Solution of the bethe-goldstone equation in a rapidly convergent polynomial expansion

Abstract

We demonstrate a simple and accurate method for obtaining the Bethe-Goldstone effective interaction G, starting from the reference interaction G R of Bethe, Brandow and Petschek. The usefulness of the method depends on only two properties of the effective interaction: (a) the intermediate-state spectrum converges to the reference spectrum for states of large single-particle momentum, and (b) the effective interaction has a smooth dependence on the relative momentum over the range of momenta for which the Bethe-Goldstone propagator is different from the reference propagator. These well-established properties allow us to solve the Bethe-Goldstone equation for the difference G− G R within a subspace containing only states of modest momentum. Within this subspace, an expansion in a complete set of orthonormal polynomials may be truncated at a low order, without sacrificing numerical accuracy. Accuracy of a few percent is obtained using a basis of quadratic polynomials, which for the angle-average Pauli operator requires inverting a 3 × 3 matrix (or 6 × 6 in the case of tensor-coupled channels). The accuracy of the results increases rapidly with the order of the polynomials used. We study the accuracy of the angle-average approximation by comparing it with explicit solutions including the angular couplings. The maximal errors introduced (in the deuteron channel) are comparable to the maximal errors in truncating our orthogonal expansion at quadratic order, except for very small relative momenta.