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Mathematical Justification of the Aharonov-Bohm Hamiltonian

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Abstract

It is presented, in the framework of nonrelativistic quantum mechanics, a justification of the usual Aharonov-Bohm hamiltonian (with solenoid of radius greater than zero). This is obtained by way of increasing sequences of finitely long solenoids together with a natural impermeability procedure; further, both limits commute. Such rigorous limits are in the strong resolvent sense and in both ℝ2 and ℝ3 spaces.

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References

  1. Adami, R., Teta, A.: On the Aharonov-Bohm hamiltonian. Lett. Math. Phys. 43, 43–54 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aharonov, Y., Bohm, D.: Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115, 485–491 (1959)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. Aharonov, Y., Bohm, D.: Remarks on the possibility of quantum electrodynamics without potentials. Phys. Rev. 125, 2192 (1962)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. Aharonov, Y., Bohm, D.: Further discussion of the role of electromagnetic potentials in the quantum theory. Phys. Rev. 130, 1625 (1963)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. Babiker, M., Loudon, R.: Gauge invariance of the Aharonov-Bohm effect. J. Phys. A: Math. Gen. 17, 2973–2982 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  6. Ballesteros, M., Weder, R.: High-velocity estimates for the scattering operator and Aharonov-Bohm effect in three dimensions. Commun. Math. Phys. (2008). doi:10.1007/s00220-008-0579-1

    Google Scholar 

  7. Berry, M.V.: The Aharonov-Bohm effect is real physics not ideal physics. In: Gorini, V., Frigerio, A. (eds.) Fundamental Aspects of Quantum Theory, vol. 144, pp. 319–320. Plenum, New York (1986)

    Google Scholar 

  8. Bocchieri, P., Loinger, A.: Nonexistence of the Aharonov-Bohm effect. Nuovo Cim. A 47, 475–482 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  9. Bocchieri, P., Loinger, A., Siracusa, G.: Nonexistence of the Aharonov-Bohm effect. 2. Discussion of the experiments. Nuovo Cim. A 51, 1–16 (1979)

    Article  ADS  Google Scholar 

  10. Casati, G., Guarneri, I.: Aharonov-Bohm effect from the hydrodynamical viewpoint. Phys. Rev. Lett. 42, 1579–1581 (1979)

    Article  ADS  Google Scholar 

  11. Davies, E.B.: One-Parameter Semigroups. Academic Press, London (1980)

    MATH  Google Scholar 

  12. Da̧browski, L., Š ťovíček, P.: Aharonov-Bohm effect with δ-type interaction. J. Math. Phys. 39, 47–62 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  13. de Oliveira, C.R., Pereira, M.: (2008, in preparation)

  14. Ehrenberg, W., Siday, R.E.: The refractive index in electron optics and the principles of dynamics. Proc. Phys. Soc. Lond. Sect. B 62, 8–21 (1949)

    Article  ADS  Google Scholar 

  15. Eskin, G.: Inverse boundary value problems and the Aharonov-Bohm effect. Inverse Probl. 19, 49–62 (2003)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  16. Franz, W.: Elektroneninterferenzen im Magnetfeld. Verh. Dtsch. Phys. Ges. 2, 65 (1939)

    Google Scholar 

  17. Greenberger, D.M.: Reality and significance of the Aharonov-Bohm effect. Phys. Rev. D 23, 1460–1462 (1981)

    Article  ADS  Google Scholar 

  18. Home, D., Sengupta, S.: A critical re-examination of the Aharonov-Bohm effect. Am. J. Phys. 51, 942–947 (1983)

    Article  ADS  Google Scholar 

  19. Huerfano, R.S., López, M.A., Socolovsky, M.: Geometry of the Aharonov-Bohm effect (2007). Preprint: arXiv:math-ph/0701050

  20. Jackson, J.D.: Classical Electrodynamics, 3rd edn. Wiley, New York (1999)

    MATH  Google Scholar 

  21. Klein, U.: Comment on “Condition for nonexistence of Aharonov-Bohm effect”. Phys. Rev. D 23, 1463–1465 (1981)

    Article  ADS  Google Scholar 

  22. Kretzschmar, M.: Aharonov-Bohm scattering of a wave packet of finite extension. Z. Phys. 185, 84–96 (1965)

    Article  MATH  ADS  Google Scholar 

  23. Lipkin, H.J.: Fringing fields and criticisms of the Aharonov-Bohm effect. Phys. Rev. D 23, 1466–1467 (1981)

    Article  ADS  Google Scholar 

  24. Magni, C., Valz-Gris, F.: Can elementary quantum mechanics explain the Aharonov-Bohm effect? J. Math. Phys. 36, 177–186 (1995)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  25. Martin, C.: A mathematical model for the Aharonov-Bohm effect. Lett. Math. Phys. 1, 155–163 (1976)

    Article  ADS  Google Scholar 

  26. Olariu, S., Popesku, I.I.: The quantum effects of electromagnetic fluxes. Rev. Mod. Phys. 57, 339–436 (1985)

    Article  ADS  Google Scholar 

  27. Peshkin, M., Tonomura, A.: The Aharonov-Bohm Effect. LNP, vol. 340. Springer, Berlin (1989)

    Google Scholar 

  28. Reed, M., Simon, B.: Functional Analysis. Academic Press, New York (1980). Revised edition

    MATH  Google Scholar 

  29. Roy, S.M.: Condition for nonexistence of Aharonov-Bohm effect. Phys. Rev. Lett. 44, 111–114 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  30. Ruijsenaars, S.N.M.: The Aharonov-Bohm effect and scattering theory. Ann. Phys. 146, 1–34 (1983)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  31. Weder, R.: The Aharonov-Bohm effect and time-dependent inverse scattering theory. Inverse Probl. 18, 1041–1056 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  32. Weisskopf, V.F.: In: Brittin, W.E. (ed.) Lectures in Theoretical Physics, vol. III, pp. 67–70. Interscience, New York (1961)

    Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemática, UFSCar, São Carlos, SP, 13560-970, Brazil

    César R. de Oliveira & Marciano Pereira

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  1. César R. de Oliveira
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  2. Marciano Pereira
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Correspondence to César R. de Oliveira.

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M. Pereira on leave of absence from Universidade Estadual de Ponta Grossa, PR, Brazil.

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de Oliveira, C.R., Pereira, M. Mathematical Justification of the Aharonov-Bohm Hamiltonian. J Stat Phys 133, 1175–1184 (2008). https://doi.org/10.1007/s10955-008-9631-y

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