Uniqueness of the solution to inverse obstacle scattering with non-over-determined data

It is proved that the scattering amplitude A(β,α0,k0), known for all β∈S2, where S2 is the unit sphere in R3, α0∈S2 is fixed, k0>0 is fixed, determines the surface S of the obstacle and the boundary condition on S uniquely. The boundary condition on S is either the Dirichlet, or Neumann, or the impedance one. The uniqueness theorems for the solution of inverse scattering problems with non-over-determined data were not known for many decades. Such a theorem is proved in this paper for inverse scattering by obstacles for the first time.