Inverse Sturm-Liouville problem with discontinuity conditions

This paper deals with the boundary value problem involving the differential equation ℓy:=−y′′+qy=λy, subject to the standard boundary conditions along with the following discontinuity conditions at a point a∈(0,π) y(a+0)=a1y(a−0),y′(a+0)=a1−1y′(a−0)+a2y(a−0), where q(x), a1, a2q(x), a1, a2 are real, q∈L2(0,π) and λ 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)−q~(x) in the finite interval where q(x) and q~(x) are analogous functions.