Relativistic effects in one-dimensional hydrogen atom

Abstract

The dependence of stationary levels of a Dirac particle in the regularized Coulomb potential Vδ(z) = −q/(|z| + δ) on the cutoff parameter δ is studied. It is shown that, in 1 + 1 D, the energy spectrum of a Dirac particle in such a potential reveals some specific features which nonanalytically depend on the coupling constant q and are essentially relativistic in nature. These properties turn out to be most important for δ ≪ 1, explicitly demonstrating the existence of a physically reasonable energy spectrum for an arbitrarily small δ > 0 and, at the same time, the absence of regular limit δ → 0 (hence, the absence of a well-defined spectral problem for the Dirac equation without regularization for arbitrary q in 1 + 1 D).