Operator Reduction of Three-Body Scattering Equations

Operator multipliers of the Lippmann-Schwinger (LS) equation are used to obtain an uncoupled equation with calculable and unique solutions. The multiplier method is used as a basis for determining the relation between various formulations of the three-body problem. The formal but calculable solutions of the formulations of Faddeev, Lovelace, Rosenberg, Noble, and Newton are found to be identical in the sense that the different formulations are completely equivalent because they all use the same multiplier. They are also identical to the solution of the equation obtained by multiplying the L-S equation by an operator whose inverse exists. We also present an equation for the exact three-body bound-state wave function and put it into calculable form with the use of a multiplier. In a calculation that uses an incomplete set of basis states (such as in the shell model), we find on rigorous grounds that it is appropriate to use the $t$ operator of the residual interaction rather than the residual interaction itself. To indicate the wide usefulness of the multiplier method, the exact distorted-wave formulation is obtained and put into calculable form with the use of a multiplier.