Abstract
We analyze spin-0 relativistic scattering of charged particles propagating in the exterior, \({\Lambda \subset \mathbb{R}^3}\), of a compact obstacle \({K \subset \mathbb{R}^3}\). The connected components of the obstacle are handlebodies. The particles interact with an electromagnetic field in Λ and an inaccessible magnetic field localized in the interior of the obstacle (through the Aharonov–Bohm effect). We obtain high-momenta estimates, with error bounds, for the scattering operator that we use to recover physical information: we give a reconstruction method for the electric potential and the exterior magnetic field and prove that, if the electric potential vanishes, circulations of the magnetic potential around handles (or equivalently, by Stokes’ theorem, magnetic fluxes over transverse sections of handles) of the obstacle can be recovered, modulo 2π. We additionally give a simple formula for the high momenta limit of the scattering operator in terms of certain magnetic fluxes, in the absence of electric potential. If the electric potential does not vanish, the magnetic fluxes on the handles above referred can be only recovered modulo π and the simple expression of the high-momenta limit of the scattering operator does not hold true.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adams, R.A., Fournier, J.J.F.: Sobolev spaces, 2nd edn. Pure and Applied Mathematics (Amsterdam), vol. 140. Elsevier/Academic Press, Amsterdam, p. xiv + 305 (2003)
Aharonov Y., Bohm D.: Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115, 485–491 (1959)
Arians S.: Geometric approach to inverse scattering for the Schrödinger’s equation with magnetic and electric potentials. J. Math. Phys. 38, 2761–2773 (1997)
Ballesteros M., Weder R.: High-velocity estimates for the scattering operator and Aharonov–Bohm effect in three dimensions. Commun. Math. Phys. 285, 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 equation. Commun. Math. Phys. 303, 175–211 (2011)
Ballesteros M., Weder R.: High-Velocity Estimates for Schrödinger Operators in Two Dimensions: Long-Range Magnetic Potentials and Time-Dependent Inverse-Scattering Rev. Math. Phys. 27, 1550006 (2015)
Enss, V., Jung, W.: Geometrical approach to inverse scattering. In: Proceedings of the first maphysto workshop on inverse problems, 22–24 April 1999, Aarhus. MaPhySto Miscellanea, vol. 13. ISSN 1398-5957 (1999)
Enss V., Weder R.: The geometrical approach to multidimensional inverse scattering. J. Math. Phys. 36, 3902–3921 (1995)
Eskin G.: Inverse boundary value problems and the Aharonov–Bohm effect. Inverse Probl. 19, 49–62 (2003)
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., Isozaki H., O’Dell S.: Gauge equivalence and inverse scattering for Aharonov–Bohm effect. Commun. Partial Differ. Equ. 35, 2164–2194 (2010)
Eskin G., Isozaki H.: Gauge equivalence and inverse scattering for long-range magnetic potentials. Russ. J. Math. Phys. 18(1), 54–63 (2011)
Eskin G., Ralston J.: The Aharonov–Bohm effect in spectral asymptotics of the magnetic Schrödinger operator. Anal. PDE 7, 245–266 (2014)
Franz W.: Elektroneninterferenzen im Magnetfeld. Verh. D. Phys. Ges. (3) 20(2), 65–66 (1939)
Franz W.: Elektroneninterferenzen im Magnetfeld. Physikalische Berichte 21, 686 (1940)
Feshbach H., Villars F.: Elementary relativistic wave mechanics of spin 0 and spin \({\frac{1}{2}}\) particles. Rev. Mod. Phys. 30, 24–45 (1958)
Gérard C.: Scattering theory for Klein–Gordon equations with non-positive energy. Ann. Henri Poincaré 13, 883–941 (2012)
Greenberg M.J., Harper J.R.: Algebraic topology, A first course. Addison-Wesley, New York (1981)
Greiner W.: Relativistic quantum mechanics, 3rd edn. Springer, Berlin (1987)
