Limiting behaviors of off-shell scattering wave functions and T matrices for centrally modified Coulomb potentials

In the formal theory of scattering, the limiting behavior of off-energy-shell scattering wave functions and T matrices as the energy shell is approached is of considerable importance in deriving an impulse approximation to the full scattering amplitudes. On-shell limits do not exist for the pure Coulomb potential, but sufficiently near the shell the off-shell wave function and T matrix are approximated by the continuum eigenstate and scattering amplitude, respectively, multiplied by certain ‘‘off-shell’’ factors. For central potentials which are Coulomb-like at large distances, but modified at smaller radii, it is shown that the on-shell limits again do not exist and the near-shell approximations mimic the pure Coulomb case with the asymptotic charge appearing in the off-shell factors. Numerical results for a realistic atomic potential covering a broad range of defects from the energy shell give a picture of the approximations involved and show that the errors arising from the use of near-shell forms are comparable in magnitude to the energy defects divided by the energy.