Fixed energy potentials through an auxiliary inverse eigenvalue problem

TL;DR

This paper extends an inverse scattering method based on an auxiliary inverse Sturm-Liouville problem to reconstruct spherically symmetric fixed energy potentials in 3D Schrödinger equations, introducing parameterized procedures and solution techniques.

Contribution

It generalizes the existing method by introducing two parameters affecting bound states and proposes new solution techniques for the inverse spectral problem.

Findings

Successfully reconstructs potentials from theoretical and experimental phase shifts.

Introduces parameter control to reduce bound states in the auxiliary problem.

Provides multiple solution methods including exact matrix inversion and bound state management.

Abstract

An inverse scattering method based on an auxiliary inverse Sturm-Liouville problem recently proposed by Horv\'ath and Apagyi [Mod. Phys. Lett. B 22, 2137 (2008)] is examined in various aspects and developed further to (re)construct spherically symmetric fixed energy potentials of compact support realized in the three-dimensional Schr\"odinger equation. The method is generalized to obtain a family of inverse procedures characterized by two parameters originating, respectively, from the Liouville transformation and the solution of the inverse Sturm-Liouville problem. Both parameters affect the bound states arising in the auxiliary inverse spectral problem and one of them enables to reduce their number which is assessed by a simple method. Various solution techniques of the underlying moment problem are proposed including exact Cauchy matrix inversion method, usage of spurious bound state and assessment of the number of bound states. Examples include (re)productions of potentials from phase shifts known theoretically or derived from scattering experiments.