Supersymmetric inversion of effective-range expansions

A complete and consistent inversion technique is proposed to derive an accurate interaction potential from an effective-range function for a given partial wave in the neutral case. First, the effective-range function is Taylor or Padé expanded, which allows high precision fitting of the experimental scattering phase shifts with a minimal number of parameters on a large energy range. Second, the corresponding poles of the scattering matrix are extracted in the complex wave-number plane. Third, the interaction potential is constructed with supersymmetric transformations of the radial Schrödinger equation. As an illustration, the method is applied to the experimental phase shifts of the neutron-proton elastic scattering in the ${}^1{S}_0$ and ${}^1{D}_2$ channels on the $[0–350]$ MeV laboratory energy interval.

Figure 1

(a) Experimental neutron-proton singlet $S$-wave phase shifts [2] and theoretical phase shifts [Eq. (8)] deduced from effective-range-function fits. (b) Same as in (a) but for singlet $D$-wave.

Figure 2

Effective-range functions corresponding to the phase shifts of Fig. 1. (a) $S$-wave: experiment [2], three-term Taylor expansion, [$3/2$] Padé expansion [left (right) axis is used for the left (right) part of the figure]; (b) $D$-wave: experiment [2], three-term Taylor expansion.

Figure 3

(a) Neutron-proton inversion potentials for the singlet $S$- and $D$-waves (central and effective potentials). (b) Plot of the asymptotic behavior of the central $S$- and $D$-wave potential together with one pion exchange potential (OPEP) in logarithmic scale. Both figures are plotted after multiplying ${\hslash }^2/\left(2\mu \right)=41.47$ MeV ${\mathrm{fm}}^2$ to the corresponding potentials.