Inverse Sturm-Liouville problem with discontinuity conditions

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 $ain (0,pi)$  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, $qin L^{2}(0,pi)$ 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.