Two-tier formulation of multichannel scattering theory and hypervirial theorems

Complex dynamics of a multichannel scattering may be treated by a variational procedure. But the conventional variational principles are not readily applicable because of (i) the intrinsic difficulties of unstable fluctuations in the calculated amplitudes as functions of nonlinear parameters and (ii) lack of criteria to optimize the solutions. A two-tier theory is formulated in which the complex dynamical mixing and the asymptotic channel sector are treated separately, but as a coupled system, such that the instability problem (i) is resolved naturally in a mathematically consistent way, even when most of the weakly coupled open channels are neglected. The resulting solution is stable, but not necessarily optimal. Modified forms of hypervirial theorems are introduced to optimize the solutions, thus rectifying the shortcoming (ii). Thus, the reformulated theory for the scattering states, coupled with a properly chosen hypervirial theorem, can be applied effectively to many-body, multichannel scattering systems.