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.
Similar content being viewed by others
References
Aharonov Y., Bohm D.: Significance of electromagnetic potentials in quantum theory. Phys. Rev. 115, 485 (1959)
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)
Ballesteros M., Weder R.: The Aharonov-Bohm effect and Tonomura et al. experiments. Rigorous results. J. Math. Phys. 50, 122108 (2009)
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)
Eskin G.: Inverse problems for the Schrödinger equations with time-dependent electromagnetic potentials and the Aharonov-Bohm effect. J. Math. Phys. 49, 022105 (2008)
Eskin G.: Optical Aharonov-Bohm effect: inverse hyperbolic problem approach. Commun. Math. Phys. 284(2), 317–343 (2008)
Eskin G.: Inverse boundary problems and the Aharonov-Bohm effect. Inverse Problems 19, 49–63 (2003)
Eskin G.: Inverse problems for the Schrödinger operators with electromagnetic potentials in domains with obstacles. Inverse Problems 19, 985–996 (2003)
Eskin G.: Inverse boundary value problems in domains with several obstacles. Inverse Problems 20, 1497–1516 (2004)
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)
Eskin G., Isozaki H.: Gauge equivalence and Inverse Scattering for Louge-Range Magnetic Potentials. Russ. J. Math. Phys. 18(1), 54–63 (2010)
Eskin G., Isozaki H., O’Dell S.: Gauge equivalence and inverse scattering for Aharonov-Bohm effect. Comm. in PDE 35, 2164–2194 (2010)
Isakov V.: Carleman type estimates in an anisotropic case and applications. J. Diff. Eq. 105, 217–238 (1993)
Kannai Y.: Off diagonal short time asymptotics for fundamental solutions of diffusions equations. Commun. in PDE 2(8), 781–830 (1977)
Nicoleau F.: An inverse scattering problem with the Aharonov-Bohm effect. J. Math. Phys. 41, 5223–5237 (2000)
Ruijsenaars S.N.M.: The Aharonov-Bohm effect and scattering theory. Ann. Phys. 146, 1–34 (1983)
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)
Roux Ph., Yafaev D.: On the mathematical theory of the Aharonov-Bohm effect. J. Phys. A: Math. Gen. 35, 7481–7492 (2002)
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)
Weder R.: The Aharonov-Bohm effect and time-dependent inverse scattering theory. Inverse Problems 18, 1041–1056 (2002)
Weder, R.: The electric Aharonov-Bohm effect. J. Math. Phys. 52, no. 5, 052109, 17 pp. (2011)
Weder R.: Inverse scattering at fixed quasi-energy for potentials periodic in time. Inverse problems 20, 893 (2004)
Yafaev D.: Scattering by magnetic fields. St. Petersburg Math. J. 17, 675–695 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by I.M. Sigal
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-013-1727-9