Effective-potential method for relativistic electron-ion scattering.

Abstract

The problem of electron scattering by a heavy hydrogenic ion is formulated relativistically by using a simplified version of hole theory in which the effects of virtual-pair creation and exchange of transverse photons are ignored. An effective-potential formalism is developed and a minimum principle for the effective potential is derived, which is valid in a range of energies below the ionization threshold, through a generalization of the projection-operator procedure developed some time ago in the context of the nonrelativistic scattering problem. A version of the minimum principle is shown to hold, even in cases in which an infinite Rydberg series of autoionizing resonances lies below the scattering energy. Concepts derived from quantum-defect theory are used, in conjuction with the effective-potential formalism, to describe the effect of these resonances on the scattering. Positive-energy projection operators appear naturally in the theory. Their presence ensures that ``variational-collapse`` difficulties are avoided, as required for the validity of the minimum principle. Some care is given to the analysis of different features of the scattering theory associated with the appearance of these projection operators.