Inverse Scattering Problem for the Sturm-Liouville Equation with Infinite Range of Discontinuous Conditions

Abstract

In this paper, we construct the new integral representation of the Jost solution of Sturm-Liouville equation with impuls in the semi axis $[0,+\infty )$ and we give this type of relation, examine the properties of the Kernel function and their partial derivatives with $x$ and $\ t$, constructed integral representation and obtain the partial differential equation provided by this Kernel function. Finally, in the paper we prove uniqueness of the determination of the potential by the scattering data.