On the Jost solutions for a class of Schrödinger equations with piecewise constant coefficients

In this paper, the new integral representations for the Jost solutions of the one-dimensional Schrödinger equation with the piecewise-constant leading coefficient are obtained. The connections, obtained between the kernel functions of the integral representations and the potential function of the Schrödinger equation, enable to solve the inverse scattering problem on the entire real line.