Unitarity and the scattering phase shifts for inversion studies

The first requirement in application of most global inverse scattering methods seeking (local) interaction potentials from scattering data, is to specify the scattering phase shifts, δl, or equivalently, the S matrix, Sl ≡ e 2iδl. With fixed energy inverse scattering problems, knowledge of those phase shifts for all physical values of the angular momentum l allow application of the Newton-Sabatier scheme as modified by Scheid and his collaborators. The aim of this work has been to obtain phase shifts by a more global means, namely by using the unitarity (generalized flux) theorem in application to real cases. Below the first noneleastic threshold and for the scattering of spinless particles (or if one simply ignores any spin dependent attributes in the scattering), this theorem translates to an integral equation to determine the phase function of the scattering amplitude f(θ) = √dσ/dΩ(θ) exp (i φ (θ)). A solution to that integral equation not only exists but also, under particular conditions, it is unique. Furthermore, with one of those conditions (hereafter defined as the Martin condition) being valid, an iterative method of Newton gives that solution. When conditions for uniques and stability of solution by an iterated fixed point method are not met, a numerical procedure has been proposed. But whatever be the chosen method of solution, the cross section data dσ/dΩ(θ) must be known at all (real) scattering angles. With actual data sets then, interpolation and extrapolation must be used. For scattering in which spin-orbit interactions are important, the generalized flux theorem leads to coupled integral equations for two unknown phase functions, and while there are still conditions for uniqueness of the solution, those conditions are now rather complex. 10 refs., 6 figs