  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.
