Skip to main content

Asymptotic and approximate formulas in the inverse scattering problem for the Schrödinger operator

Part of the Lecture Notes in Mathematics book series (LNM,volume 1218)

Keywords

  • Asymptotic Formula
  • Inverse Scattering
  • Approximation Formula
  • Eikonal Equation
  • Inverse Scattering Problem

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Agmon, S.: Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Nor. Sup. Pisa (4) 2(1975), 151–218.

    MathSciNet  MATH  Google Scholar 

  2. Agranovich, Z. S. and V. A. Marchenko: The Inverse Problem of Scattering Theory (English translation), Gordon and Breach, New York, 1963.

    MATH  Google Scholar 

  3. Amrein, W., J. Jauch and K. Sinha: Scattering theory in quantum mechanics, Lecture Notes and Supplements in Physics, Benjamin, Reading, 1977.

    MATH  Google Scholar 

  4. Balres, G. (CEREMADE, Université de Paris IX, Paris, France): Private communications, 1985.

    Google Scholar 

  5. Cheney, M.: Inverse scattering in dimension two, J. Math. Phys. 25 (1984), 94–107.

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  6. Faddeev, L. D.: Uniqueness of the inverse scattering problem, Vestn. Leningr. Univ. 7(1956), 126–130.

    MathSciNet  Google Scholar 

  7. Friedman, A.: On the properties of a singular Strum-Liouville equation determined by its spectral functions, Michigan Math. J. 4(1957), 137–145.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Gel'fand, I. M. and B. M. Levitan: On the determination of a differential equation by its spectral function, Izv. Akad. Nauk. SSR ser. Math. 15(1951), 309–360 (English translation: Amer. Math. Soc. Transl. ser. 2, 1(1955), 253–304).

    MathSciNet  Google Scholar 

  9. Isozaki, H. and H. Kitada: Asymptotic behavior of the scattering amplitude at high energies, to appear in Scientific Papers of the College of Arts and Sciences, The University of Tokyo, 1986.

    Google Scholar 

  10. Kato, T.: Perturbation theory for linear operators, 2nd edition, Springer, New York, 1976.

    CrossRef  MATH  Google Scholar 

  11. Kuroda, S. T.: On the existence and the unitary property of the scattering operator, Nuovo Cimento 12(1959), 431–454.

    CrossRef  MATH  Google Scholar 

  12. Mochizuki, K. and J. Uchiyama: Radiation conditions and spectral theory for 2-body Schrödinger operators with "oscillating" long-range potentials I, J. Math. Kyoto Univ., 18(1978), 377–408.

    MathSciNet  MATH  Google Scholar 

  13. _____: Radiation conditions and spectral theory for 2-body Schrödinger operators with "oscillating" long-range potentials II, J. Math. Kyoto Univ., 19(1979), 47–70.

    MathSciNet  MATH  Google Scholar 

  14. _____: Radiation conditions and spectral theory for 2-body Schrödinger operators with "oscillating" long-range potentials III, J. Math. Kyoto Univ. 21(1981), 605–618.

    MathSciNet  MATH  Google Scholar 

  15. _____: Time dependent representations of the stationary wave operators for "oscillating" long-range potentials, J. Math. Kyoto Univ., 18(1982), 947–972.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. Newton, R.: Inverse scattering, II, J. Math. Phys. 21(1980), 1698–1715.

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  17. Newton, R.: Inverse scattering, III, J. Math. Phys. 22(1981), 2191–2200.

    CrossRef  ADS  MathSciNet  Google Scholar 

  18. _____: Inverse scattering, IV, J. Math. Phys. 23(1982), 592–604.

    ADS  Google Scholar 

  19. Prosser, R.: Formal solutions of inverse scattering problems, IV. Error estimates, J. Math. Phys. 23(1982), 2127–2130.

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  20. Saitō, Y.: Spectral and scattering theory for second-order differential operators with operator-valued coefficients, Osaka J. Math. 9(1972), 463–498.

    MathSciNet  MATH  Google Scholar 

  21. _____: On the S-matrix for Schrödinger operators with long-range potentials, J. Reine Angew. Math. 314(1980), 99–116.

    MathSciNet  MATH  Google Scholar 

  22. _____: Some properties of the scattering amplitude and the inverse scattering problem, Osaka J. Math. 19(1982), 527–547.

    ADS  MathSciNet  MATH  Google Scholar 

  23. _____: An inverse problem in potential theory and the inverse scattering problem, J. Math. Kyoto Univ. 22(1982), 315–329.

    MathSciNet  Google Scholar 

  24. _____: An asymptotic behavior of S-matrix and the inverse scattering problem, J. Math. Phys. 25(1984), 3105–3111.

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  25. ______: Schrödinger operators with a nonspherical radiation condition, 1985, to appear in Pacific J. Math.

    Google Scholar 

  26. _____: An approximation formula in the inverse scattering problem, 1985, to appear in J. Math. Phys.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1986 Springer-Verlag

About this paper

Cite this paper

Saitō, Y. (1986). Asymptotic and approximate formulas in the inverse scattering problem for the Schrödinger operator. In: Balslev, E. (eds) Schrödinger Operators, Aarhus 1985. Lecture Notes in Mathematics, vol 1218. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073051

Download citation

  • DOI: https://doi.org/10.1007/BFb0073051

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16826-3

  • Online ISBN: 978-3-540-47119-6

  • eBook Packages: Springer Book Archive