh, P.O. Box 55181-83111, Maragheh, Iran.

2 Faculty of Mathematical Sciences, University of Tabriz, Tabriz 51664, Iran.


This paper deals with the boundary value problem involving the differential equation \begin{equation*}     \ell y:=-y''+qy=\lambda y,  \end{equation*}  subject to the standard boundary conditions along with the following discontinuity  conditions at a point $a\in (0,\pi)$  \begin{equation*}     y(a+0)=a_1 y(a-0),\quad y'(a+0)=a_1^{-1}y'(a-0)+a_2 y(a-0), \end{equation*} where $q(x),  \ a_1 ,\ a_2$ are  real, $q\in L^{2}(0,\pi)$ and $\lambda$ is a parameter independent of $x$. We develop the Hochestadt's result based on the transformation operator for inverse Sturm-Liouville problem when there are discontinuous conditions.  Furthermore, we establish a formula for $q(x) - \tilde{q}(x)$  in the finite interval where $q(x)$ and $\tilde{q}(x)$ are analogous functions.


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