Asymptotics of atomic ground states: The relation between the ground state of helium and the ground state of He+

Abstract
The asymptotic behavior of ground states of two‐electron atoms is investigated. Suppose ψ(x1,x2) is the ground state of helium, ρ(x1) = Fψ2(x1,x2) dx2 the corresponding electron density, and Φ(x2) the ground state of He+. We show that in the L2(dx2)‐sense, lim‖x1‖→∞ ψ(x1,x2)[ρ(x1)]−1/2 = Φ(x2), and that ψ[ρ(x1)]−1/2 solves for large ‖x1‖ the Schrödinger equation for He+ in the quadratic form sense. The rate of convergence of these limits is also discussed.