Inverse Scattering Problem for Sturm--Liouville Operators

TL;DR

This paper proves that Sturm-Liouville operators with specific potential behaviors are uniquely determined by their scattering data, and provides a method to recover the operator via a linear integral equation.

Contribution

It establishes the uniqueness of inverse scattering for a class of Sturm-Liouville operators and offers a practical recovery method.

Findings

Uniqueness of the inverse scattering problem for the specified operators.

Reduction of the recovery process to solving a linear integral equation.

Confirmation that scattering data fully determine the operator.

Abstract

On the space $L^{2}(\mathbb{R})$ the Sturm-Liouville operator $L$ with certain behavior of the potential at infinity is considered. It is proved that $L$ is uniquely determined by its scattering data. The recovery of $L$ is reduced to the solving of a certain linear integral equation.