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 Holder continuous. In order to work in weighted Holder 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.
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