Distorted-Wave Born Expansions

Abstract

A successive approximation procedure is presented for calculating transition amplitudes for direct and rearrangement collision processes. The essential feature of the method is that the base problem about which the solution is expanded exhibits all the bound states (of subsystems of particles) which appear in initial or final scattering states. While no proof is given that our expansion converges, arguments which have been presented indicating that the ordinary Born expansion diverges for a wide class of problems no longer apply to the expansion proposed here. A particular choice of base problem leads directly to the well-known distorted-wave Born approximation. The variational nature of this approximation is exhibited and, as a consequence of the general formalism, a procedure for systematic improvement is presented. Circumstances are described under which the first term in the modified Born expansion has an error of known sign. The use of separable potentials to generate solutions to the base problem is discussed and is shown, in the three-body case, to lead to a model proposed recently by Amado. As a by-product of our work a variational principle for transition amplitudes is developed which is a generalization of the Kohn principle for the two-body elastic amplitude and is valid for any scattering process described by the Schrödinger equation.