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TMF, 2007, Volume 150, Number 1, Pages 41–84 (Mi tmf5965)  

This article is cited in 43 scientific papers (total in 43 papers)

The Dirac Hamiltonian with a superstrong Coulomb field

B. L. Voronova, D. M. Gitmanb, I. V. Tyutina

a P. N. Lebedev Physical Institute, Russian Academy of Sciences
b Universidade de São Paulo

Abstract: We consider the quantum mechanical problem of a relativistic Dirac particle moving in the Coulomb field of a point charge $Ze$. It is often declared in the literature that a quantum mechanical description of such a system does not exist for charge values exceeding the so-called critical charge with $Z=\alpha^{-1}=137$ because the standard expression for the lower bound-state energy yields complex values at overcritical charges. We show that from the mathematical standpoint, there is no problem in defining a self-adjoint Hamiltonian for any charge value. Furthermore, the transition through the critical charge does not lead to any qualitative changes in the mathematical description of the system. A specific feature of overcritical charges is a nonuniqueness of the self-adjoint Hamiltonian, but this nonuniqueness is also characteristic for charge values less than critical $($and larger than the subcritical charge with $Z=(\sqrt{3}/2)\alpha^{-1}=118)$. We present the spectra and $($generalized$)$ eigenfunctions for all self-adjoint Hamiltonians. We use the methods of the theory of self-adjoint extensions of symmetric operators and the Krein method of guiding functionals. The relation of the constructed one-particle quantum mechanics to the real physics of electrons in superstrong Coulomb fields where multiparticle effects may be crucially important is an open question.

Keywords: Dirac Hamiltonian, Coulomb field, self-adjoint extensions, spectral analysis

DOI: https://doi.org/10.4213/tmf5965

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English version:
Theoretical and Mathematical Physics, 2007, 150:1, 34–72

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Received: 08.08.2006

Citation: B. L. Voronov, D. M. Gitman, I. V. Tyutin, “The Dirac Hamiltonian with a superstrong Coulomb field”, TMF, 150:1 (2007), 41–84; Theoret. and Math. Phys., 150:1 (2007), 34–72

