Arrangement-channel quantum mechanics: A general time-dependent formalism for multiparticle scattering

A general time-dependent formulation of many-body, multichannel scattering using the channel-coupling-array theory is presented. The formalism is based on the use of the channel-component states, previously introduced by Hahn, Kouri, and Levin. These states obey a set of non-Hermitian matrix equations, the non-Hermiticity arising from the presence of the channel-coupling array $W$. Despite the lack of Hermiticity, it is shown that the eigenvalues will always be real if the channel-component states obey a fixed-phase convention under time reversal. Use of the channel-component states leads in a straightforward way to an interaction picture and a single $S$ operator. The analog of the usual two-body (single-channel) result connecting the time-dependent and time-independent descriptions is demonstrated. It is also shown that the channel-component states have the remarkable property that only the component in channel $j$ gives rise to (two-body) outgoing waves in that channel: Components in channels $m\ne j$ do not yield outgoing waves in channel $j$. This is explicitly seen from the time-dependent development, but is inherent in the time-independent description. Such a property is known for the Faddeev decomposition in the three-body case; the present work extends and generalizes this result for the case of an arbitrary number of particles. In particular, the present formulation is seen to yield precisely the Faddeev equations for the three-body problem.