Numerical quasiparticle representation of the nucleon-nucleonTmatrix

Abstract

In this paper we develop a practical ``hybrid'' numerical representation of the nucleon-nucleon T matrix. Part of the expression contains nonseparable terms which are easily calculated, and the rest consists of a separable representation of small rank in terms of Weinberg states. The method rests on the observation that when a set of positive-energy Weinberg states is used to obtain a separable representation of the potential V then the residue \ensuremath{\Delta}V, due to the basis-set truncation, has very special properties: (1) The contribution \ensuremath{\Delta}T to the T matrix due to \ensuremath{\Delta}V is identical to the undistorted T matrix for \ensuremath{\Delta}V alone, ${\mathit{T}}_{\mathrm{\ensuremath{\Delta}}}$, i.e., the usual Moeller distortion factors in the two-potential formula are unity in this case. (2) A perturbative-iterative treatment of ${\mathit{T}}_{\mathrm{\ensuremath{\Delta}}}$ in powers of \ensuremath{\Delta}V is found to be equivalent to the finite-rank representation of operators of the type T, T-V, T-V-${\mathit{VG}}_{0}$V, and so on. This equivalence has both practical and theoretical implications. On the one hand, it provides a reliable method for calculating the T matrix and for analyzing the corresponding accuracy properties. On the other hand, a connection is established between each order of the quasiparticle method and the different variational principles which underlie the finite-rank representation of operators such as T, T-V, T-V-${\mathit{VG}}_{0}$V, etc. Numerical examples are provided for two different nucleon-nucleon singlet potentials (Reid soft core and Malfliet-Tjon). In the Malfliet-Tjon case, for instance, two Weinberg states are found to be sufficient in order to give an accuracy of 0.1% for the calculation of T-V-${\mathit{VG}}_{0}$V, while for T-V and T the same two states give an accuracy of 1% and 10%, respectively, in an interval of 6 ${\mathrm{fm}}^{\mathrm{\ensuremath{-}}1}$ around the on-shell point.