Variational Bounds in N-Particle Scattering Using the Faddeev–Yakubovskii Equations: Deuteron-Deuteron S=2 Scattering

Abstract

A variational-bound formulation of the N-particle scattering problem has been developed based on the Yakubovskii–Faddeev chain-of-partition equations. It is shown that the scattering amplitude for the elastic, rearrangement or break-up processes satisfies a Lippmann–Schwinger type integral equation in which the kernel integration is over the open channels and the closed channels enter through the effective potential. In the case where only two (three)-cluster open channels are allowed, the integral equation for the transition amplitude involves integration over only one (two) momentum vector(s). A variational estimate for the effective potential input to the integral equation is obtained when the closed channel partition Green’s functions are estimated variationally. It is shown that the variational estimates for the closed-channel Green’s function also provide upper and lower bounds that can be used as a subsidiary extremum principle to determine optimum parameters in the trial function. Several methods are provided for simplifying the determination of the effective potential. The inclusion of Coulomb potentials in the Yakubovskii–Faddeev (YF) chain-of-partition formalism is also described. In this approach the inter-particle potential is not assumed to be separable as in typical quasi-particle schemes. The many-body dependence of the effective potential is included via expectation values involving an inter-particle potential and spatially decaying trial functions. The N-body scattering problem is therefore reduced to: (a) solving a two-body scattering problem (three-body in the case of break-up) and (b) a bound-state type calculation to determine the effective potential. As an initial application, the method is applied to the case of low-energy elastic deuteron-deuteron scattering (including the Coulomb force) and compared to a recent cluster-reduction calculation.