Bound-state methods for low-energy electron-ion scattering

An effective-potential formalism, previously developed for electron scattering by a neutral target, is extended to apply to electron-ion scattering, with the requirement of antisymmetrization now accounted for explicitly. A minimum principle for the effective potential is derived, valid for scattering below the ionization threshold and applicable when, as is usually the case, the target wave functions are imprecisely known. The basis for the minimum principle is the Rayleigh-Ritz property that is satisfied by the modified Hamiltonian in terms of which the effective potential is defined. An analysis of single-channel, zero-energy scattering for a particular partial wave is presented; it is based on the effective-potential formalism and leads to an absolute definition of the zero-energy phase shift δ(0) of the form δ(0)=μ(∞)π, where μ(n) is the quantum defect of the nth energy level. This result may be thought of as an extension of Levinson close-quote s theorem for scattering by short-range potentials. copyright 1996 The American Physical Society