Analytic T matrices for Coulomb plus rational separable potentials

Abstract

The l=0 partial wave projected Coulomb off‐shell T matrix T_c,l=0 in momentum representation is obtained in closed form. Problems existing in the literature concerning the half‐ and on‐shell behavior of T_c and T_c,l are discussed and clarified by means of explicit formulas. The remaining derivations in this paper are based on T_c,l=0. We consider the class of N‐term separable potentials where the form factors are rational functions of p² (in momentum representation). We prove that the l=0 T matrix corresponding to the Coulomb potential plus any such so‐called rational separable potential has a very simple form, namely, it can be written in terms of rational functions and the (simple) hypergeometric function with parameters (1, iγ; 1+iγ), where γ is the well‐known Coulomb parameter. Explicit analytic formulas are derived for a number of simple members of the class, the Yamaguchi potential being one of them. In this particular case the expressions of Zachary and of Bajzer are reproduced who used a method based on the O₄ symmetry.