Some properties of Jost functions for Schrödinger equation with distribution potential

Abstract

The work is devoted to the substantial extension of the space of the potentials in the inverse scattering problem for the linear Schrödinger equation on the real axis. We consider the Schrödinger operator with a potential in the space of generalized functions. This extension includes not only the potential like delta function, but also exotic cases like Cantor functions. In this way we establish the conditions on existence and uniqueness of Jost solutions. We study their analytic properties. We provide some estimates for the Jost solutions and their derivatives. We show that the Schrödinger equation with the distribution potential can be uniformly approximated by the equations with smooth potentials.