Full Salpeter Equation with Confining Interactions: Analytic Stability Proof

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.