Hyperspherical Coulomb spheroidal representation in the Coulomb three-body problem

Abstract

A new representation of the Coulomb three-body wavefunction via the well-known solutions of the separable Coulomb two-centre problem ϕj(ξ, η) = Xj(ξ)Yj(η) is obtained, where Xj(ξ) and Yj(η) are the Coulomb spheroidal functions. Its distinguishing characteristic is the coordination with the asymptotic conditions of the scattering problem below the three-particle breakup. That is, the wavefunction of two colliding clusters in any open channel is the asymptotics of the single, corresponding to that channel, term of the suggested expansion. The effect is achieved due to a new relation between three internal coordinates of a three-body system and the parameters of ϕj(ξ, η). It ensures the orthogonality of ϕj(ξ, η) on a sphere of constant hyperradius, ρ = const, in place of the surface R = |x2 − x1| = const appearing in the traditional Born–Oppenheimer approach. The independent variables ξ and η are the orthogonal coordinates on this sphere with three poles in the coalescence points. They are connected with the elliptic coordinates on the plane by means of a stereographic projection. For the total angular momentum J ⩾ 0 the products of ϕj and the Wigner D-functions form a hyperspherical Coulomb spheroidal (HSCS) basis on a five-dimensional hypersphere, ρ being a parameter. The system of the differential equations and the boundary conditions for the radial functions fJi(ρ), the coefficients of the HSCS decomposition of the three-body wavefunction, are presented.