Hydrogen atom with a Yukawa potential: Perturbation theory and continued-fractions–Padé approximants at large order

A simple power-series method is developed to calculate to large order the Rayleigh-Schrödinger perturbation expansions for energy levels of a hydrogen atom with a Yukawa-type screened Coulomb potential. Perturbation series for the 1s, 2s, and 2p levels, shown not to be of the Stieltjes type, are calculated to 100th order. Nevertheless, the poles of the Padé approximants to these series generally avoid the region of the positive real axis 0<λ<${\lambda }^{\mathrm{*}}$, where ${\lambda }^{\mathrm{*}}$ represents the coupling constant threshold. As a result, the Padé sums afford accurate approximations to E(λ) in this domain. The continued-fraction representations to these perturbation series have been accurately calculated to large (100th) order and demonstrate a curious ‘‘quasioscillatory,’’ but non-Stieltjes, behavior. Accurate values of E(λ) as well as ${\lambda }^{\mathrm{*}}$ for the 1s, 2s, and 2p levels are reported.