Considering scattering amplitudes as integral operators, the formal summation of their perturbation expansions can be done using operator Padé approximants. The lowest-order approximant can be considered a natural improvement of the Bethe-Salpeter equation in ladder approximation if one includes one-loop diagrams other than the direct box graph. The problem of how to evaluate the approximants arises. Variational principles for their calculation have been proposed earlier but yielded ambiguous results. A new variational technique, the method of the variational gradient, is presented which provides a unique though elaborate procedure. The applicability of the method is demonstrated in two cases: a simple potential model and the Bethe-Salpeter equation in ladder approximation for nucleon-nucleon scattering.

• Received 20 October 1980

DOI:https://doi.org/10.1103/PhysRevD.24.1978