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A Simple Proof of Magnetic and Electric Aharonov-Bohm Effects

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Abstract

Magnetic Aharonov-Bohm effect (AB effect) was studied in hundreds of papers starting with the seminal paper of Aharonov and Bohm (Phys Rev 115:485, 1959). We give a new proof of the magnetic Aharonov-Bohm effect without using the scattering theory and the theory of inverse boundary value problems. We consider separately the cases of one and several obstacles. The electric AB effect was studied much less. We give the first proof of the electric AB effect in domains with moving boundaries. When the boundary does not move with the time the electric AB effect is absent.

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References

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. Ballesteros M., Weder R.: High-velocity estimates for the scattering operator and Aharonov-Bohm effect in three dimensions. Commun. Math. Phys. 283, 345–398 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  3. Ballesteros M., Weder R.: The Aharonov-Bohm effect and Tonomura et al. experiments. Rigorous results. J. Math. Phys. 50, 122108 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  4. Ballesteros M., Weder R.: Aharonov-Bohm effect and high-velocity estimates of solutions to the Schrödinger equations. Commun. Math. Phys. 303(1), 175–211 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Eskin G.: Inverse problems for the Schrödinger equations with time-dependent electromagnetic potentials and the Aharonov-Bohm effect. J. Math. Phys. 49, 022105 (2008)

    Article  MathSciNet  Google Scholar 

  6. Eskin G.: Optical Aharonov-Bohm effect: inverse hyperbolic problem approach. Commun. Math. Phys. 284(2), 317–343 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Eskin G.: Inverse boundary problems and the Aharonov-Bohm effect. Inverse Problems 19, 49–63 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Eskin G.: Inverse problems for the Schrödinger operators with electromagnetic potentials in domains with obstacles. Inverse Problems 19, 985–996 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Eskin G.: Inverse boundary value problems in domains with several obstacles. Inverse Problems 20, 1497–1516 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. Eskin G.: Inverse problems for Schrödinger equations with Yang-Mills potentials in domains with obstacles and the Aharonov-Bohm effect. J. Phys. Conf. Ser. 12, 23–32 (2005)

    Article  ADS  Google Scholar 

  11. Eskin G., Isozaki H.: Gauge equivalence and Inverse Scattering for Louge-Range Magnetic Potentials. Russ. J. Math. Phys. 18(1), 54–63 (2010)

    Article  MathSciNet  Google Scholar 

  12. Eskin G., Isozaki H., O’Dell S.: Gauge equivalence and inverse scattering for Aharonov-Bohm effect. Comm. in PDE 35, 2164–2194 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Isakov V.: Carleman type estimates in an anisotropic case and applications. J. Diff. Eq. 105, 217–238 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kannai Y.: Off diagonal short time asymptotics for fundamental solutions of diffusions equations. Commun. in PDE 2(8), 781–830 (1977)

    Article  MathSciNet  Google Scholar 

  15. Nicoleau F.: An inverse scattering problem with the Aharonov-Bohm effect. J. Math. Phys. 41, 5223–5237 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Roux Ph., Yafaev D.: The scattering matrix for the Schrödinger operator with a long-range electro-magnetic potential. J. Math. Phys. 44, 2762–2786 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. 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 (1986)

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. Weder, R.: The electric Aharonov-Bohm effect. J. Math. Phys. 52, no. 5, 052109, 17 pp. (2011)

  22. Weder R.: Inverse scattering at fixed quasi-energy for potentials periodic in time. Inverse problems 20, 893 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Yafaev D.: Scattering by magnetic fields. St. Petersburg Math. J. 17, 675–695 (2006)

    MathSciNet  Google Scholar 

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Correspondence to G. Eskin.

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Communicated by I.M. Sigal

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Eskin, G. A Simple Proof of Magnetic and Electric Aharonov-Bohm Effects. Commun. Math. Phys. 321, 747–767 (2013). https://doi.org/10.1007/s00220-013-1727-9

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