Helffer B.: Effet d’Aharonov–Bohm sur un état borné de l’équation de Schrödinger (French) [The Aharonov–Bohm effect on a bound state of the Schrödinger equation] . Commun. Math. Phys. 119, 315–329 (1988)
Ito H.T., Tamura H.: Aharonov–Bohm effect in scattering by point-like magnetic fields at large separation. Ann. Henri Poincaré 2, 309–359 (2001)
Ito H.T., Tamura H.: Aharonov–Bohm effect in scattering by a chain of point-like magnetic fields. Asymptot. Anal. 34, 199–240 (2003)
Ito H.T., Tamura H.: Semiclassical analysis for magnetic scattering by two solenoidal fields. J. London Math. Soc. (2) 74, 695–716 (2006)
Jollivet A.: On inverse scattering for the multidimensional relativistic Newton equation at high energies. J. Math. Phys. 47, 062902 (2006)
Jollivet A.: On inverse problems for the multidimensional relativistic Newton equation at fixed energy. Inverse Probl. 23, 231–242 (2007)
Jollivet A.: On inverse scattering in electromagnetic field in classical relativistic mechanics at high energies. Asymptot. Anal. 55, 103–123 (2007)
Jung W.: Geometrical approach to inverse scattering for the Dirac equation. J. Math. Phys. 38, 39–48 (1997)
Newton T.D., Wigner E.P.: Localized states for elementary particles. Rev. Mod. Phys. 21, 400–406 (1949)
Nicoleau F.: A stationary approach to inverse scattering for Schrödinger operators with first order perturbation. Commun. Partial Differ. Equ. 22, 527–553 (1997)
Nicoleau F.: An inverse scattering problem with the Aharonov–Bohm effect. J. Math. Phys. 41, 5223–5237 (2000)
Reed M., Simon B.: Methods of modern mathematical physics II Fourier analysis, Self-adjointness. Academic Press, New York (1975)
Reed M., Simon B.: Methods of modern mathematical physics III scattering theory. Academic Press, New York (1979)
Roux Ph.: Scattering by a toroidal coil. J. Phys. A 36, 5293–5304 (2003)
Roux Ph., Yafaev D.R.: On the mathematical theory of the Aharonov–Bohm effect. J. Phys. A 35, 7481–7492 (2002)
Roux Ph., Yafaev D.R.: The scattering matrix for the Schrödinger operator with a long-range electromagnetic potential. J. Math. Phys. 44, 2762–2786 (2003)
Schechter, M.: Spectra of partial differential operators, 2nd edn. North-Holland Series in Applied Mathematics and Mechanics, vol. 14. North-Holland Publishing Co., Amsterdam, p. xiv+310 (1986)
Warner F.W.: Foundations of differentiable manifolds. Springer-Verlag, Berlin (1983)
Weder R.: The Aharonov–Bohm effect and time-dependent inverse scattering theory. Inverse Probl. 18, 1041–1056 (2002)
Weder R.: Selfadjointness and invariance of the essential spectrum for the Klein–Gordon equation. Helv. Phys. Acta 50, 105–115 (1977)
Weder R.: Scattering theory for the Klein–Gordon equation. J. Funct. Anal. 27, 100–117 (1978)
Weder, R.: High-velocity estimates, inverse scattering and topological effects. In: Edited by: Khruslov, E., Pastur, L., Shepelsky, D. (eds.) Spectral theory and differential equations: V. A. Marchenko’s 90th Anniversary Collection. American Mathematical Society Translations–Series 2. Advances in the Mathematical Sciences. vol. 233. AMS, Providence, pp. 225–251 (2014)
Yafaev D.R.: Scattering matrix for magnetic potentials with Coulomb decay at infinity. Integral Equ. Oper. Theory 47, 217–249 (2003)
Yafaev D.R.: Scattering by magnetic fields. St. Petersburg Math. J. 17, 875–895 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jan Dereziński.
Research partially supported by the project PAPIIT-DGAPA UNAM IN102215.
Miguel Ballesteros and Ricardo Weder: Fellows of the Sistema Nacional de Investigadores.
Rights and permissions
About this article
Cite this article
Ballesteros, M., Weder, R. Aharonov–Bohm Effect and High-Momenta Inverse Scattering for the Klein–Gordon Equation. Ann. Henri Poincaré 17, 2905–2950 (2016). https://doi.org/10.1007/s00023-016-0466-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-016-0466-9