# Inverse backscattering in two dimensions

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## Abstract

This article extends the authors' previous results (Commun. Math. Phys. where

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

*b*(0)=0 and*b*(ζ) is Hölder continuous. In order to work in weighted Hölder spaces as before, the constant β and the function*b*(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
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© Springer-Verlag 1991