Variational method for off-shell three- body amplitudes

Abstract

A variational scheme of the Schwinger type, used successfully in a previous paper for calculating two-body $t$ matrix elements of all kinds (on shell, half shell, and off shell), is here applied to the three-body collision problem. Numerical results are given for the case of the Amado model of the $N\ensuremath{-}d$ system. These results show excellent convergence properties, even for the case of fully off-shell amplitudes in the region where the analytic structure is most troublesome. It is shown theoretically that the three-body variational method, if restricted to the case of physical on-shell amplitudes, is formally equivalent to the several variational principles given by Pieper, Schlessinger, and Wright, though the methods may differ considerably in practice. The formal equivalence is used to show that Pieper's recent perturbative calculations of $N\ensuremath{-}d$ polarization, in which the zero-order problem was approached variationally, have a property not previously noted, namely that even the perturbed solution is fully variational. More generally, a useful degree of cooperation is shown to occur between the present variational method and the perturbation method, making it quite easy to preserve the variational property in a perturbative calculation.[NUCLEAR REACTIONS $^{2}\mathrm{H}(n,n)$, $E=0\ensuremath{-}40$ MeV; calculated off-shell amplitudes. Variational method, separable-potential model.]