Resonances in positron scattering on a supercritical nucleus and spontaneous production of $e^{+}{e}^{-}$ pairs

We re-examine the physics of supercritical nuclei, specially focusing on the scattering phase ${\delta }_{\mathrm{\varkappa }}$ and its dependence on the energy $\epsilon$ of the diving electronic level, for which we give both exact and approximate formulas. The Coulomb potential $Z\alpha \mathrm{/}r$ is rounded to the constant $Z\alpha \mathrm{/}R$ for $r<R$. We confirm the resonant behavior of ${\delta }_{\mathrm{\varkappa }}$ that we investigate in details. In addition to solving the Dirac equation for an electron, we solve it for a positron, in the field of the same nucleus. This clarifies the interpretation of the resonances. Our results are compared with claims made in previous works.