Nucleon-nucleon potentials from Gel’fand-Levitan and Marchenko inversions

Inverse scattering theory for the uncoupled channels of neutron-proton systems is developed from both the Gel’fand-Levitan and Marchenko fundamental equations. A most practical form of that theory is deduced by starting with a rational function representation of the phase shift data. By using Padé approximants for the exponential function $e^z$, rational function representations for the scattering, Jost and spectral functions result. They facilitate accurate numerical solutions of both fundamental equations and from which local, energy-independent channel potentials are obtained. The Reid soft-core potential phase shifts when used as input data give potentials in excellent agreement with the original. Inversion potentials have also been generated by using as input empirical phase shifts and also those from the Paris and Bonn meson exchange interactions. Results are computed for the ${}_{}{}^1{}_{}{}^{}\mathrm{S}_0^{}{}_{}{}^{}$, ${}_{}{}^3{}_{}{}^{}\mathrm{P}_0^{}{}_{}{}^{}$, ${}_{}{}^3{}_{}{}^{}\mathrm{P}_1^{}{}_{}{}^{}$, ${}_{}{}^1{}_{}{}^{}\mathrm{D}_2^{}{}_{}{}^{}$, and ${}_{}{}^1{}_{}{}^{}\mathrm{P}_1^{}{}_{}{}^{}$ channels specifically and the potentials are transformed into central, tensor, spin-orbit, and quadratic spin-orbit radial form factors.