Inverse Scattering Problem for the Schr\"odinger Operator with a Separable Potential

TL;DR

This paper demonstrates that the inverse scattering problem for Schrödinger operators with separable potentials can be simplified to solving a singular integral equation, establishing uniqueness of the potential from scattering data.

Contribution

It introduces a reduction of the inverse scattering problem to a singular integral equation and proves the uniqueness of the potential within separable potentials.

Findings

Inverse scattering problem reduces to a singular integral equation.

Uniqueness of the potential is established.

Applicable to Schrödinger operators with separable potentials.

Abstract

We prove that the inverse scattering problem for the Schr\"odinger operator with the separable potential can be reduced to the solving of a certain singular integral equation. We establish the uniqueness of the potential corresponding to given scattering amplitude in the class of separable potentials.