Skip to main content

On the resonances and the inverse scattering problem for perturbed wave equations

  • 849 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 1324)

Abstract

We consider finite energy solutions of the perturbed wave equation □u+q(x,t)u=0 where x ε ℝ3, t ε ℝ. We analyse two type of problems: First, we give suitable conditions on q and we prove that there exist infinite many "resonances" λj associated with q. Secondly, we study the problem of determining q from the scattering operator associated with the above equation. We describe a uniqueness result on the inverse scattering problem and state some open problems on the subject.

Keywords

  • Wave Equation
  • Green Function
  • Compact Operator
  • Bounded Solution
  • Complex Pole

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 an expanded version on a one-hour invited Lecture presented by the author at the Latin American School of Mathematics (ELAM) held at IMPA (July 1986), Rio de Janeiro, RJ, Brasil.

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   54.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.95
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. C.E. BAUM-Emerging technology for transient and broad-band analysis and synthesis of antennas and scatterers, Proc. IEEE 64(1976) 1598–1616.

    CrossRef  MathSciNet  Google Scholar 

  2. J. COOPER and W.A. STRAUSS-The leading singularity of a wave reflected by a moving boundary, J. Diff. Equations, 54(2), (1984), 175–203.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. J. COOPER and W.A. STRAUSS-Abstract scattering theory for time-periodic systems with applications to electromagnetism, Ind. Univ.Math.J., 34(1) (1985), 33–84.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. J. COOPER; G. PERLA MENZALA & W.A. STRAUSS-On the scattering frequencies of time dependent potentials (to appear).

    Google Scholar 

  5. J.A. FERREIRA and G. PERLA MENZALA-Time dependent approach to the inverse scattering problem for wave equations with time dependent coefficients (to appear).

    Google Scholar 

  6. J.A. FERREIRA and G. PERLA MENZALA-Inverse scattering for wave equations with time-dependent coefficients: The even dimensional case (to appear).

    Google Scholar 

  7. P. LAX and R. PHILLIPS-Decaying models for the wave equation in the exterior of an obstacle, Comm. Pure Appl.Math. 22 (1969), 737–787.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. G. PERLA MENZALA-Sur l'opérateur de diffusion pour l'equation des ondes avec des potentiels dependant du temps, C.R.Sc. Paris, t. 300, no 18, (1985), 621–624.

    MathSciNet  MATH  Google Scholar 

  9. G. PERLA MENZALA-Scattering properties of wave equations with time-dependent potentials, Comp. and Math. with Appls., 12A, (1986), 457–475.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. G. PERLA MENZALA and T. SCHONBEK-Scattering frequencies for the wave equation with a potential term, J. Funct.Anal., 55(3), (1984), 297–322.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. S. STEINBERG-Meromorphic families of compact operators, Arch.Rat. Mech.Anal., 13, (1968), 372–379.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. D. THOE-On the exponential decay of solutions of the wave equation, J.Math.Anal.Appl., 16, (1966), 333–346.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. M. WEI; G. MADJA and W.A. STRAUSS-Numerical computation of the scattering frequencies for acoustic wave equations (to appear).

    Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1988 Springer-Verlag

About this paper

Cite this paper

Menzala, G.P. (1988). On the resonances and the inverse scattering problem for perturbed wave equations. In: Cardoso, F., de Figueiredo, D.G., Iório, R., Lopes, O. (eds) Partial Differential Equations. Lecture Notes in Mathematics, vol 1324. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0100795

Download citation

  • DOI: https://doi.org/10.1007/BFb0100795

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-50111-4

  • Online ISBN: 978-3-540-45928-6

  • eBook Packages: Springer Book Archive