Spectrum and Nonrelativistic Limit of a Bethe-Salpeter Equation

The eigenvalue spectrum of the relativistic two-body equation introduced by Wick is investigated in detail for arbitrary binding energy. It is verified that for any nonzero binding energy there is a complete discrete spectrum of allowed coupling constants and that this spectrum is labeled by the four quantum numbers $n$, $l$, $m$, and $\kappa$. In the limit of zero binding energy, it is shown that all eigenvalues tend to zero. The set of coupling constants with $\kappa =0\mathrm{}$ correspond best to the Balmer formula. The wave functions are also investigated with respect to certain proposed boundary conditions and are compared with the nonrelativistic hydrogen atom wave functions. Again it is found that all states with $\kappa =0\mathrm{}$ correspond best to the nonrelativistic states, but that the "spurious" states with $\kappa >0\mathrm{}$ are not unacceptable. Possible generalizations of these results are discussed.