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

It is proved that the scattering amplitude A(β,α0,k0)A(β,α0,k0), known for all β∈S2β∈S2, where S2S2 is the unit sphere in R3R3, α0∈S2α0∈S2 is fixed, k0>0k0>0 is fixed, determines the surface SS of the obstacle and the boundary condition on SS uniquely. The boundary condition on SS 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.