Improved Minimum Principle for Single-Channel Scattering

The minimum principle for single-channel scattering obtained previously required the introduction of channel radii for an incident relative kinetic energy $E$ greater than zero. The channel radii, which greatly complicate the numerical calculations, no longer appear in recent formal theories of reactions and they need not appear in the minimum principle formulation. By projecting out all of the open-channel components from the full wave function, it is possible to eliminate the continuum states of the total Hamiltonian $H$ originally present below $E$. The closed-channel Hamiltonian $\mathcal{H}$ has only a discrete spectrum below the inelastic threshold. By appropriately subtracting the contributions from the discrete portion of the spectrum, $\mathcal{H}$ can be used to obtain a minimum principle for $kcot\eta$, where $\eta$ is the phase shift. This minimum principle provides a rigorous criterion for determining the parameters in a trial function and for choosing among the numerical results obtained with different trial functions. The method requires that one solve exactly the static approximation equation, which plays a particular role in uncoupling the closed-channel equations from the open-channel equation. Finally, contact is made with the results previously obtained at zero energy and a generalized Levinson's theorem is briefly considered.