  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.
