Nonperturbative vacuum polarization effects are explored for a supercritical Coulomb source with Z > Zcr in 1+1 D. Both the vacuum charge density ρvac(x) and vacuum energy Evac are considered. It is shown that in the overcritical region the behavior of vacuum energy could be significantly different from perturbative quadratic growth up to decrease reaching large negative values.

Energy spectrums of a nonrelativistic particle and an H-like atom in a spherical box of size R with general conditions of "not going out" through the box surface are explored. The lowest energy levels reconstruction is described from the point of view of their asymptotical behavior for large R. The role of von Neumann-Wigner level reflection/avoided crossing effect in this spectrum reconstruction is emphasized. The properties of atomic H ground state in a cell, formed by a spherical cavity with an outer potential shell and Neumann condition on the outward boundary, are studied in detail. Some of them turn out to be quite new. The relevance of such a cell to a cubic lattice of cavities, occupied by H, is discussed be means of first principles and assumptions of the Wigner-Seitz model.

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Non-perturbative vacuum polarization effects are explored for a supercritical Dirac-Coulomb system with Z > Zcr,1 in 2+1 D, based on the original combination of analytical methods, computer algebra and numerical calculations, proposed recently in Refs.[1]- [3]. Both the vacuum charge density ρV P ( r) and vacuum energy EV P are considered. Due to a lot of details of calculation the whole work is divided into two parts I and II. Taking account of results, obtained in the part I [4] for ρV P , in the present part II the evaluation of the vacuum energy EV P is investigated with emphasis on the renormalization and convergence of the partial expansion for EV P . It is shown that the renormalization via fermionic loop turns out to be the universal tool, which removes the divergence of the theory both in the purely perturbative and essentially non-perturbative regimes of the vacuum polarization. The main result of calculation is that for a wide range of the system parameters in the overcritical region EV P turns out to be a rapidly decreasing function ∼ −η ef f Z 3 /R with η ef f > 0 and R being the size of the external Coulomb source. To the end the similarity in calculations of EV P in 2+1 and 3+1 D is discussed, and qualitative arguments are presented in favor of the possibility for complete screening of the classical electrostatic energy of the Coulomb source by the vacuum polarization effects for Z Zcr,1 in 3+1 D.