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Schrödinger and Dirac Operators with Aharonov–Bohm and Magnetic-Solenoid Fields

  • D. M. Gitman
  • I. V. Tyutin
  • B. L. Voronov
Chapter
Part of the Progress in Mathematical Physics book series (PMP, volume 62)

Abstract

We examine the Dirac Hamiltonian with the Aharonov–Bohm field and with the so-called magnetic-solenoid field. We construct systematically all the self-adjoint Schrödinger and Dirac operators both with pure AB field and with magnetic-solenoid field. We perform a complete spectral analysis for these Hamiltonians, which includes finding spectra and inversion formulas.

References

  1. 2.
    Adami, R., Teta, A.: On the Aharonov–Bohm effect. Lett. Math. Phys. 43, 43–54 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 4.
    Alford, M.G., March-Russel, J., Wilczek, F.: Enhanced baryon number violation due to cosmic strings. Nucl. Phys. B 328, 140–158 (1989)CrossRefGoogle Scholar
  3. 6.
    Aharonov, Y, Bohm, D.: Significance of electromagnetic potentials in quantum theory. Phys. Rev. 115, 485–491 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 10.
    Araujo, V.S., Coutinho, F.A.B., Perez, J.F.: On the most general boundary conditions for the Aharonov–Bohm scattering of a Dirac particle: helicity and Aharonov–Bohm symmetry conservation. J. Phys. A 34, 8859–8876 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 14.
    Bagrov, V.G., Gavrilov, S.P., Gitman, D.M., Meira Filho D.P.: Coherent states of non-relativistic electron in magnetic-solenoid field. J. Phys. A 43, 3540169 (2010); Coherent and semiclassical states in magnetic field in the presence of the Aharonov–Bohm solenoid. J. Phys. A: Math. Theor. 44, 055301 (2011)Google Scholar
  6. 15.
    Bagrov, V.G., Gitman, D.M., Tlyachev, V.B.: Solutions of relativistic wave equations in superpositions of Aharonov–Bohm, magnetic, and electric fields. J. Math. Phys. 42, 1933–1959 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 16.
    Bagrov, V.G., Gitman, D.M., Levin, A., Tlyachev, V.B.: Impact of Aharonov–Bohm solenoid on particle radiation in magnetic field. Mod. Phys. Lett. A 16, 1171–1179 (2001)CrossRefGoogle Scholar
  8. 17.
    Bagrov, V.G., Gitman, D.M., Levin, A., Tlyachev, V.B.: Aharonov–Bohm effect in cyclotron and synchrotron radiations. Nucl. Phys. B 605, 425–454 (2001)zbMATHCrossRefGoogle Scholar
  9. 18.
    Bagrov, V.G., Gitman, D.M., Tlyachev, V.B.: l-Dependence of particle radiation in magnetic-solenoid field and Aharonov–Bohm effect. Int. J. Mod. Phys. A 17, 1045–1048 (2002)CrossRefGoogle Scholar
  10. 33.
    Breitenecker, M., Grümm, H.-R.: Remarks on the paper by Bocchieri, P., Loinger, A.: “Nonexistence of the Aharonov–Bohm effect ”Nuovo Cim. A 55, 453–455 (1980)Google Scholar
  11. 40.
    Coutinho, F.A.B., Nogami, Y., Perez, J.F.: Self-adjoint extensions of the Hamiltonian for a charged-particle in the presence of a thread of magnetic-flux. Phys. Rev. A 46, 6052–6055 (1992); Self-adjoint extensions of the Hamiltonian for a charged spin-1/2 particle in the Aharonov–Bohm field. J. Phys. A 27, 6539–6550 (1994)Google Scholar
  12. 41.
    Coutinho, F.A.B., Perez, J.F.: Boundary-conditions in the Aharonov–Bohm scattering of Dirac particles and the effect of Coulomb interaction. Phys. Rev. D 48, 932–939 (1993)CrossRefGoogle Scholar
  13. 42.
    Coutinho, F.A.B., Perez, J.F.: Helicity conservation in the Aharonov–Bohm scattering of Dirac Particles. Phys. Rev. D 49, 2092–2097 (1994)CrossRefGoogle Scholar
  14. 46.
    Dirac, P.A.M.: The theory of magnetic poles. Phys. Rev. 74, 817–830 (1948)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 47.
    Dirac, P.A.M.: Quantized singularities in the electromagnetic field. Proc. Royal Soc. (London) A 133, 60–72 (1931)Google Scholar
  16. 53.
    Ehrenberg, W, Siday, R.E.: The refractive index in electron optics and the principles of dynamics. Proc. Phys. Soc. Lond., B 62, 8–21 (1949)Google Scholar
  17. 55.
    Exner, P., Št’oviček, P., Vytřas, P.: Generalized boundary conditions for the Aharonov–Bohm effect combined with a homogeneous magnetic field. J. Math. Phys. 43, 2151–2168 (2002)Google Scholar
  18. 58.
