Essentially non-perturbative and peculiar polarization effects in planar QED with strong coupling

The essentially non-perturbative polarization effects are considered for a planar supercritical Dirac-Coulomb system with strong coupling (similar to graphene and graphene-based heterostructures) in terms of induced charge density $\rho_{VP}(\vec{r})$. The main attention is paid to the renormalization, convergence of the partial expansion and the behavior of $\rho_{VP}(\vec{r})$ and the integral induced charge $Q_{VP}$ in the overcritical region. The dependence of the induced density on the screening of the Coulomb asymptotics of the external source is also explored in detail. Some peculiar effects in the discrete spectrum with the lowest rotational numbers $m_j=\pm 1/2\, , \pm3/2$ in the screened case are detected and their possible role in the transition through corresponding $Z_{cr}$ is also discussed.