Abstract
Specific features of the behavior of the spectrum of steady states of the Dirac particle in a regularized “Coulomb” potential Vδ(z) = −q/(|z| + δ) as a function of the cutting parameter of δ in 1 + 1 D are investigated. It is shown that in such a one-dimensional relativistic “hydrogen atom” at δ ≪ 1, the discrete spectrum becomes a quasi-periodic function of δ; 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 δ > 0 and simultaneously the absence of the regular limiting transition to δ → 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 = Zα > 1, i.e., when the selfconsistent extension is required for the Dirac Hamiltonian with an irregularized potential.
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Original Russian Text © K.A. Sveshnikov, D.I. Khomovskii, 2012, published in Vestnik Moskovskogo Universiteta. Fizika, 2012, No. 4, pp. 41–46.
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Sveshnikov, K.A., Khomovskii, D.I. The Dirac particle in a one-dimensional “hydrogen atom”. Moscow Univ. Phys. 67, 358–363 (2012). https://doi.org/10.3103/S0027134912040157
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DOI: https://doi.org/10.3103/S0027134912040157