    Flekkøy, E.G., Leinaas, J.M.: Vacuum currents around a magnetic fluxstring. Int. J. Mod. Phys. A 6, 5327–5347 (1991)CrossRefGoogle Scholar
  19. 66.
    Gavrilov, S.P., Gitman, D.M., Smirnov, A.A.: Dirac equation in the magnetic-solenoid field. Euro. Phys. J. C 30, 009 (2003); 32(Suppl.) 119–142 (2003)Google Scholar
  20. 67.
    Gavrilov, S.P., Gitman, D.M., Smirnov, A.A.: Self-adjoint extensions of Dirac Hamiltonian in magnetic-solenoid field and related exact solutions. Phys. Rev. A 67(4) 024103 (2003)CrossRefGoogle Scholar
  21. 68.
    Gavrilov, S.P., Gitman, D.M., Smirnov, A.A.: Green functions of the Dirac equation with magnetic-solenoid field. J. Math. Phys. 45, 1873–1886 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 69.
    Gavrilov, S.P., Gitman, D.M., Smirnov, A.A., Voronov, B.L.: Dirac fermions in a magnetic-solenoid field. In: Benton, C.V. (ed.) Focus on Mathematical Physics Research, pp. 131–168. Nova Science Publishers, New York (2004)Google Scholar
  23. 72.
    Gerbert, Ph. de S., Jackiw, R.: Classical and quantum scattering on a spinning cone. Commun. Math. Phys. 124, 229–260 (1989)Google Scholar
  24. 73.
    Gerbert, Ph. de S.: Fermions in an Aharonov–Bohm field and cosmic strings. Phys. Rev. D 40, 1346–1349 (1989)Google Scholar
  25. 78.
    Gitman, D.M., Tyutin, I.V., Smirnov, A., Voronov, B.L.: Self-adjoint Schrödinger and Dirac operators with Aharonov–Bohm and magnetic-solenoid fields. Phys. Scr. 85 (2012) 045003CrossRefGoogle Scholar
  26. 85.
    Hagen, C.R.: Aharonov–Bohm scattering of particles with spin. Phys. Rev. Lett. 64, 503–506 (1990); Spin dependence of the Aharonov–Bohm effect. Int. J. Mod. Phys. A 6, 3119–3149 (1991)Google Scholar
  27. 86.
    Hagen, C.R.: Effects of nongauge potentials on the spin-1/2 Aharonov–Bohm problem. Phys. Rev. D 48, 5935–5939 (1993)CrossRefGoogle Scholar
  28. 89.
    Hamilton, J.: Aharonov–Bohm and Other Cyclic Phenomena. Springer Tracts in Modern Physics. Springer, New York (1997)zbMATHGoogle Scholar
  29. 107.
    Lewis, R.R.: Aharonov–Bohm effect for trapped ions. Phys. Rev. A 28, 1228–1236 (1983)CrossRefGoogle Scholar
  30. 110.
    Lisovyy, O.: Aharonov–Bohm effect on the Poincaré disk. J. Math. Phys. 48, 052112-17 (2007). doi:10.1063/1.2738751MathSciNetCrossRefGoogle Scholar
  31. 117.
    Nambu, Y.: The Aharonov–Bohm problem revisited. Nucl. Phys. B 579, 590–616 (2000); Hirokawa, M., Ogurisu, O.: Ground state of a spin-1/2 charged particle in a two-dimensional magnetic field. J. Math. Phys. 42, 3334–3343 (2001)Google Scholar
  32. 120.
    Olariu, S., Popescu, I.I.: The quantum effects of electromagnetic fluxes. Rev. Mod. Phys. 57, 339–436 (1985)CrossRefGoogle Scholar
  33. 121.
    Oliveira C.R. de, Pereira, M.: Mathematical justification of the Aharonov–Bohm Hamiltonian. J. Stat. Phys. 133, 1175–1184 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 124.
    Peshkin M., Tonomura, A.: The Aharonov–Bohm Effect. Lecture Notes in Physics. Springer, New York (1989)CrossRefGoogle Scholar
  35. 135.
    Ruijsenaars, S.N.M.: The Aharonov–Bohm effect and scattering theory. Ann. Phys. 146, 1–34 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 150.
    Tyutin, I.V.: Electron scattering on a solenoid. Preprint FIAN (P.N. Lebedev Physical Institute, Moscow) no. 27. arXiv:0801.2167 (quant-ph) (1974)Google Scholar
  37. 152.
    Villalba, V.M.: Exact solutions of the Dirac equation for a Coulomb and scalar potential in the presence of an Aharonov–Bohm and magnetic monopole fields. J. Math. Phys. 36, 3332–3344 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 159.
    Voropaev, S.A., Galtsov, D.V., Spasov, D.A.: Bound states for fermions in the gauge Aharonov–Bohm field. Phys. Lett. B 267, 91–94 (1991)MathSciNetCrossRefGoogle Scholar
  39. 165.
    Wu, T.T., Yang, C.N.: Concept of nonintegrable phase factors and global formulation of gauge fields. Phys. Rev. D 12, 3845–3857 (1975)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • D. M. Gitman
    • 1
  • I. V. Tyutin
    • 2
  • B. L. Voronov
    • 2
  1. 1.Instituto de FísicaUniversidade de São PauloSão PauloBrasil
  2. 2.Department of Theoretical PhysicsP.N. Lebedev Physical InstituteMoscowRussia

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