The inverse backscattering problem in three dimensions
This article is a study of the mapping from a potential q(x) on \({\mathbb{R}}^ 3\) to the backscattering amplitude associated with the Hamiltonian \(-\Delta +q(x)\). The backscattering amplitude is the restriction of the scattering amplitude a(\(\theta\),\(\omega\),k), \((\theta,\omega,k)\in S^ 2\times S^ 2\times {\mathbb{R}}_+\), to a(\(\theta\),-\(\theta\),k). We show that in suitable (complex) Banach spaces the map from q(x) to a(x/\(| x|,-x/| x|,| x|)\) is usually a local diffeomorphism. Hence in contrast to the overdetermined problem of recovering q from the full scattering amplitude the inverse backscattering problem is well posed. When one wishes to obtain analogous results for real-valued potentials q, it is necessary to further restrict the scattering data. We consider the mapping from real-valued q to \[ (a(x/| x|,-x/| x|,| x|)+\bar a(-x/| x|,x/| x|,| x|)). \] However, the proof that this is usually a diffeomorphism is only given for q without bound states in this paper. The general proof will appear in the Proceedings of the Workshop on Spectral and Scattering Theory of Partial Differential Operators, Jerusalem, June 12-15, 1990.
scientific article; zbMATH DE number 4159309
scientific article