Super-critical QED-effects via VP-energy

Abstract: The properties of the QED-vacuum energy $\hbox{$\cal E $}{VP}$ under Coulomb super-criticality condition $Z>Z{cr,1}$ are explored in essentially non-perturbative approach. It is shown that in the supercritical region $\hbox{$\cal E $}{VP}$ is the decreasing function of the Coulomb source parameters, resulting in decay into the negative range as $\sim - Z^4/R$. The conditions, under which the emission of vacuum positrons can be unambiguously detected on the nuclear conversion pairs background, are also discussed. In particular, for a super-critical dummy nucleus with charge $Z$ and $R(Z) \simeq 1.2 \ (2.5\, Z)^{1/3}$ fm the lowest $1s$-level dives into the lower continuum at $Z{cr,1} \simeq 170$ (the model of uniformly charged ball) or $Z_{cr,1} \simeq 173$ (the spherical shell model), whereupon there appears the vacuum shell with the induced charge (-$2|e|$) and the QED-vacuum becomes charged, but in both cases the actual threshold for reliable spontaneous positrons detection is not less than $Z^\ast \simeq 210$.