Large-order perturbation expansion of three-dimensional Coulomb systems from four-dimensional anharmonic oscillators.

Abstract

The three-dimensional Coulomb system can be mapped onto the four-dimensional harmonic oscillator. Additional perturbations of the type ${\mathit{gr}}^{\mathit{p}}$ [r=(${\mathit{x}}^{2}$+${\mathit{y}}^{2}$+${\mathit{z}}^{2}$${)}^{1/2}$] translate into anharmonic perturbations \ensuremath{\lambda}(${\mathbf{x}}^{2}$${)}^{\mathit{p}+1}$ (${\mathbf{x}}^{2}$=${\mathit{x}}_{1}^{2}$+${\mathit{x}}_{2}^{2}$+${\mathit{x}}_{3}^{2}$+${\mathit{x}}_{4}^{2}$) to the oscillator. We use this observation to relate the large-order behavior of perturbation series of the perturbed Coulomb systems to well-known large-order formulas for anharmonic oscillators. In this way, the leading behavior of Coulomb systems can be understood by simple scaling and symmetry arguments within the oscillator systems. Applications to more physical Stark- and Zeeman-type perturbations are briefly discussed.