STABILITY ANALYSIS OF THE INSTANTANEOUS BETHE-SALPETER-EQUATION AND THE CONSEQUENCES FOR MESON SPECTROSCOPY

We investigate the light and heavy meson spectra in the context of the instantaneous approximation to the Bethe-Salpeter equation (Salpeter' s equation). We use a static kernel consisting bf a one-gluon-exchange component and a confining contribution, Salpeter's equation is known to be formally equivalent to a random-phase-approximation equation; as such, it can develop imaginary eigenvalues. Thus our study cannot be complete without first discussing the stability of Salpeter's equation . The stability analysis limits the form of the kernel and reveals tha t a Lorentz scalar confining interaction in the Salpeter equation lead s to instabilities (imaginary eigenvalues), whereas one transforming a s the time component of a vector does not. Moreover, the stability ana lysis sets an upper limit on the size of the one-gluon-exchange compon ent; the value for the critical coupling, is determined through a solu tion of the ''semirelativistic'' Coulomb problem. These limits place i mportant constraints on the interaction and suggest that a more sophis ticated model is needed to describe the light and heavy quarkonia.