Stability in the instantaneous Bethe-Salpeter formalism: reduced exact-propagator bound-state equation with harmonic interaction

Several numerical investigations of the Salpeter equation with static confining interactions of Lorentz-scalar type revealed that its solutions are plagued by instabilities of presumably Klein-paradox nature. By proving rigorously that the energies of all predicted bound states are part of real, entirely discrete spectra bounded from below, these instabilities are shown, for confining interactions of harmonic-oscillator shape, to be absent for a reduced version of an instantaneous Bethe-Salpeter formalism designed to generalize the Salpeter equation towards an approximate inclusion of the exact propagators of all bound-state constituents.