Eigenvalues of the Breit Equation

Eigenvalues of the Breit Equation {eqnarray*} [(\vec{\alpha}{1} \vec{p} + \beta{1}m){\alpha \alpha^{\prime}} \delta{\beta \beta^{\prime}} + \delta_{\alpha \alpha^{\prime}} (-\vec{\alpha}{2} \vec{p} + \beta{2}M){\beta \beta^{\prime}} - \frac{e^{2}}{r} \delta{\alpha \alpha^{\prime}} \delta_{\beta \beta^{\prime}}] \Psi_{\alpha^{\prime} \beta^{\prime}} = E \Psi_{\alpha \beta}, {eqnarray*} in which only the static Coulomb potential is considered, have been found. Here the detailed discussion on the simple caces, ${}^1{S}_0,m=M$ and $m\mathrm{\ne }M$ is given deriving the exact energy eigenvalues. The ${\alpha }^2$ expansion is used to find radial wave functions. The leading term is given by classical Coulomb wave function. The technique used here can be applied to other cases.