Analytic perturbation theory for screened Coulomb potentials: Nonrelativistic case

We have developed an analytic perturbation theory for screened Coulomb radial wave functions, based on an expansion of the potential of the form $V\left(r\right)=-\left(\frac{a}{r}\right)[1+V_1\lambda r+V_2{\left(\lambda r\right)}^2+V_3{\left(\lambda r\right)}^3+\cdots ]$, where $\lambda \simeq \alpha Z^{\frac{1}{3}}$ is a small parameter characterizing the screening. The coefficients $V_k$ may be chosen such that the above form converges rapidly and gives a good approximation to realistic numerical potentials, such as those of Herman and Skillman, in the interior of the atom. The screened radial wave functions are obtained as a series in $\lambda$ with simple analytic coefficients, owing to the special symmetries of the unperturbed Coulomb problem. Both bound and continuum shapes are correctly treated in the region $\lambda r<\mathrm{}1\mathrm{}$. For inner bound states, this includes all of the region where the wave function is large. Similarly, high-energy continuum wave functions will have completed several oscillations in this interval so that by $r\simeq {\lambda }^{-1\mathrm{}}$ one has reached the asymptotic region. Consequently, expressions for bound-state normalizations can be given as series in $\lambda$, which are accurate, in general, for the $K$ shell and for other low-lying levels of high-$Z$ elements. The continuum normalizations which we obtain are valid for energies on the order of the $K$-shell binding energy above threshold. Bound-state energies and continuum phase shifts are also obtained in these circumstances.