Coulomb problem for a $Z>Z_{\rm cr}$ nucleus

A closed-form equation is derived for the critical nucleus charge $Z=Z_{\rm cr}$ at which a discrete level with the Dirac quantum number touches the lower continuum of the Dirac equation solutions. For the Coulomb potential cut off rectangularly at the short distance $r_{0} = R{\hbar}/(mc)$, $R \ll {1}$, the critical nucleus charge values are obtained for several values of $\kappa$ and $R$. It is shown that the partial scattering matrix of elastic positron–nucleus scattering, $S_{\kappa} = \exp(2i\delta_{\kappa}(\varepsilon_{\rm p}))$, is also unitary for $Z>Z_{\rm cr}$. For this range, the scattering phase $\delta _{\kappa }(\varepsilon _{\rm p})$ is calculated as a function of the positron energy $E_{\rm p}$ = $\varepsilon_{\rm p} mc^{2}$, as are the positions and widths of quasidiscrete levels corresponding to the scattering matrix poles. The implication is that the single-particle approximation for the Dirac equation is valid not only for $Z<Z_{\rm cr}$ but also for $Z>Z_{\rm cr}$ and that there is no spontaneous creation of ${\rm e}^+{\rm e}^-$ pairs from the vacuum.