Vacuum effects for a one-dimensional “hydrogen atom” with Z > Zcr

Abstract

For a supercritical Coulomb source with a chargeZ > Z_cr in 1+1 dimensions, we study the nonperturbative properties of the vacuum density ρ_VP(x) and the energy ɛ_VP. We show that for corresponding problem parameters, nonlinear effects in the supercritical region can lead to behavior of the vacuum energy differing significantly from the perturbative quadratic growth, to the extent of an (almost) quadratic decrease of the form −|η|Z² into the negative region. We also show that although approaches for calculating vacuum expectations values and the behavior of ρ_VP(x) in the supercritical region for various numbers of spatial dimensions indeed have many common features, ɛ_VP for 1+1 dimensions in the supercritical region nevertheless has several specific features determined by the one-dimensionality of the problem.