Restricted quatum-mechanical three-body problems

On the basis of a perturbative procedure in which the eigenfunctions of a helium-like ion are expanded in the Hilbert space built up from the eigenfunctions of an electron in two fixed Coulomb charges,all asymptotic eigenfunctions are constructed for the Schrödinger equation of helium-like ions. If the nuclear chargeZ₁ is not less than 2, then our asymptotic considerations clarify the singularities of the Schrödinger equation of a helium-like ion (atr₁=0, ,r₂=0, ,r₁₂=0, ), while in the case ofZ₁=1 (negative hydrogen ion)r₂=0 will not be treated in this paper. The established order (inr₂) of asymptotics at 0 or in an exceptional case the zeroth-order term of the functions (actually the coefficients of an expansion of the desired eigenfunctions) as one of the electron coordinates (r₂, the distance of the two fixed Coulomb charges, a parameter of the set of the basic functions) enables us to classify the eigenfunctions of a helium-like ion. This classification resembles the classification scheme for one-electron configurations. The asymptotics forr₂ indicate bounded, pseudobounded (auto-ionizing) and free states. (Doubly ionized continuum states are not discussed here.) The use of the asymptotic solutions is indicated for the complete solution of the problem which may be either numerical integration or a variational procedure.Neutral muonic helium is included in the discussion.