Properties of an analytically solvable model of multichannel scattering

A study is made of an analytically solvable model of multichannel scattering, leading to the derivation of various properties of the solutions. In this model the intrinsic potential in each channel and the interchannel coupling operators are all taken to be separable interactions. In an especially simple version of the model an assumption is also made concerning a factorization property of the coupling coefficients that pertain to these interactions. Extremely simple explicit expressions are obtained for the elastic- and the inelastic-scattering amplitudes, as well as for the Feshbach potential, the effective interaction in the elastic channel owing to the presence of all other channels. The expressions obtained are transparent to analysis, and in the simple version of the model they do not involve the evaluation of inverses or determinants of large matrices, as is the case in general for formal explicit solutions of Fredholm equations. The simple version of the model can also be extended to the case of a continuum of channels, thus being capable of describing systems with open rearrangement or dissociation channels. Within the framework of this model, a rigorous analysis is given for the case of an infinite number of channels, and conditions are obtained for convergence of the close-coupling scheme. Error bounds are obtained for an $N$-channel truncated calculation on an infinite-channel system. Other results obtained include: (a) establishing the asymptotic behavior of the scattering amplitude at extremely high energies, and (b) deriving some especially simple features, "scaling properties," of the scattering amplitudes.