Separable operator expansions for the t-matrix

Abstract

Abstract A general scheme is developed for constructing finite-rank approximations to thet-matrix. The emphasis is on obtaining approximations that are good in a particular context (specifically, in the kernel of few-body problems), rather than on attaining good pointwise accuracy. The scheme in its most general form makes use of two arbitrary sets of expansion functions, and various choices for these functions are considered, and their virtues assessed. Other expansion methods (Weinberg series, unitary pole expansion, and the expansion of Ernst, Shakin and Thaler) can be obtained from the formalism through special choices of the expansion functions, but the methods proposed here, unlike those methods, do not require the solution of an eigenvalue problem. Numerical calculations, using simple choices for the expansion functions, are carried out for the Reid 1S0 soft-core N-N potential. They show that the convergence of the t-matrix expansion can be very good indeed, especially if it is used in the operator context for which it is designed.