Advertisement

Communications in Mathematical Physics

, Volume 138, Issue 3, pp 451–486 | Cite as

Inverse backscattering in two dimensions

  • G. Eskin
  • J. Ralston
Article

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.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [BC] Beals, R., Coifman, R.: Multidimensional inverse scattering and nonlinear partial differential equations. Proc. Symp. Pure Math.43, 45–70. Providence, RI: AMS, 1985Google Scholar
  2. [BML] Bayliss, A., Li, Y., Morawetz, C.: Scattering by potentials using hyperbolic methods. Math. Comp.52, 321–338 (1989)Google Scholar
  3. [C] Cheney, M.: Inverse scattering in dimension 2. J. Math. Phys.25, 94–107 (1984)Google Scholar
  4. [ER] Eskin, G., Ralston, J.: The inverse backscattering problem in three dimensions. Commun. Math. Phys.124, 169–215 (1989)Google Scholar
  5. [F] Faddeev, L. D.: Inverse problem of quantum scattering theory, II. J. Sov. Math.5, 334–396 (1976)Google Scholar
  6. [M] Melin, A.: The Lippman-Schwinger equation treated as a characteristic Cauchy problem. Seminaire sur EDP 1988–1989. Exposé IV. Ecole Polytech. PalaiseauGoogle Scholar
  7. [MU] Melrose, R., Uhlmann, G.: Introduction to microlocal analysis with applications to scattering theory, preprint 1989Google Scholar
  8. [NA] Nachman, A., Ablowitz, M.: A multidimensional inverse scattering method. Stud. Appl. Math.71, 243–250 (1984)Google Scholar
  9. [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
  10. [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
  11. [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
  12. [W] Weder, R.: Generalized limiting absorption method and multidimensional inverse scattering theory. preprint (1990)Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • G. Eskin
    • 1
  • J. Ralston
    • 1
  1. 1.Department of MathematicsUCLALos AngelesUSA

Personalised recommendations