Inverse scattering with non-over-determined data

It is proved that the scattering amplitude A(β,α
₀
, k
₀
), known for all β ∈ S
²
, where S
²
is the unit sphere in ℝ
³
, and fixed α
₀
∈ S
²
and k
₀
> 0, determines uniquely the surface S of the obstacle D and the boundary condition on S. The boundary condition on S is assumed to be the Dirichlet, or Neumann, or the impedance one. The uniqueness theorems for the solution of multidimensional inverse scattering problems with non-over-determined data was not known for many decades. Such theorems are presented in this paper.