Secular perturbation theory of long-range interactions.

Abstract

Solutions of Schroedinger's equation for systems interacting with long-range power-law potentials, [ital V][sub int]([ital r])[proportional to][ital r][sup [minus][ital n]] with [ital n][ge]2, are cast in the form of a power series in [ital V][sub int]. These perturbations lead to secular divergences that are eliminated by renormalizing the angular-momentum quantum number. Long-known perturbation techniques in classical mechanics and quantum-field theory yield modified effective-range formulas, quantum-defect functions, and solutions of close-coupling equations for wave propagation in long-range fields. As an example, we extract in first-order perturbation theory a modified effective-range expansion for the phase shift of an electron interacting with an atomic 1/[ital r][sup 4] polarization potential. Near thresholds, the method is applicable to all power-law potentials with [ital n][ge]2, and to their combinations, as well as to multichannel problems involving anisotropic potentials.