Perturbation Theory in High Order for Generalized Charmonium Potentials

This work is concerned with the Rayleigh-Schrödinger perturbation expansions associated with Hamiltonians of the form $\hat{H}\left(\lambda \right)=\frac{1}{2}{\hat{p}}^2-\frac{Z}{r}+\lambda r^m$, $m=1,2,3,\dots$. These Hamiltonians are important in atomic and molecular physics as well as in confinement studies of quantum chromodynamics. A method of calculating the Rayleigh-Schrödinger coefficients $E^{\mathrm{}\left(n\right),m}$, based on an so(4,2) Lie algebraic reformulation of the Schrödinger equation, is outlined. Nonrigorous Bender-Wu WKB methods are employed to determine the high-order behavior of the $E^{\mathrm{}\left(n\right),m}$.