Systematic use is made of the Riemann function to find the conditions for the existence of the kernel of the inverse problem at fixed value of the angular momentum. When the reference potential is the centrifugal one, only an l-dependent condition on the potential must be required in the Marchenko case; in the Gel’fand–Levitan case, the condition is l-independent. When the Coulomb potential is included in the reference potential, an exponential decrease of the potential is needed in both instances.
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 Pade approximants for the exponential function e', 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 'So, Po, P l, 'D2, and 'P l channels specifically and the potentials are transformed into central, tensor, spin-orbit, and quadratic spin-orbit radial form factors.
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