Self-energy operator for an electron in an external Coulomb potential. II

Abstract

Relativistic Coulomb Sturmian matrix elements of the operator O≡ln(1−ρ)/ρ, ρ=−[π⋅(1+iσ)⋅π]/m2, in terms of which the self-energy operator for an electron in an external Coulomb potential has been expressed, are studied. The operator O is dealt with on a term by term basis in a Sturmian expansion. Each term of the Sturmian expansion is separated into a part whose matrix elements are analytic functions of Zα, plus a remainder evaluated in closed form by use of the Cauchy residue theorem. All ignorance about the matrix element of the general term in the Sturmian expansion of O is thereby placed entirely in the analytic part, for which an explicit integral representation is derived.