Unified Wronskian formulation of inverse scattering with supersymmetric quantum mechanics

TL;DR

This paper introduces a unified Wronskian-based approach to inverse scattering in supersymmetric quantum mechanics, simplifying the construction of phase-equivalent potentials and solving the fixed-angular-momentum inversion problem.

Contribution

It presents a novel unified Wronskian formulation that combines two steps of SUSYQM inversion into a single, elegant method, improving the construction of phase-equivalent potentials.

Findings

Successfully applied to neutron-proton scattering data

Provides a complete solution to the fixed-angular-momentum inversion problem

Numerical implementation demonstrates practical effectiveness

Abstract

The Wronskian formulation of supersymmetric quantum mechanics (SUSYQM) confluent transformation pairs is applied to the construction of phase-equivalent potentials with different bound spectra, replacing integral formulas. This allows to unify the two steps of a SUSYQM inversion scheme consisting in (i) the construction of a unique bound-state-less potential, possibly singular, from phase-shift inversion by a chain of non-confluent SUSYQM transformations, and (ii) the phase-equivalent addition of bound states by confluent SUSYQM pairs. Both steps are now combined in a single Wronskian formula, providing an elegant complete solution to the fixedangular-momentum inversion problem. This formalism is applied to the inversion of 3S1 and 1S0 neutron-proton data and its numerical implementation is discussed.