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 $\delta_{\varkappa}$ and its dependence on the energy $\varepsilon$ of the diving electronic level, for which we give both exact and approximate formulas. The Coulomb potential $Z\alpha/r$ is rounded to the constant $Z\alpha/R$ for $r < R$. We confirm the resonant behavior of $\delta_{\varkappa}$ 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.