Quantum-mechanical few-body scattering equations with half-on-shell energy-independent subsystem input

New equations are presented for three- and four-body scattering, within the context of nonrelativistic quantum mechanics and a Hamiltonian scattering theory. For the three-body case Faddeev-type equations are presented which, although obtained from the rigorous Faddeev theory, only require two-body bound state wave functions and half-off-shell transition amplitudes as input. In addition, their ''effective potentials'' are independent of the three-body energy, and can easily be made real after an angular momentum decomposition. The equations are formulated in terms of physical transition amplitudes for three-body processes, except that in the breakup case the partial-wave amplitudes differ from the corresponding full amplitudes by a Watson final-state-interaction factor. New equations are also presented for four-body scattering, obtained by generalizing our three-body formalism to the four-body case. These equations, although equivalent to those of Faddeev-Yakubovskii, are expressed in terms of singularity-free transition amplitudes, and their energy-independent effective potentials require only half-on-shell subsystem transition amplitudes (and bound state wavefunctions) as input. However, due to the detailed index structure of the Faddeev-Yakubovskii formalism, the result of the generalization is considerably more complicated than in the three-body case.