SeparabletMatrices and the Three-Body Problem

A new, separable expansion of the two-body $t$ matrix is presented. Such an expansion reduces the Faddeev equations to coupled equations in one continuous variable. The leading term of the expansion is the separable approximation suggested by Kowalski and Noyes. Any separable approximation to the $t$ matrix obtained by truncating the expansion is exact half-off the energy shell, exactly satisfies the off-shell unitarity relation, and duplicates the exact $t$ matrix in the neighborhood of two-body bound-state and resonance energies. The rate of convergence of the expansion is tested by means of examples. Two terms give a good approximation to the $S$-wave part of the $t$ matrix arising from a square-well potential, which fits the low-energy two-nucleon scattering data in an average way. It is also shown that the first term of the expansion gives a very good approximation to the $t$ matrix arising from a pure hard-core potential. Results are given for the binding energy of a system of three identical spinless particles interacting via square-well potentials with and without hard cores. The two potentials have the same scattering length and effective range. The potential without a core produces a three-body binding energy of 10.1 MeV; the potential with a core produces a three-body binding energy of 8.00 MeV.