inverse sturm-liouville problem with discontinuity conditions

this paper deals with the boundary value problem involving the differential equationbegin{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.