Communications in Mathematical Physics

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

Inverse backscattering in two dimensions

  • G. Eskin
  • J. Ralston


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.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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Copyright information

© Springer-Verlag 1991

Authors and Affiliations

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

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