  A popular three-dimensional reduction of the Bethe-Salpeter formalism for the
description of bound states in quantum field theory is the Salpeter equation,
derived by assuming both instantaneous interactions and free propagation of all
bound-state constituents. Numerical (variational) studies of the Salpeter
equation with confining interaction, however, observed specific instabilities
of the solutions, likely related to the Klein paradox and rendering (part of
the) bound states unstable. An analytic investigation of this problem by a
comprehensive spectral analysis is feasible for the reduced Salpeter equation
with only harmonic-oscillator confining interactions. There we are able to
prove rigorously that the bound-state solutions correspond to real discrete
energy spectra bounded from below and are thus free of any instabilities.
