Generalized Faddeev Integral Equations for Multiparticle Scattering Amplitudes

The Faddeev integral equations for nonrelativistic three-body scattering amplitudes are generalized to apply to the multiparticle-scattering problem. The equations obtained are similar in structure to the $N$-body formulation of Weinberg and have the same desirable feature, namely, that $\delta$-function singularities in the kernel due to disconnected processes have been removed, so that Fredholm theory may be expected to apply. A possible advantage of the present (Faddeev-type) approach to the $N$-body problem is that the potentials do not appear explicitly; they have been formally eliminated in favor of the two-body scattering amplitudes. In addition to the integral equations for the scattering amplitudes, in all channels, we obtain integral equations for the wave function and for the effective (or optical) potential. A minimum principle for the effective potential obtained previously for $N=3\mathrm{}$ is generalized to arbitrary $N$. To illustrate the possible utility of these equations for a wider class of problems where potentials are not defined, we have shown that the Lee model in the three-particle sector can be reformulated in terms of the Faddeev equations, without introducing field operators.