Radius of convergence and analytic behavior of the1Zexpansion

We have performed a 401-order perturbation calculation to resolve the controversy over the radius of convergence of the $\frac{1}{Z}$ expansion for the ground-state energy $E\left(\lambda \right)$ of heliumlike ions, where $\lambda =\frac{1}{Z}$ and $H\left(\lambda \right)=-\frac{1}{2}{\nabla }_1^2-\frac{1}{r_1}-\frac{1}{2}{\nabla }_2^2-\frac{1}{r_2}+\frac{\lambda }{r_{12}}$. Such high-order calculations followed by Neville-Richardson extrapolation of the ratios of the coefficients are necessary to study the asymptotic behavior of the perturbation series. We find (i) that ${\lambda }_c$, the critical value of $\lambda$ for which $H\left(\lambda \right)$ has a bound state with zero binding energy, is approximately 1.097 66, (ii) that ${\lambda }^{*}$, the radius of convergence of the perturbation series, is equal to ${\lambda }_c$, and (iii) that the nearest singularity of $E\left(\lambda \right)$ in the complex plane, which determines ${\lambda }^{*}$, is on the positive real axis at ${\lambda }_c$. Thus our results confirm Reinhardt's analysis Phys. Rev. A 15 802 1977 of this problem using the theory of dilatation analyticity (complex scaling). We also find that the perturbation series for $E\left(\lambda \right)$ is convergent at $\lambda ={\lambda }_c$. The same statements hold for the perturbation series for the square of the norm of the corresponding eigenfunction ${\parallel \psi \left(\lambda \right)\parallel }^2$. We find numerically that $E\left(\lambda \right)$ has a complicated branch-point singularity at $\lambda ={\lambda }_c$ of the same type as the function ${(1-\frac{\lambda }{{\lambda }^{*}})}^{-a}U(a,c;\frac{x}{(l-\frac{\lambda }{{\lambda }^{*}})})$, where $U$ is the irregular solution of the confluent hypergeometric equation, and that ${\parallel \psi \left(\lambda \right)\parallel }^2$ has a similar but even more complicated singularity at ${\lambda }^{*}$. We also discuss the $\frac{1}{Z}$ expansions for excited states of the helium isoelectronic sequence and for states of multielectron atomic ions. Byproducts of our calculation include the most accurate estimates so far of the nonrelativistic ground-state energies of the ${\mathrm{H}}^{-}$ ion and of the helium atom, as...