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TMF, 2012, Volume 173, Number 2, Pages 293–313 (Mi tmf8349)  

This article is cited in 6 scientific papers (total in 6 papers)

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

K. A. Sveshnikov, D. I. Khomovskii

Lomonosov Moscow State University, Moscow, Russia

Abstract: 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.

Keywords: relativistic effect, Dirac equation, regularized Coulomb potential, one-dimensional hydrogen atom

DOI: https://doi.org/10.4213/tmf8349

Full text: PDF file (668 kB)
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English version:
Theoretical and Mathematical Physics, 2012, 173:2, 1587–1603

Bibliographic databases:

Received: 24.04.2012
Revised: 05.06.2012

Citation: K. A. Sveshnikov, D. I. Khomovskii, “Schrödinger and Dirac particles in quasi-one-dimensional systems with a Coulomb interaction”, TMF, 173:2 (2012), 293–313; Theoret. and Math. Phys., 173:2 (2012), 1587–1603

Citation in format AMSBIB
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\paper Schr\"odinger and Dirac particles in quasi-one-dimensional systems
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. A. Rajabi, M. Hamzavi, “Spin-1/2 particle in scalar-vector-pseudoscalar spatially dependent mass Coulomb fields: 1+1 dimensions”, Few-Body Syst., 54:11 (2013), 2067–2071  crossref  adsnasa  isi  elib
    2. V. A. Harutyunyan, “Semiconductor nanotube in the field of uniformly charged ring: additional quantization in the form of one-dimensional hydrogen-type levels”, Physica E, 57 (2014), 69–75  crossref  adsnasa  isi  elib
    3. Yu. S. Voronina, A. S. Davydov, K. A. Sveshnikov, “Vacuum effects for a one-dimensional “hydrogen atom” with $Z>Z_{\mathrm{cr}}$”, Theoret. and Math. Phys., 193:2 (2017), 1647–1674  mathnet  crossref  crossref  adsnasa  isi  elib
    4. A. Davydov, K. Sveshnikov, Yu. Voronina, “Vacuum energy of one-dimensional supercritical Dirac-Coulomb system”, Int. J. Mod. Phys. A, 32:11 (2017), 1750054  crossref  mathscinet  zmath  isi
    5. A. Novoselov, O. Pavlovsky, “Critical charge in gapped graphene: the role of screening of the interaction potential by $\sigma$-orbitals”, Int. J. Mod. Phys. B, 31:9 (2017), 1750068  crossref  isi
    6. Majorosi S., Benedict M.G., Czirjak A., “Improved One-Dimensional Model Potentials For Strong-Field Simulations”, Phys. Rev. A, 98:2 (2018), 023401  crossref  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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