The problem of generating nonrelativistic three-body scattering amplitudes which satisfy unitarity exactly, at all energies, is studied in the context of an effective-potential theory. It is shown how the nonlinear unitarity relations can be replaced by linear integral equations, simpler in structure than the original Faddeev equations, such that for any set of input amplitudes satisfying standing-wave boundary conditions the output will be unitary. Variational principles for these input amplitudes are derived from the Faddeev equations which define them, so that approximations can be systematically improved. Attention is drawn to a particular class of approximations for which the integral equations to be solved are all of the twobody type. These approximations have the additional virtue that the trial functions which enter into the Schwinger-type variational expression are square-integrable. This property allows for a choice of trial functions based on an "effective-range" type of argument. The Schwinger principle can then be though of as providing an analytic continuation of the effective potential from an energy domain below the breakup threshold to a limited range above it.

• Received 22 May 1972

DOI:https://doi.org/10.1103/PhysRevD.7.1113