The inverse scattering problem on a fixed energy level for the two-dimensional Schrödinger operator

This paper contains a solution with complete proofs of the main problems of the inverse scattering at fixed energy E for the Schrödinger two-dimensional operator with decreasing coefficients (uniqueness of the reconstruction, a reconstruction procedure, characterization of the scattering data). Most completed results are obtained under the condition of "smallness" of coefficients in comparison with the number E. This paper develops and generalizes several preceded papers.