Schrödinger and Dirac particles in quasi-one-dimensional systems with a Coulomb interaction

We consider specific features and principal distinctions in the behavior of the energy spectra of Schrödinger and Dirac particles in the regularized “Coulomb”; potential $V_\delta(z)=-q/(|z|+\delta)$ as functions of the cutoff parameter $\delta$ in $1{+}1$ dimensions. We show that the discrete spectrum becomes a quasiperiodic function of $\delta$ for $\delta\ll1$ in such a one-dimensional “hydrogen atom” in the relativistic case. This effect is nonanalytically dependent on the coupling constant and has no nonrelativistic analogue in this case. This property of the Dirac spectral problem explicitly demonstrates the presence of a physically informative energy spectrum for an arbitrarily small $\delta>0$, but also the absence of a regular limit transition $\delta\to0$ for all nonzero $q$. We also show that the three-dimensional Coulomb problem has a similar property of quasiperiodicity with respect to the cutoff parameter for $q=Z\alpha>1$, i.e., in the case where the domain of the Dirac Hamiltonian with the nonregularized potential must be especially refined by specifying boundary conditions as $r\to0$ or by using other methods.