Skip to main content
SpringerLink
Log in

Inverse Scattering for the Magnetic Schrödinger Operator on Surfaces with Euclidean Ends

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We prove a fixed frequency inverse scattering result for the magnetic Schrödinger operator (or connection Laplacian) on surfaces with Euclidean ends. We show that, under suitable decaying conditions, the scattering matrix for the operator determines both the gauge class of the connection and the zeroth order potential.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Albin P., Guillarmou C., Tzou L., Uhlmann G.: Inverse boundary problems for systems in two dimensions. Annales Henri Poincaré 6, 1551–1571 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Ballesteros M., Weder R.: High-velocity estimates for the scattering operator and Aharonov–Bohm ffect in three dimensions. Commun. Math. Phys. 1, 345–398 (2009)

    Article  ADS  MATH  Google Scholar 

  3. 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, 54 (2015)

    Article  MATH  Google Scholar 

  4. 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  ADS  MATH  Google Scholar 

  5. Farkas, H.M., Kra, I.: Riemann surfaces. In: Graduate Texts in Mathematics, 2nd edn, vol. 71. Springer, New York (1992)

  6. Guillarmou, C., Tzou, L.: Calderón inverse problem for Schrodinger operator on Riemann surfaces. In: Proceedings ANU (2009)

  7. Guillarmou C., Tzou L.: Identification of a connection from cauchy data on a Riemann surface with boundary. GAFA 21(2), 393–418 (2011)

    MathSciNet  MATH  Google Scholar 

  8. Guillarmou C., Salo M., Tzou L.: Inverse scattering at fixed energy on surfaces with Euclidean ends. Commun. Math. Phys. 303(3), 761–784 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Imanuvilov O.Y., Uhlmann G., Yamamoto M.: Partial Cauchy data for general second order elliptic operators in two dimensions. Publ. Res. Inst. Math. Sci. 48, 971–1055 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Joshi M.S., Sá Barreto A.: Recovering asymptotics of metrics from fixed energy scattering data. Invent. Math. 137, 127–143 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Joshi M.S., Sá Barreto A.: Inverse scattering on asymptotically hyperbolic manifolds. Acta Math. 184, 41–86 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kobayashi, S.: Differential geometry of complex vector bundles. Publications of the Mathematical Society of Japan, 15. Kan Memorial Lectures, 5. Princeton University Press, Princeton, NJ; Iwanami Shoten, Tokyo, pp. xii+305 (1987)

  13. Krantz S.G.: Function Theory of Several Complex Variables. AMS Chelsea Publishing, Providence, Rhode Island (2001)

    Book  MATH  Google Scholar 

  14. Melrose R.B.: The Atiyah–Patodi–Singer Index Theorem. AK Peters, Wellesley (1993)

    MATH  Google Scholar 

  15. Melrose R.B.: Geometric Scattering Theory. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  16. Nakamura G., Sun Z., Uhlmann G.: Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field. Math. Ann. 3, 377–388 (1995)

    Article  MATH  Google Scholar 

  17. Päivärinta L., Salo M., Uhlmann G.: Inverse scattering for the magnetic Schroedinger operator. J. Funct. Anal. 259, 1771–1798 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Salo M.: Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field. Commun. Partial Differ. Equ. 31(10–12), 1639–1666 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Tzou, L.: The reflection principle and Calderón problems with partial data, preprint

  20. Uhlmann G., Vasy A.: Fixed energy inverse problem for exponentially decreasing potentials. Methods Appl. Anal. 9(2), 239–247 (2002)

    MathSciNet  MATH  Google Scholar 

  21. Weder R., Yafaev D.: On inverse scattering at a fixed energy for potentials with a regular behaviour at infinity. Inverse Probl. 21, 1937–1952 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Zworski M.: Semiclassical Analysis. American Mathematical Society, Providence, Rhode Island (2012)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Jyvaskyla, Jyväskylä, Finland

    Valter Pohjola

  2. University of Sydney, Sydney, Australia

    Leo Tzou

Authors
  1. Valter Pohjola
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leo Tzou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leo Tzou.

Additional information

Communicated by H.-T. Yau

Leo Tzou is supported by Australian Research Council FT-130101346. Valter Pohjola is employed under Academy of Finland grant AKA-251469.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pohjola, V., Tzou, L. Inverse Scattering for the Magnetic Schrödinger Operator on Surfaces with Euclidean Ends. Commun. Math. Phys. 356, 107–142 (2017). https://doi.org/10.1007/s00220-017-2982-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-017-2982-y

Search

Navigation