Dilatation analyticity and the radius of convergence of the1Zperturbation expansion: Comment on a conjecture of Stillinger

Concepts from the theory of dilatation analytic operators are applied to the problem of determining the radius of convergence of the $\frac{1}{Z}$ perturbation expansion for atomic bound-state energies. It is shown that if the radius of convergence is determined by a singularity on the positive real $Z$ axis, it will occur for a value of $Z$ such that $E\left(\frac{1}{Z}\right)$ becomes degenerate with a threshold. This result has serious consequences regarding conjectured bound states embedded in electronic continua of the same symmetry.