On the Inverse Scattering Problem for the One-Dimensional Schrödinger Equation with Increasing Potential

We consider a one-dimensional Schrödinger equation on the entire axis whose potential rapidly decreases on the left-hand side and infinitely increases on the right-hand side. By the method of transformation operators, we study the inverse scattering problem. We establish conditions for the scattering data under which the inverse problem is solvable. The main Marchenko-type integral equations are investigated and their unique solvability is established.