Sturmian function approach and [Formula Presented] bound states

A suitable numerical approach based on Sturmian functions is employed to solve the [Formula Presented] bound state problem for local and nonlocal potentials. The approach accounts for both the strong short-range nuclear potential and the long-range Coulomb force and provides directly the wave function of protonium and [Formula Presented] deep bound states with complex eigenvalues [Formula Presented]. The spectrum of [Formula Presented] bound states has two parts, the atomic states bound by several keV, and the deep bound states which are bound by several hundred MeV. The observed very small hyperfine splitting of the [Formula Presented] level and the [Formula Presented] and [Formula Presented] decay widths are reasonably well reproduced by both the Paris and Bonn potentials (supplemented with a microscopically derived quark annihilation potential), although there are differences in magnitude and level ordering. We present further arguments for the identification of the [Formula Presented] deep bound state with the exotic tensor meson [Formula Presented]. Both investigated models can accommodate the [Formula Presented] but differ greatly in the total number of levels and in their ordering. The model based on the Paris potential predicts the [Formula Presented] level slightly below 1.1 GeV while the model based on the Bonn potential puts this state below 0.8 GeV. It remains to be seen if this state can be identified with a scalar partner of the [Formula Presented]. © 1997 The American Physical Society.