Minimum Principle for Multi-Channel Scattering

The usual variational principles of scattering theory are simply stationary principles. In a recent series of papers, all restricted to single-channel scattering, conditions were established under which variational principles can be found which are much more powerful in that the functional that represents the variational estimate is not simply stationary in the neighborhood of the exact scattering solution but is rather an extremum. A lower bound was obtained on the single real parameter, $tan{\eta }_c$, which characterizes the scattering in the (uncoupled) channel $c$. The conditions previously established allowed for composite bound states, for the Pauli principle, for arbitrary angular momenta, and for long-range and in particular Coulomb potentials; at nonzero energies, the various potentials had to be truncated. The present paper deals with the extension to multi-channel scattering in which each of the open channels contains only two systems. The bounds are now on linear combinations of the elements of the reactance matrix or of the derivative matrix. The potentials must be truncated in a fashion very similar to that used in Wigner-Eisenbud theory.