The Dirac particle in a one-dimensional "hydrogen atom"

Specific features of the behavior of the spectrum of steady states of the Dirac particle in a regularized "Coulomb" potential V delta(z) = -q/(|z| + delta) as a function of the cutting parameter of delta in 1 + 1 D are investigated. It is shown that in such a one-dimensional relativistic "hydrogen atom" at delta a parts per thousand(a) 1, the discrete spectrum becomes a quasi-periodic function of delta; this effect depends on the bonding constant analytically and has no nonrelativistic analog. This property of the Dirac spectral problem clearly demonstrates the presence of a physically containable energy spectrum at arbitrary small delta > 0 and simultaneously the absence of the regular limiting transition to delta -> 0. Thus, the necessity of extension of a definition for the Dirac Hamiltonian with irregularized potential in 1 + 1 D is confirmed at all nonzero values of the bonding constant q. It is also noted that the three-dimensional Coulomb problem possesses a similar property at q = Z alpha > 1, i.e., when the selfconsistent extension is required for the Dirac Hamiltonian with an irregularized potential.