Analytic solution of the relativistic Coulomb problem for a spinless Salpeter equation

Abstract

We construct an analytic solution to the spinless $S$-wave Salpeter equation for two quarks interacting via a Coulomb potential, $[2{(\ensuremath{-}{\ensuremath{\nabla}}^{2}+{m}^{2})}^{\frac{1}{2}}\ensuremath{-}M\ensuremath{-}\frac{\ensuremath{\alpha}}{r}] \ensuremath{\psi}(r)=0$, by transforming the momentum-space form of the equation into a mapping or boundary-value problem for analytic functions. The principal part of the three-dimensional wave function is identical to the solution of a one-dimensional Salpeter equation found by one of us and discussed here. The remainder of the wave function can be constructed by the iterative solution of an inhomogeneous singular integral equation. We show that the exact bound-state eigenvalues for the Coulomb problem are ${M}_{n}=\frac{2m}{{(1+\frac{{\ensuremath{\alpha}}^{2}}{4{n}^{2}})}^{\frac{1}{2}}}$, $n=1, 2, \dots{}$, and that the wave function for the static interaction diverges for $r\ensuremath{\rightarrow}0$ as $C{(\mathrm{mr})}^{\ensuremath{-}\ensuremath{\nu}}$, where $\ensuremath{\nu}=(\frac{\ensuremath{\alpha}}{\ensuremath{\pi}})(1+\frac{\ensuremath{\alpha}}{\ensuremath{\pi}}+\ensuremath{\cdots})$ is known exactly.