  The most popular 3-dimensional reduction of the Bethe-Salpeter formalism for
the description of bound states within quantum field theory is the Salpeter
equation, found as the instantaneous limit of the Bethe-Salpeter framework if
allowing, in addition, for free propagation of the bound-state constituents.
Unfortunately, depending on the Lorentz nature of the Bethe-Salpeter kernel,
supposedly stable results of Salpeter's equation with confining interactions
exhibit (un-)expected instabilities, probably related to Klein's paradox.
Clearly, bound states may be regarded as stable if, for appropriate
interactions, their mass eigenvalues belong to a real and discrete (part of
the) spectrum that is bounded from below. Some general features of the
eigenvalue spectra of any Salpeter equation are well-established: all
bound-state masses squared are real and, for a large class of sufficiently
reasonable Bethe-Salpeter kernels - which includes all interactions of
phenomenological relevance for QCD -, the spectrum of mass eigenvalues
consists, in the complex bound-state mass plane, at most of real pairs of
opposite sign and imaginary points. For time-component vector
harmonic-oscillator kernels, a straightforward stability proof is given.
