Secular perturbation theory of long-range interactions

Solutions of Schrödinger’s equation for systems interacting with long-range power-law potentials, ${\mathit{V}}_{\mathrm{int}}$(r)∝${\mathit{r}}^{\mathrm{-}\mathit{n}}$ with n≥2, are cast in the form of a power series in ${\mathit{V}}_{\mathrm{int}}$. 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/${\mathit{r}}^4$ polarization potential. Near thresholds, the method is applicable to all power-law potentials with n≥2, and to their combinations, as well as to multichannel problems involving anisotropic potentials.