Skip to main content
SpringerLink
Log in

Three-dimensional Direct and Inverse Scattering for the Schrödinger Equation with a General Nonlinearity

  • Conference paper
  • First Online:
Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 236))

Abstract

We discuss the direct and inverse scattering theory for the nonlinear Schrödinger equation \(-\triangle u(x)+h(x,|u(x)|)u(x)=k^{2}u(x),\quad x \epsilon\mathbb{R}^{3}\) where h is a very general and possibly singular combination of potentials. We prove first that the direct scattering problem has a unique bounded solution. We establish also the asymptotic behaviour of scattering solutions. A uniqueness result and a representation formula is proved for the inverse scattering problem with general scattering data. The method of Born approximation is applied for the recovery of jumps in the unknown function from general scattering data and fixed angle data.

Mathematics Subject Classification (2010). Primary 35P25; Secondary 35R30.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin, Heidelberg, 1998.

    Google Scholar 

  2. M. Harju, On the direct and inverse scattering problems for a nonlinear threedimensional Schrödinger equation, Doctoral Dissertation, University of Oulu, 2010.

    Google Scholar 

  3. P. Ola, L. Päivärinta and V. Serov, Recovering singularities from backscattering in two dimensions. Commun. Partial Differ. Equ. 26 (2001), 697–715.

    Article  MATH  Google Scholar 

  4. L. Päivärinta and V. Serov, Recovery of singularities of a multidimensional scattering potential. SIAM J. Math. Anal. 29 (1998), 697–711.

    Article  MathSciNet  MATH  Google Scholar 

  5. L. Päivärinta and V. Serov, New mapping properties for the resolvent of the Laplacian and recovery of singularities of a multi-dimensional scattering potential. Inverse Problems 17 (2001), 1321–1326.

    Article  MathSciNet  MATH  Google Scholar 

  6. L. Päivärinta and V. Serov, Recovery of jumps and singularities in the multidimensional Schrödinger operator from limited data. Inverse Probl. Imaging 1 (2007), 525–535.

    Article  MathSciNet  MATH  Google Scholar 

  7. L. Päivärinta, V. Serov and E. Somersalo, Reconstruction of singularities of a scattering potential in two dimensions. Adv. Appl. Math. 15 (1994), 97–113.

    Article  MATH  Google Scholar 

  8. L. Päivärinta and E. Somersalo, Inversion of discontinuities for the Schrödinger equation in three dimensions. SIAM J. Math. Anal. 22 (1991), 480–499.

    Article  MathSciNet  MATH  Google Scholar 

  9. J.M. Reyes, Inverse backscattering for the Schrödinger equation in 2D. Inverse Problems 23 (2007), 625–643.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Ruiz, Recovery of the singularities of a potential from fixed angle scattering data. Commun. Partial Differ. Equ. 26 (2001), 1721–1738.

    Article  MATH  Google Scholar 

  11. A. Ruiz and A. Vargas, Partial recovery of a potential from backscattering data. Commun. Partial Differ. Equ. 30 (2005), 67–96.

    Article  MathSciNet  MATH  Google Scholar 

  12. Y. Saito, Some properties of the scattering amplitude and the inverse scattering problem. Osaka J. Math. 19 (1982), 527–547.

    MathSciNet  MATH  Google Scholar 

  13. V. Serov, Inverse Born approximation for the nonlinear two-dimensional Schrödinger operator. Inverse Problems 23 (2007), 1259–1270.

    Article  MathSciNet  MATH  Google Scholar 

  14. V. Serov, Inverse Born approximation for the generalized nonlinear Schrödinger operator in two dimensions. Modern Phys. Lett. B 22 (2008), 2257–2275.

    Article  MathSciNet  MATH  Google Scholar 

  15. V. Serov, An inverse Born approximation for the general nonlinear Schrödinger operator on the line. J. Phys. A: Math. Theor. 42 (2009), 332002.

    Article  MathSciNet  Google Scholar 

  16. V. Serov, Inverse fixed energy scattering problem for the generalized nonlinear Schrödinger operator. Inverse Problems 28 (2012), 025002.

    Article  MathSciNet  Google Scholar 

  17. V. Serov and M. Harju, Reconstruction of discontinuities in the nonlinear onedimensional Schrödinger equation from limited data. Inverse Problems 23 (2007), 493–506.

    Article  MathSciNet  MATH  Google Scholar 

  18. V. Serov and M. Harju, Partial recovery of potentials in generalized nonlinear Schrödinger equations on the line. J. Math. Phys. 48 (2007), 083512.

    Article  MathSciNet  Google Scholar 

  19. V. Serov and M. Harju, A uniqueness theorem and reconstruction of singularities for a two-dimensional nonlinear Schrödinger equation. Nonlinearity 21 (2008), 1323– 1337.

    Article  MathSciNet  MATH  Google Scholar 

  20. V. Serov, M. Harju and G. Fotopoulos, Direct and inverse scattering for nonlinear Schrödinger equation in 2D, J. Math. Phys. 53 (2012), 123522.

    Article  Google Scholar 

  21. V. Serov and J. Sandhu, Inverse backscattering problem for the generalized nonlinear Schrödinger operator in two dimensions. J. Phys. A:Math. Theor. 43 (2010), 325206.

    Article  MathSciNet  Google Scholar 

  22. E. Zeidler, Applied Functional Analysis: Part I. Applications to Mathematical Physics, Springer, New York, 1995.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Oulu, 3000, FIN-90014, Oulu, Finland

    Markus Harju & Valery Serov

Authors
  1. Markus Harju
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Valery Serov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Markus Harju .

Editor information

Editors and Affiliations

  1. Departamento de Análisis Matemático, Universidad de Sevilla, Sevilla, Spain

    Manuel Cepedello Boiso

  2. Department of Mathematics, The Royal Institute of Technology, Stockholm, Sweden

    Håkan Hedenmalm

  3. Department of Mathematics & Computer Sci, Free University Amsterdam, Amsterdam, The Netherlands

    Marinus A. Kaashoek

  4. Departamento de Análisis Matemático, Universidad de Sevilla, Sevilla, Spain

    Alfonso Montes Rodríguez

  5. Department of Mathematics, Brown University, Providence, Rhode Island, USA

    Sergei Treil

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Basel

About this paper

Cite this paper

Harju, M., Serov, V. (2014). Three-dimensional Direct and Inverse Scattering for the Schrödinger Equation with a General Nonlinearity. In: Cepedello Boiso, M., Hedenmalm, H., Kaashoek, M., Montes Rodríguez, A., Treil, S. (eds) Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation. Operator Theory: Advances and Applications, vol 236. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0648-0_16

Download citation

Publish with us

Policies and ethics

Search

Navigation