Variational separable expansion scheme for two-body Coulomb-scattering problems

We present a separable expansion approximation method for Coulomblike potentials which is based on Schwinger variational principle and uses Coulomb-Sturmian functions as basis states. The new scheme provides faster convergence with respect to our formerly used non-variational approach. Both variational approaches and separable expansion schemes are extensively used in solving few-body problems. Some time ago Adhikari and Tomio presented an unified treatment of separable expansion schemes based on Schwinger variational principles [1]. They proposed various approximation schemes for the transition operator t satisfying the LippmannSchwinger equation t = v + vgt, (1) where v is the potential and g is the free Green’s operator. It was found that using these schemes with appropriate choice of expansion functions a rapid convergence could be obtained. However, in this paper, not a single word was devoted to Coulomb-like potentials. At about the same time in a series of papers [2] a separable expansion scheme for Coulomb-like potentials was proposed by one of us. The Coulomb interaction was kept in the Green’s operator and only the short range part of the potential was subject to the separable expansion. This approach uses Coulomb-Sturmian (CS) functions as basis allowing an exact analytical calculation of the matrix elements of the Coulomb Green’s operator (see [2], and recently [3]). Thereby only the short-range part of the interaction is approximated and the correct Coulomb asymptotic properties of all quantities are guaranteed. The method has also been applied in three-body Faddeev calculations for bound-state and scattering problems with Coulomb interactions [4]. In this paper we generalize one of the separable expansion schemes proposed by Adhikari and Tomio for two-body Coulomb-scattering problems. The expansion schemes in Ref. [1] are based on a finite rank N approximation of the product of operators