Dispersion Relations for Three-Particle Scattering Amplitudes. II

We continue our discussion of the scattering of three nonrelativistic spinless particles interacting via two-body Yukawa potentials. The on-energy-shell T matrix is studied as a function of the total center-of-mass energy E for fixed physical values of the vectors yᵢ=kᵢ(2mᵢE)^-1/2, yᵢ^'=kᵢ^'(2mᵢE)^-1/2, $i=1,2,3$. Here kᵢ and kᵢ^' are the initial and final momenta of the particles, respectively, and mᵢ are the masses. We show that $T(E)$ can be written as the ratio of two Fredholm series, each of which is uniformly convergent with respect to E for all values of E on the physical sheet including the real axis. Since we have previously seen that each term in these series satisfies a dispersion relation in E with no complex singularities, it follws that the full three-particle amplitude satisfies such a dispersion relation.