Convergence study of the 1/Z expansion for the energy levels of two-electron atoms

摘要

We perform numerical analysis of the first 20 and 14 coefficients for 1 ~1S and 2~3S states of the 1/Z expansion of the energy of two-electron atoms, respectively. The radius of convergence and large-order behavior of the coefficients are determined. The results obtained are in disagreement with those given so far in the literature. We sum the terms of the series with known coefficients and the remainder of the series where we replace the actual coefficients by their large-order values. We show that inclusion of the remainder improves agreement with variational results by more than three orders of magnitude. We argue that the energy is at least three times and most likely infinitely degenerate at the singularity. Numerical result for the effective characteristic polynomial supports this conclusion.