Hyperspherical theory of three-particle fragmentation and Wannier's threshold law

A representation of three particle wave functions well adapted to computations of low-energy fragmentation states of systems interacting electrostatically is derived. A basis called an angle-Sturmian basis, is introduced. Exact wave functions are represented by sums over the angle-Sturmian functions and integrals over the index of Bessel functions. Equations for the coefficients of the Sturmian functions are derived. Solutions of these equations are given in the approximation that one Sturmian is employed. Integral representations of the approximate three-particle wave functions are obtained. Evaluation of the integral for large hyper-radius R gives the hidden-crossing theory, familiar from representations of ion-atom interactions at low energy. It is shown that ionization components emerge simply only for complex values of R. Such components conform to Wannier's threshold law.