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Optical Aharonov-Bohm Effect: An Inverse Hyperbolic Problems Approach

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Abstract

We describe the general setting for the optical Aharonov-Bohm effect based on the inverse problem of the identification of the coefficients of the governing hyperbolic equation by the boundary measurements. We interpret the inverse problem result as a possibility in principle to detect the optical Aharonov-Bohm effect by the boundary measurements.

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References

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

    MATH  ADS  MathSciNet  Google Scholar 

  2. Belishev M.: Boundary control in reconstruction of manifolds and metrics (the BC method). Inverse Problems 13, R1–R45 (1997)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. Berry M., Chambers R., Large M., Upstill C., Walmsley J.: Eur. J. Phys. 1, 154 (1980)

    Article  MathSciNet  Google Scholar 

  4. Cook R., Fearn H., Millouni P.: Am. J. Phys. 63, 705 (1995)

    Article  ADS  Google Scholar 

  5. Eskin G.: A new approach to the hyperbolic inverse problems. Inverse problems 22(3), 815–831 (2006)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. Eskin G.: A new approach to the hyperbolic inverse problems II: global step. Inverse Problems 23, 2343–2356 (2007)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Eskin G.: Inverse hyperbolic problems with time-dependent coefficients. Comm. in PDE 32, 1737–1758 (2007)

    MATH  MathSciNet  Google Scholar 

  8. 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  ADS  MathSciNet  Google Scholar 

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. Eskin G., Ralston J.: Inverse scattering problem for the Schrödinger equation with magnetic and electric potentials. The IMA Volumes in Mathematics and its applications, Vol 90., pp. 147–166. Springer, New York (1997)

    Google Scholar 

  11. Eskin G., Ralston J.: Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy. Commun. Math. Phys. 173, 199–224 (1995)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. Gordon W.: Ann. Phys. (Leipzig) 72, 421 (1923)

    ADS  Google Scholar 

  13. Hirsch M.: Differential Topology. New York, Springer (1976)

    MATH  Google Scholar 

  14. Katchalov A., Kurylev Y., Lassas M.: Inverse Boundary Spectral Problems. Chapman & Hall, Boca Baton (2001)

    MATH  Google Scholar 

  15. Katchalov A., Kurylev Y., Lassas M.: Energy measurements and equivalence of boundary data for inverse problems on noncompact manifolds. IMA Volumes 137, 183–214 (2004)

    MathSciNet  Google Scholar 

  16. Kurylev Y., Lassas M.: Hyperbolic inverse problems with data on a part of the boundary. AMS/1P Stud. Adv. Math 16, 259–272 (2000)

    MathSciNet  Google Scholar 

  17. Katchalov A., Kurylev Y., Lassas M., Mandache N.: Equivalence of time-domain inverse problems and boundary spectral problems. Inverse Problems 20(2), 419–436 (2004)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. Leonhardt V., Philbin T.: General relativity in Electrical Engineering. New J. Phys. 8, 247 (2006)

    Article  ADS  Google Scholar 

  19. Leonhardt V., Piwnicki P.: Phys. Rev. A60, 4301 (1999)

    ADS  Google Scholar 

  20. Leonhardt V., Piwnicki P.: Phys. Rev. Lett. 84, 822 (2000)

    Article  ADS  Google Scholar 

  21. Lee J., Uhlmann G.: Determining anisotropic real-analytic conducivity by boundary measurements. Comm. Pure Appl. Math. 42, 1097–1112 (1989)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

  23. Nakamura G., Sun Z., Uhlmann G.: Global identifiability for inverse problem for the Schrödinger equation in a magnetic field. Math. Ann. 303(1), 377–88 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  24. Novello, M., Visser, M., Volovik, G. (eds): Artificial Black Holes. World Scientific, Singapore (2002)

    Google Scholar 

  25. O’Dell S.: Inverse scattering for the Laplace-Beltrami operators with complex-valued electromagnetic potentials and embedded obstacles. Inverse Problems 22(5), 1579–1603 (2006)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. Olarin S., Popescu I. Iovitzu: The quantum effects of electromagnetic fluxes. Rev. Mod. Phys. 57(N2), 339–436 (1985)

    Article  ADS  Google Scholar 

  27. Quan Pham Mau: Arch. Rat. Mech. Anal. 1, 54 (1957)

    Article  Google Scholar 

  28. Roux P., de Rosny J., Tanter M., Fink M.: Phys. Rev. Lett. 79, 317 (1997)

    Article  Google Scholar 

  29. Vivanco F., Melo F., Coste C., Lund F.: Surface Wave Scattering by a Vertical Vortex and the Symmetry of the Aharonov-Bohm Wave Function. Phys. Rev. Lett. 83, 1966–1969 (1999)

    Article  ADS  Google Scholar 

  30. Weder R.: The Aharonov-Bohm effect and time-dependent inverse scattering theory. Inverse Problems 18(4), 1041–1056 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  31. Wu T., Yang C.: Phys. Rev. D 12, 3845 (1975)

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, UCLA, Los Angeles, CA, 90095-1555, USA

    G. Eskin

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  1. G. Eskin
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Correspondence to G. Eskin.

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Communicated by P. Constantin

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Eskin, G. Optical Aharonov-Bohm Effect: An Inverse Hyperbolic Problems Approach. Commun. Math. Phys. 284, 317–343 (2008). https://doi.org/10.1007/s00220-008-0647-6

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  • DOI: https://doi.org/10.1007/s00220-008-0647-6

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