  In the framework of instantaneous approximations to the Bethe-Salpeter
formalism for the description of bound states within quantum field theories,
depending on the Lorentz structure of the Bethe-Salpeter interaction kernel the
solutions of the full Salpeter equation with some confining interaction may
exhibit certain instabilities, which are possibly related to the Klein paradox
and signal the decay of states assumed to be bound by the confining
interactions. They are observed in numerical (variational) studies of the
Salpeter equation. The presumably simplest scenario allowing for the fully
analytic investigation of this problem is set by the reduced Salpeter equation
with harmonic-oscillator interaction. In this case, Salpeter's integral
equation simplifies to either an algebraic relation or a second-order
homogeneous linear ordinary differential equation, immediately accessible to
standard techniques. There one may hope to be able to decide unambiguously
whether this setting poses a well-defined eigenvalue problem the solutions of
which correspond to stable bound states associated to real energy eigenvalues
bounded from below. By analytical spectral analysis the bound-state solutions
of this "harmonic-oscillator reduced Salpeter equation" can be shown to be free
of such instabilities.
