Adiabatic representation of three-body problem in hyperspherical coordinates

For a three-body Coulomb problem a hyperspherical parametrization is given for independent variables on a five-dimensional sphere S5 with a hyperradius RH, the first linear invariant of the tensor of inertia of three particles. The hyperspherical adiabatic basis is defined as a complete set of eigenfunctions and eigenvalues of the Hamiltonian angular part on the sphere S5 for every fixed value of the slow variable RH. The partial wave analysis in the total momentum J representation allows us to separate three Euler angles and to reduce the hyperspherical problem on S5 to a system of (J+1) two-dimensional problems. Classification is given of states of the hyperspherical adiabatic basis for small and large values of the hyperradius RH and the logarithmic Fock singularity at the point of three-body collision is explicitly shown. The approach is assigned at computing the cross sections of mesic-atomic processes in the muon-catalysis problem