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A New Version of the Aharonov–Bohm Effect


We propose a simple situation in which the magnetic Aharonov–Bohm potential influences the values of the deficiency indices of the initial Schrödinger operator, so determining whether the particle interacts with the solenoid or not. Even with the particle excluded from the magnetic field, the number of self-adjoint extensions of the initial Hamiltonian depends on the magnetic flux. This is a new point of view of the Aharonov–Bohm effect.

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  1. 1.

    Abramowitz, A.: Handbook of Mathematical Functions with Formulas, Graphs Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55 (1964)

  2. 2.

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

  3. 3.

    Ahari, M.T., Ortiz, G., Seradjeh, B.: On the role of self-adjointness in the continuum formulation of topological quantum phases. Am. J. Phys. 84, 858–868 (2016)

  4. 4.

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

  5. 5.

    Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space. Dover Publications, New York (1993)

  6. 6.

    Audretsch, J., Skarzhinsky, V.D., Voronov, B.L.: Elastic scattering and bound states in the Aharonov–Bohm potential superimposed by an attractive \(\rho ^{-2}\) potential. J. Phys. A 34, 235–250 (2001)

  7. 7.

    Ávila-Aoki, M., Cisneros, C., Martínez-y-Romero, R.P., Núñez-Yépez, H.N., Salas-Brito, A.L.: Classical and quantum motion in an inverse square potential. Phys. Lett. A 373, 418–421 (2009)

  8. 8.

    Ballesteros, M., Weder, R.: High-Velocity for the scattering operator Aharonov–Bohm effect in three dimensions. Commun. Math. Phys. 285, 345–398 (2009)

  9. 9.

    Basu-Mallick, B., Gupta, K.S.: Bound states in one-dimensional quantum N-body systems with inverse square interaction. Phys. Lett. A 292, 36–42 (2001)

  10. 10.

    Batelaan, H., Tonomura, A.: The Aharonov–Bohm effects: variations on a subtle theme. Phys. Today 62, 38–43 (2009)

  11. 11.

    Becker, M., Batelaan, H.: Experimental test for approximately dispersionless forces in the Aharonov–Bohm effect. Europhys. Lett. 115, 10011 (2016)

  12. 12.

    Bonneau, G., Faraut, J., Valent, G.: Self-adjoint extensions of operators the teaching of quantum mechanics. Am. J. Phys. 69, 322–331 (2001)

  13. 13.

    Brattan, D.K., Ovdat, O., Akkermans, E.: Scale anomaly of a Lifshitz scalar: a universal quantum phase transition to discrete scale invariance. Phys. Rev. D 97, 061701 (2018)

  14. 14.

    Brattan, D.: \({\cal{N}}=2\) supersymmetry anisotropic scale invariance. Phys. Rev. D 98, 036005 (2018)

  15. 15.

    Camblong, H.E., Epele, L.N., Fanchiotti, H., García-Canal, C.A.: Renormalization of the inverse square potential. Phys. Rev. Lett. 85, 1590–1593 (2000)

  16. 16.

    Caprez, A., Barwick, B., Batelaan, H.: Macroscopic test of the Aharonov–Bohm effect. Phys. Rev. Lett. 99, 210401 (2007)

  17. 17.

    Dabrowski, L., Šťovíček, P.: Aharonov–Bohm effect with \(\delta \)-type interaction. J. Math. Phys. 39, 47–62 (1998)

  18. 18.

    de Oliveira, C.R., Pereira, M.: Mathematical justification of the Aharonov–Bohm Hamiltonian. J. Stat. Phys. 133, 1175–1184 (2008)

  19. 19.

    de Oliveira, C.R., Pereira, M.: Scattering self-adjoint extensions of the Aharonov–Bohm Hamiltonian. J. Phys. A 43, 354011 (2010)

  20. 20.

    de Oliveira, C.R., Pereira, M.: Impenetrability of Aharonov–Bohm solenoids. Proof of norm resolvent convergence. Lett. Math. Phys. 95, 41–51 (2011)

  21. 21.

    de Oliveira, C.R., Romano, R.G.: Aharonov–Bohm effect without contact with the solenoid. J. Math. Phys. 58, 102102 (2017)

  22. 22.

    Earman, J.: The role of idealizations in the Aharonov–Bohm effect. Synthese 196, 1991–2019 (2019)

  23. 23.

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

  24. 24.

    Gitman, D.M., Tyutin, I.V., Voronov, B.L.: Self-adjoint Extensions in Quantum Mechanics, PMP 62. Birkhäuser, New York (2012)

  25. 25.

    Gupta, K.S., Rajeev, S.G.: Renormalization in quantum mechanics. Phys. Rev. D 48, 5940–5945 (1993)

  26. 26.

    Helffer, B.: Effet d’Aharonov Bohm sur un état borné de l’équation de Schrödinger. Commun. Math. Phys. 119, 315–329 (1988)

  27. 27.

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

  28. 28.

    Maeda, H.: Unitary evolution of the quantum universe with a Brown–Kuchar dust. Class. Quantum Gravity 32, 235023 (2015)

  29. 29.

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

  30. 30.

    Martínez-y-Romero, R.P., Núñez-Yépez, H.N., Salas-Brito, A.L.: The two dimensional motion of a particle in an inverse square potential: classical and quantum aspects. J. Math. Phys. 54, 053509 (2013)

  31. 31.

    Peshkin, M.: Aharonov–Bohm effect in bound states: theoretical experimental status. Phys. Rev. A 23, 360–363 (1981)

  32. 32.

    Peshkin, M., Tonomura, A.: The Aharonov–Bohm Effect, LNP 340. Springer, New York (1989)

  33. 33.

    Roux, Ph, Yafaev, D.: On the mathematical theory of the Aharonov–Bohm effect. J. Phys. A 35, 7481–7492 (2002)

  34. 34.

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

  35. 35.

    Seradjeh, B., Vennettilli, M.: Surface spectra of Weyl semimetals through self-adjoint extensions. Phys. Rev. B 97, 075132 (2018)

  36. 36.

    Tonomura, A., Osakabe, N., Matsuda, T., Kawasaki, T., Endo, J., Yano, S., Yamada, H.: Evidence for Aharonov–Bohm effect with magnetic field completely shielded from electron wave. Phys. Rev. Lett. 56, 792–795 (1986)

  37. 37.

    Wei, H., Han, R., Wei, X.: Quantum phase of induced dipoles moving in a magnetic field. Phys. Rev. Lett. 75, 2071–2073 (1995)

  38. 38.

    Wilkens, M.: Quantum phase of a moving dipole. Phys. Rev. Lett. 72, 5–8 (1994)

  39. 39.

    Zhou, L.J., et al.: Smooth sharp creation of a pointlike source for a (\(3+1\))-dimensional quantum field. Phys. Rev. D 95, 085007 (2017)

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CRdO thanks the partial support by CNPq (a Brazilian government agency, under contract 303503/2018-1).

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Correspondence to César R. de Oliveira.

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de Oliveira, C.R., Romano, R.G. A New Version of the Aharonov–Bohm Effect. Found Phys (2020). https://doi.org/10.1007/s10701-020-00328-6

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  • Aharonov–Bohm effect
  • Self-adjoint extensions
  • Magnetic potential