Generalized Variational Bounds on the Positron-Hydrogen Reaction Matrix. I. The Coupled-Static Approximation

The theory of generalized variational bounds is applied to two-channel collisions of positrons on hydrogen leading to positronium formation. The formalism is developed in detail for this problem, and the zero partial-wave solution is obtained in the coupled-static approximation. The nonlocal coupling kernels and the behavior of the wave functions are exhibited in detail. Eigenphase shifts and coupling parameters, and the stripping and pickup cross sections, as well as the elastic amplitudes for positron-hydrogen and positronium-proton collisions, are studied as functions of scattering energies. The unitarity limits are exhibited, one of which is nearly saturated in the Ps-p channel. A useful iteration procedure for solving the coupled equations with large coupling kernels is developed.