A method for relating N-body scattering- and bound-state equations of motion to effective-few-body equations involving either energy-dependent or energy-independent potentials is developed.
The existence of an energy-independent potential for problems that satisfy time-translation invariance is proven. A detailed derivation of the equations that relate the three-body problem to an effective-two-body problem is presented:
Equations for an energy-dependent effective-two-body potential and a class of energy-independent effective-two-body potentials are derived, a general expression for the class of energy-independent potentials is given, it is shown that these potentials are proportional to the inverse of the effective-two-body inelasticity parameter, and an expression for a Hermitian energy-independent potential is derived.
A derivation of the equations that relate the N-body problem to an effective three-body problem is also presented. This reduction is shown to generate a three-body energy-dependent potential, and further reduction to equations with energy-independent potentials is shown to generate an additional three-body potential.
As an example, linear integral equations for equivalent two- and three-body energy-independent potentials are obtained from a set of linear energy-dependent two-body potentials. These equations are solved exactly when the energy-dependent potentials are also local. Perturbative expansion of the equations developed here is shown to be the same as obtained by folded-diagram techniques.