Uniqueness of the solution to inverse scattering problem with non-overdetermined data

Let q(x) be a real-valued compactly supported sufficiently smooth function. It is proved that the scattering data A(−β, β, k) ∀β ∈ S, ∀k > 0, determine q uniquely. Under the same assumptions on q(x) it is proved that the scattering data A(β, α0, k) ∀β ∈ S, ∀k > 0, and a fixed α0, the direction of the incident plane wave, determine q uniquely. The above scattering data are non-overdetermined in the sense that the scattering data depends on the same number of variables as the potential q(x), that is, on three variables. earlier there were no uniqueness results for the solution of three-dimensional inverse scattering problems with non-overdetermined scattering data. MSC: 35P25, 35R30, 81Q05;