Hyperspherical Coulomb spheroidal representation in the Coulomb three-body problem

Abstract

The new representation of the Coulomb three-body wave function via the well-known solutions of the separable Coulomb two-centre problem $\phi_j(\xi,\eta)=X_j(\xi)Y_j(\eta)$ is obtained, where $X_j(\xi)$ and $Y_j(\eta)$ are the Coulomb spheroidal functions. Its distinguishing characteristic is the coordination with the boundary conditions of the scattering problem below the three-particle breakup. That is, the wave function of the scattering particles in any open channel is the asymptotics of the single, corresponding to that channel, term of the expansion suggested. The effect is achieved due to the new relation between three internal coordinates of the three-body system and the parameters of $\phi_j(\xi,\eta)$. It ensures the orthogonality of $\phi_j(\xi,\eta)$ on the sphere of a constant hyperradius, $\rho=const$, in place of the surface $R=|\bi{x}_2-\bi{x}_1|=const$ appearing in the traditional Born-Oppenheimer approach. The independent variables $\xi$ and $\eta$ are the orthogonal coordinates on that sphere with three poles in the coalescence points. They are connected with the elliptic coordinates on the plane by means of the stereographic projection. For the total angular momentum $J\ge 0$ the products of $\phi_j$ and the Wigner $D$-functions form the hyperspherical Coulomb spheroidal (HSCS) basis on the five-dimensional hypersphere, $\rho$ being a parameter. The system of the differential equations and the boundary conditions for the radial functions $f^J_i(\rho)$, the coefficients of the HSCS decomposition of the three-body wave function, are presented.