The classification of hidden crossings in the three-body Coulomb problem: separable and conventional hyperspherical representations

A classification of hidden crossings in the three-body Coulomb problem is developed on the basis of hyperspherical adiabatic approach and approximate separability of the hyperspherical adiabatic eigenvalue problem in hyperspherical elliptic coordinates. For the case when two of three particles arc identical we found two main types of hidden crossings which generalize the well-known S and T series of hidden crossings in the two-Coulomb-centre problem and reveal essential mechanisms of nonadiabatic transitions. The unstable periodic trajectories associated with these two types of hidden crossings are discussed. The results are illustrated by the calculations for the positronium negative ion eee(+) with zero total angular momentum.