Variational Approach to the Faddeev Equations

A variational principle is formulated as a means for obtaining approximate solutions to the Faddeev integral equations for the nonrelativistic three-body scattering problem. Bound states are described by the homogeneous version of the Faddeev equations. In this case a minimum principle for the binding energies is obtained in close analogy with the familiar Rayleigh-Ritz method. An effective potential is defined as the solution of a set of modified Faddeev equations. The variational principle therefore applies to the effective potential. This is shown to be a minimum principle for the eigenphase shifts for energies below the three-particle breakup threshold. Some formal applications of the variational principle are described. These include derivations, within the Faddeev formalism, of a normalization condition for bound-state wave functions, a distorted-wave theory, and a perturbation theory for discrete states.