The inverse scattering problem at fixed energy based on the Marchenko equation for an auxiliary Sturm-Liouville operator

TL;DR

This paper introduces a novel method for solving the quantum inverse scattering problem at fixed energy by connecting phase shifts to an auxiliary Sturm-Liouville problem and using the Marchenko equation to recover the potential.

Contribution

It presents a new approach that links fixed energy phase shifts to an auxiliary Sturm-Liouville problem and employs the Marchenko equation for potential reconstruction.

Findings

New method for inverse scattering at fixed energy

Relates phase shifts to Sturm-Liouville problem

Uses Marchenko equation for potential recovery

Abstract

A new approach is proposed to the solution of the quantum mechanical inverse scattering problem at fixed energy. The method relates the fixed energy phase shifts to those arising in an auxiliary Sturm-Liouville problem via the interpolation theory of the Weyl-Titchmarsh m-function. Then a Marchenko equation is solved to obtain the potential.