Nonperturbative vacuum polarization effects in two-dimensional supercritical Dirac-Coulomb system I. Vacuum charge density

Based on the original combination of analytical methods, computer algebra tools and numerical calculations, proposed recently in Refs. 1-3, the nonperturbative vacuum polarization effects in the 2+1 D supercritical Dirac-Coulomb system with Z > Z(cr),(1) are explored. Both the vacuum charge density rho(VP)((r) over right arrow) and vacuum energy epsilon(VP) are considered. The main result of the work is that in the overcritical region epsilon(VP) turns out to be a rapidly decreasing function similar to - eta(eff)Z(3)/R with eta(eff) > 0 and R being the size of the external Coulomb source. Due to a lot of details of calculation the whole work is divided into two parts I and II. In the present part I, we consider the evaluation and behavior of the vacuum density rho(VP), which further is used in part II for evaluation of the vacuum energy, with emphasis on the renormalization, convergence of the partial expansion for pvp and behavior of the integral induced charge Q(VP) in the overcritical region.