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A New Version of the Aharonov–Bohm Effect

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Abstract

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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References

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

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

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Book  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Book  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

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

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Authors and Affiliations

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

    César R. de Oliveira

  2. Departamento de Matemática, UFSC, Blumenau, SC, 89036-004, Brazil

    Renan G. Romano

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  1. César R. de Oliveira
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  2. Renan G. Romano
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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 50, 137–146 (2020). https://doi.org/10.1007/s10701-020-00328-6

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