Off-shell sensitivity in relativistic Schrödinger and Klein-Gordon optical models of pion-nucleus interactions

The elastic scattering of pions from finite nuclei is examined theoretically within the context of the impulse approximation to the optical potential for the relativistic Schrödinger and Klein-Gordon equations. The kinematical relationships between potentials and wave functions are specified by considerations similar to those discussed by Cammarata and Banerjee. We find: (1) At low energy ($T_{\pi }\lesssim 50$ MeV) there is a large difference between the solutions of the two equations. (2) At low energy there is a strong sensitivity to the range of the pion-nucleon interaction, the Klein-Gordon equation being dramatically sensitive. (3) Near the (3-3) resonance there are small differences between the solutions of the two equations and there is only a moderate sensitivity to the range of the interaction. The low energy results are explained in terms of the interaction of pions with nuclear matter. The resonance energy results are explained in simple geometric terms. The large differences between wave functions near the resonance region found by Siciliano and Thaler are not found if one uses the relationships proposed here, although we do find significant differences at lower energies.