Adiabatic representation for the three-body problem in hyperspherical coordinates. I. Statement of the problem

Abstract

For the three-body Coulomb problem a hyperspherical parametrisation of independent variables is given on a five-dimensional sphere S5with a hyperradius RH, the first linear invariant of the inertia tensor. The hyperspherical adiabatic basis is defined as a complete set of eigenfunctions and eigenvalues of the Hamiltonian on the sphere S5for every fixed value of the slow variable RH. The partial wave analysis in the total momentum J representation allows the authors to separate three Euler angles and to reduce the hyperspherical problem on S5to a system of (J+1) two-dimensional problems. Classification is given of the hyperspherical adiabatic basis for small and large values of the hyperradius RH. The logarithmic Fock singularity at the point of triple collision (RH=0) is explicitly shown. The approach is assigned to computing the cross sections of mesic atomic processes in the muon catalysis problem.