Skip to main content
SpringerLink
Log in

Discrete spectra of the Dirac Hamiltonian in Coulomb and Aharonov-Bohm potentials in 2+1 dimensions

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We find all self-adjoint Dirac Hamiltonians in Coulomb and Aharonov-Bohm potentials in 2+1 dimensions with the fermion spin taken into account. We obtain implicit equations for the spectra and construct eigenfunctions for all self-adjoint Dirac Hamiltonians in the indicated external fields. We find explicit solutions of the equations for the spectra in some cases.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. M. J. Schakel and G. W. Semenoff, Phys. Rev. Lett., 66, 2653–2656 (1991).

    Article  ADS  Google Scholar 

  2. Y. Aharonov and D. Bohm, Phys. Rev., 115, 485–491 (1959).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. R. E. Prange and S. M. Girvin, eds., The Quantum Hall Effect, Springer, New York (1990).

    Google Scholar 

  4. F. Wilczek, Fractional Statistics and Anyon Superconductivity, World Scientific, Teaneck, N. J. (1990).

    Google Scholar 

  5. A. Neagu and A. M. J. Schakel, Phys. Rev. D, 48, 1785–1791 (1993); arXiv:hep-th/9306092v1 (1993).

    Article  ADS  Google Scholar 

  6. M. G. Alford and F. Wilczek, Phys. Rev. Lett., 62, 1071–1074 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  7. P. De Sousa Gerbert, Phys. Rev. D, 40, 1346–1349 (1989).

    Article  Google Scholar 

  8. M. G. Alford, J. March-Pussel, and F. Wilczek, Nucl. Phys. B, 328, 140–158 (1989).

    Article  ADS  Google Scholar 

  9. K. S. Novoselov A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature, 438, 197–200 (2005); arXiv:cond-mat/0509330v1 (2005).

    Article  ADS  Google Scholar 

  10. Z. Jiang, Y. Zhang, H. L. Stormer, and P. Kim, Phys. Rev. Lett., 99, 106802 (2007); arXiv:0705.1102v2 [cond-mat.mes-hall] (2007).

    Article  ADS  Google Scholar 

  11. I. F. Herbut, Phys. Rev. Lett., 104, 066404 (2010); arXiv:0909.4231v2 [cond-mat.mes-hall] (2009).

    Article  ADS  Google Scholar 

  12. I. V. Tyutin, “Electron scattering by a solenoid [in Russian],” FIAN Preprint No. 27, Lebedev Physical Institute, USSR Acad. Sci. (1974); arXiv:0801.2167v2 [quant-ph] (2008).

  13. Ya. B. Zeldovich and V. S. Popov, Sov. Phys. Usp., 14, 673–694 (1972).

    Article  ADS  Google Scholar 

  14. A. B. Migdal, Fermions and Bosons in Strong Fields [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  15. J. Rafelski, L. P. Fulcher, and A. Klein, Phys. Rep., 38, 227–361 (1978).

    Article  ADS  Google Scholar 

  16. M. Soffel, B. Müller, and W. Greiner, Phys. Rep. C, 85, 51–122 (1982).

    Article  ADS  Google Scholar 

  17. W. Greiner and J. Reinhardt, Quantum Electrodynamics, Springer, Berlin (2009).

    MATH  Google Scholar 

  18. V. R. Khalilov, Theor. Math. Phys., 116, 956–963 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  19. V. R. Khalilov, Theor. Math. Phys., 158, 210–220 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  20. B. L. Voronov, D. M. Gitman, and I. V. Tyutin, Theor. Math. Phys., 150, 34–72 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  21. D. M. Gitman, A. A. Smirnov, I. V. Tyutin, and B. L. Voronov, “Self-adjoint Schrödinger and Dirac operators with Aharonov-Bohm and magnetic-solenoid fields,” arXiv:0911.0946v1 [quant-ph] (2009).

  22. M. A. Naimark, Linear Differential Operators, Frederick Ungar, New York (1968).

    MATH  Google Scholar 

  23. S. G. Krein, ed., Functional Analysis, Noordhoff, Groningen (1972).

    Google Scholar 

  24. C. R. Hagen, Phys. Rev. Lett., 64, 503–506 (1990).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  25. V. R. Khalilov and C.-L. Ho, Ann. Phys., 323, 1280–1293 (2008); arXiv:0708.3131v2 [hep-th] (2007).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. V. R. Khalilov, Phys. Rev. A, 71, 012105 (2005); arXiv:quant-ph/0406033v1 (2004).

    Article  ADS  Google Scholar 

  27. B. L. Voronov, D. M. Gitman, and I. V. Tyutin, “Self-adjoint differential operators associated with self-adjoint differential expressions,” arXiv:quant-ph/0603187v2 (2006).

  28. V. B. Berestetskii, E. M. Lifshits, and L. P. Pitaevskii, Quantum Electrodynamics [in Russian] (Vol. 4 of Course of Theoretical Physics by L. D. Landau and E. M. Lifshits), Nauka, Moscow (1980); English transl., Pergamon, New York (1982).

    Google Scholar 

  29. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products [in Russian], Fizmatlit, Moscow (1971); English transl.: Tables of Integrals, Series, and Products, Acad. Press, New York (1980).

    Google Scholar 

  30. S. Flügge, Practical Quantum Mechanics, Springer, Berlin (1971); Russian transl.: Problems in Quantum Mechanics [in Russian], Mir, Moscow (1974).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    V. R. Khalilov & K. E. Lee

Authors
  1. V. R. Khalilov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. E. Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. R. Khalilov.

Additional information

__________

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 169, No. 3, pp. 368–390, December, 2011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Khalilov, V.R., Lee, K.E. Discrete spectra of the Dirac Hamiltonian in Coulomb and Aharonov-Bohm potentials in 2+1 dimensions. Theor Math Phys 169, 1683–1703 (2011). https://doi.org/10.1007/s11232-011-0145-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-011-0145-4

Keywords

Search

Navigation