Vacuum effects for a one-dimensional "hydrogen atom" with $Z>Z_{\mathrm{cr}}$

For a supercritical Coulomb source with a charge $Z>Z_{\mathrm{cr}}$ in $1{+}1$ dimensions, we study the nonperturbative properties of the vacuum density $\rho_{\mathrm{VP}}(x)$ and the energy $\mathcal E_{\mathrm{VP}}$. 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 $-|\eta|Z^2$ into the negative region. We also show that although approaches for calculating vacuum expectations values and the behavior of $\rho_{\mathrm{VP}}(x)$ in the supercritical region for various numbers of spatial dimensions indeed have many common features, $\mathcal E_{\mathrm{VP}}$ for $1{+}1$ dimensions in the supercritical region nevertheless has several specific features determined by the one-dimensionality of the problem.