Skip to main content
SpringerLink
Log in

Inverse backscattering in two dimensions

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

Abstract

This article extends the authors' previous results (Commun. Math. Phys.124, 169–215 (1989) to inverse scattering in two space dimensions. The new problem in two dimensions is the behavior of the backscattering amplitude near zero energy. Generically, this has the form

$$a({\xi \mathord{\left/ {\vphantom {\xi {\left| \xi \right|,}}} \right. \kern-\nulldelimiterspace} {\left| \xi \right|,}} - {\xi \mathord{\left/ {\vphantom {\xi {\left| \xi \right|,\left| \xi \right|}}} \right. \kern-\nulldelimiterspace} {\left| \xi \right|,\left| \xi \right|}}) = 2\pi (2\pi \beta + \ln \left| \xi \right|)^{ - 1} + b(\xi ),$$

whereb(0)=0 andb(ζ) is Hölder continuous. In order to work in weighted Hölder spaces as before, the constant β and the functionb(ζ) must now be interpreted as “coordinates” on the space of backscattering data. In this setting the mapping to backscattering data is again a local diffeomorphism at a dense open set in the real-valued potentials.

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

Similar content being viewed by others

References

  • [BC] Beals, R., Coifman, R.: Multidimensional inverse scattering and nonlinear partial differential equations. Proc. Symp. Pure Math.43, 45–70. Providence, RI: AMS, 1985

    Google Scholar 

  • [BML] Bayliss, A., Li, Y., Morawetz, C.: Scattering by potentials using hyperbolic methods. Math. Comp.52, 321–338 (1989)

    Google Scholar 

  • [C] Cheney, M.: Inverse scattering in dimension 2. J. Math. Phys.25, 94–107 (1984)

    Google Scholar 

  • [ER] Eskin, G., Ralston, J.: The inverse backscattering problem in three dimensions. Commun. Math. Phys.124, 169–215 (1989)

    Google Scholar 

  • [F] Faddeev, L. D.: Inverse problem of quantum scattering theory, II. J. Sov. Math.5, 334–396 (1976)

    Google Scholar 

  • [M] Melin, A.: The Lippman-Schwinger equation treated as a characteristic Cauchy problem. Seminaire sur EDP 1988–1989. Exposé IV. Ecole Polytech. Palaiseau

  • [MU] Melrose, R., Uhlmann, G.: Introduction to microlocal analysis with applications to scattering theory, preprint 1989

  • [NA] Nachman, A., Ablowitz, M.: A multidimensional inverse scattering method. Stud. Appl. Math.71, 243–250 (1984)

    Google Scholar 

  • [N] Newton, R.: Inverse scattering II, III, IV. J. Math. Phys.21, 1968–1715 (1980); J. Math. Phys.22, 2191–2200 1981 (Correction, J. Math. Phys.23, 693 (1982)) J. Math. Phys.23, 594–604 (1982)

    Google Scholar 

  • [NK] Novikov, R. G., Khenkin, G. M.: The\(\bar \partial \)-equation in multidimensional inverse scattering problems. Russ. Math. Surv.42, 109–180 (1987)

    Google Scholar 

  • [P] Prosser, R. J.: Formal solutions of inverse scattering problems I, II, III. J. Math. Phys.10, 1819–1822 (1969); J. Math. Phys.17, 1225–1779 (1976); J. Math. Phys.21, 2648–2653 (1980)

    Google Scholar 

  • [W] Weder, R.: Generalized limiting absorption method and multidimensional inverse scattering theory. preprint (1990)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, UCLA, 90024-1555, Los Angeles, CA, USA

    G. Eskin & J. Ralston

Authors
  1. G. Eskin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Ralston
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by B. Simon

Partially supported by NSF Grant DMS89-02246

Rights and permissions

Reprints and permissions

About this article

Cite this article

Eskin, G., Ralston, J. Inverse backscattering in two dimensions. Commun.Math. Phys. 138, 451–486 (1991). https://doi.org/10.1007/BF02102037

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02102037

Keywords

Search

Navigation