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 ${\mathbf{y}}_i={\mathbf{k}}_i{\left(2\mathrm{}{m}_{i}E\right)}^{-\frac{1}{2}}$, ${{\mathbf{y}}_i}^{\prime }={{\mathbf{k}}_i}^{\prime }{\left(2\mathrm{}{m}_{i}E\right)}^{-\frac{1}{2}}$, $i=1,2,3\mathrm{}$. Here ${\mathbf{k}}_i$ and ${{\mathbf{k}}_i}^{\prime }$ are the initial and final momenta of the particles, respectively, and $m_i$ are the masses. We show that $T\left(E\right)\mathrm{}$ 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.