Spontaneous emission in the supercritical QED: what is wrong and what is possible

Spontaneous positron emission, caused by the supercritical Coulomb source with charge $Z$ and size $R$, is explored in essentially non-perturbative approach with emphasis on the VP-energy $\cal E_{VP}$, considered as a function of the Coulomb source parameters $Z$ and $R$. The specific contribution to $\cal E_{VP}$, which appears in the supercritical case due to direct Coulomb interaction between VP-charge densities $\varrho_{VP}(\vec{r})$, is outlined and studied in detail. It is shown that after first levels diving into the lower continuum this contribution, being properly renormalized, turns out to be negative in contrast to the classical electrostatics and so could play an important role in spontaneous emission, especially for $Z$ just above the first $Z_{cr,1}$. The additional problems of spontaneous emission, caused by the lepton number conservation, are also discussed.