Citation in format AMSBIB
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    1. Gupta K.S., Sen S., “Bound states in gapped graphene with impurities: Effective low-energy description of short-range interactions”, Phys. Rev. B, 78:20 (2008), 205429, 7 pp.  crossref  adsnasa  isi  elib  scopus
    2. Adorno T.C., Baldiotti M.C., Chaichian M., Gitman D.M., Tureanu A., “Dirac equation in noncommutative space for hydrogen atom”, Phys. Rev. B, 682:2 (2009), 235–239  crossref  mathscinet  isi  scopus
    3. Gupta K.S., Samsarov A., Sen S., “Scattering in graphene with impurities: A low energy effective theory”, European Physical Journal B - Condensed Matter and Complex Systems, 73:3 (2010), 389–404  crossref  adsnasa  isi  scopus
    4. Gitman D.M., Tyutin I.V., Voronov B.L., “Self-adjoint extensions and spectral analysis in the Calogero problem”, J. Phys. A, 43:14 (2010), 145205  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    5. Boussaid N., Golénia S., “Limiting absorption principle for some long range perturbations of Dirac systems at threshold energies”, Comm. Math. Phys., 299:3 (2010), 677–708  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    6. Khalilov V.R., Lee K.E., “Bound fermion states in a vector $1/r$ and Aharonov-Bohm potential in $(2+1)$ dimensions”, Modern Phys. Lett. A, 26:12 (2011), 865–883  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    7. Khalilov V.R., Lee K.E., “Fermions in scalar Coulomb and Aharonov-Bohm potentials in 2+1 dimensions”, J. Phys. A, 44:20 (2011), 205303  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    8. Zinoviev Yu.M., “Causal electromagnetic interaction equations”, J. Math. Phys., 52:2 (2011), 022302  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    9. V. R. Khalilov, K. E. Lee, “Discrete spectra of the Dirac Hamiltonian in Coulomb and Aharonov–Bohm potentials in $2+1$ dimensions”, Theoret. and Math. Phys., 169:3 (2011), 1683–1703  mathnet  crossref  crossref  isi
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    13. K. A. Sveshnikov, D. I. Khomovskii, “Schrödinger and Dirac particles in quasi-one-dimensional systems with a Coulomb interaction”, Theoret. and Math. Phys., 173:2 (2012), 1587–1603  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    14. Khalilov V.R. Lee K.E., “Planar massless fermions in Coulomb and Aharonov-Bohm potentials”, Internat. J. Modern Phys. A, 27:29 (2012), 1250169, 14 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    15. Fulton Ch., Langer H., Luger A., “Mark Krein's method of directing functionals and singular potentials”, Math. Nachr., 285:14-15 (2012), 1791–1798  crossref  mathscinet  zmath  isi  elib  scopus
    16. Sveshnikov K.A., Khomovskii D.I., “The Dirac particle in a one-dimensional “hydrogen atom””, Mosc. Univ. Phys. Bull., 67:4 (2012), 358–363  crossref  adsnasa  isi  elib  elib  scopus
    17. Gitman D.M., Tyutin I.V., Voronov B.L., “Schrödinger and Dirac operators with the Aharonov-Bohm and magnetic-solenoid fields”, Phys. Scr., 85:4 (2012), 045003, 20 pp.  crossref  zmath  adsnasa  isi  elib  scopus
    18. Hogreve H., “The overcritical Dirac–Coulomb operator”, J. Phys. A, 46:2 (2013), 025301  crossref  mathscinet  adsnasa  isi  elib  scopus
    19. V. R. Khalilov, “Zero-mass fermions in Coulomb and Aharonov–Bohm potentials in 2+1 dimensions”, Theoret. and Math. Phys., 175:2 (2013), 637–654  mathnet  crossref  crossref  zmath  adsnasa  isi  elib  elib
    20. Khalilov V.R., “Creation of Planar Charged Fermions in Coulomb and Aharonov-Bohm Potentials”, Eur. Phys. J. C, 73:8 (2013), 2548  crossref  adsnasa  isi  scopus
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    22. Khalilov V.R., “Bound States of Massive Fermions in Aharonov-Bohm-Like Fields”, Eur. Phys. J. C, 74:1 (2014), 2708  crossref  adsnasa  isi  scopus
    23. Khalilov V.R., “Effect of Vacuum Polarization of Charged Massive Fermions in An Aharonov-Bohm Field”, Eur. Phys. J. C, 74:9 (2014), 3061  crossref  adsnasa  isi  scopus
    24. B. M. Karnakov, V. D. Mur, S. V. Popruzhenko, V. S. Popov, “Current progress in developing the nonlinear ionization theory of atoms and ions”, Phys. Usp., 58:1 (2015), 3–32  mathnet  crossref  crossref  adsnasa  isi  elib  elib
    25. V. M. Kuleshov, V. D. Mur, N. B. Narozhny, A. M. Fedotov, Yu. E. Lozovik, V. S. Popov, “Coulomb problem for a $Z>Z_cr$ nucleus”, Phys. Usp., 58:8 (2015), 785–791  mathnet  crossref  crossref  adsnasa  isi  elib  elib
    26. B. L. Voronov, D. M. Gitman, A. D. Levin, R. Ferreira, “Peculiarities of the electron energy spectrum in the Coulomb field of a superheavy nucleus”, Theoret. and Math. Phys., 187:2 (2016), 633–648  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    27. I. V. Mamsurov, V. R. Khalilov, “Induced vacuum charge of massless fermions in Coulomb and Aharonov–Bohm potentials in $2+1$ dimensions”, Theoret. and Math. Phys., 188:2 (2016), 1181–1196  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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    29. Bagci A., Hoggan P.E., “Solution of the Dirac equation using the Rayleigh-Ritz method: Flexible basis coupling large and small components. Results for one-electron systems”, Phys. Rev. E, 94:1 (2016), 013302  crossref  isi  elib  scopus
    30. Andrzejewski K., Ann. Phys., 367 (2016), 227–250  crossref  mathscinet  zmath  isi  elib  scopus
    31. Gitman D.M., Gavrilov S.P., “QFT Treatment of Processes in Strong External Backgrounds”, Russ. Phys. J., 59:11 (2017), 1723–1730  crossref  zmath  isi  scopus
    32. Khalilov V.R., “Quasi-Stationary States and Fermion Pair Creation From a Vacuum in Supercritical Coulomb Field”, Mod. Phys. Lett. A, 32:38 (2017), 1750200  crossref  mathscinet  isi  scopus
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    34. Morozov S., Mueller D., “On the Virtual Levels of Positively Projected Massless Coulomb-Dirac Operators”, Ann. Henri Poincare, 18:7 (2017), 2467–2497  crossref  mathscinet  zmath  isi  scopus
    35. Kuleshov V.M. Mur V.D. Narozhny N.B., “Coulomb Problem For Graphene With Supercritical Impurity”, V International Conference on Problems of Mathematical and Theoretical Physics and Mathematical Modelling, Journal of Physics Conference Series, 788, IOP Publishing Ltd, 2017, UNSP 012044  crossref  isi  scopus
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    41. Gallone M., Michelangeli A., “Self-Adjoint Realisations of the Dirac-Coulomb Hamiltonian For Heavy Nuclei”, Anal. Math. Phys., 9:1 (2019), 585–616  crossref  mathscinet  isi  scopus
    42. Cassano B., Pizzichillo F., “Boundary Triples For the Dirac Operator With Coulomb-Type Spherically Symmetric Perturbations”, J. Math. Phys., 60:4 (2019), 041502  crossref  mathscinet  zmath  isi  scopus
    43. Neznamov V.P. Safronov I.I., “Second-Order Stationary Solutions For Fermions in An External Coulomb Field”, J. Exp. Theor. Phys., 128:5 (2019), 672–683  crossref  isi
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