Nonautonomous Volterra Series Expansion of the Variable Phase Approximation and its Application to the Nucleon-Nucleon Inverse Scattering Problem

Abstract

In this paper, the non-linear Volterra series expansion is extended and used to describe certain types of non-autonomous differential equations related to the inverse scattering problem in nuclear physics. The non-autonomous Volterra series expansion lets us determine a dynamic, polynomial approximation of the variable phase approximation (VPA), which is used to determine the phase shifts from nuclear potentials through first-order non-linear differential equations. By using the first-order Volterra expansion, a robust approximation is formulated to the inverse scattering problem for weak potentials and/or high energies. The method is then extended with the help of radial basis function neural networks by applying a nonlinear transformation on the measured phase shifts to be able to model the scattering system with a linear approximation given by the first order Volterra expansion. The method is applied to describe the 1S0 NN potentials in neutron+proton scattering below 200 MeV laboratory kinetic energies, giving physically sensible potentials and below $1\%$ averaged relative error between the re-calculated and the measured phase shifts.