Comparison of Plane-Wave Approximations in Potential Scattering

We compare four different plane-wave approximations for scattering by short-range potentials. These four are the first and second Born approximations, the plane-wave Schwinger approximation and a restricted form of Ford's renormalized Born approximation. For the Yukawa and Gaussian potentials, the differential and total cross sections are calculated in the energy range [Formula: see text] (arbitrary units) and for dimensionless coupling constants ranging between 0.1 and 10. The first Born and renormalized Born are 'first-order' approximations, requiring an order of magnitude less calculational labor than the second Born and Schwinger approximations. The results suggest that the restricted renormalized Born approximation is rarely better than the ordinary first Born approximation. The Schwinger plane-wave approximation is always better than the second Born approximation and since these two involve the evaluation of identical expressions, the Schwinger approximation is always to be preferred over the second Born. The Schwinger plane-wave approximation gives quite consistently the best results of the four methods considered, but even it is not be be trusted for calculating differential cross sections at large angles or for reproducing "resonance structure" in total cross sections.