Separable-Expansion Method for the Three-Nucleon System

A previously developed separable expansion for the two-particle $T$ matrix is applied to calculating the properties of the three-nucleon system, using spin-dependent static central potentials. The force model is identical to one that was previously employed by Brayshaw and Buck. Results are presented for the triton binding energy, the doublet and quartet scattering lengths, and for the doublet and quartet $s$-wave phase shifts below and above the three-body breakup threshold. It is found that two or three terms of the expansion in each two-particle spin state give satisfactory convergence in the doublet state of the three-nucleon system. Only one term is needed for the quartet state. The over-all agreement with experiment is good, but some possibly significant discrepancies are found between the theoretical and experimental quartet phase shifts above the breakup threshold. Also, it is shown that a two-particle $T$ matrix or an approximation to one which satisfies the off-shell unitarity relation and has the correct behavior at the bound-state poles leads to three-particle scattering amplitudes which satisfy the three-particle off-shell unitarity relations. The proof does not depend on the existence of a two-particle energy-independent Hermitian potential.