Inverse obstacle scattering with non-over-determined data

It is proved that the scattering amplitude $A(\beta ,{\alpha }_0,k_0)$, known for all $\beta \in S^2$, where $S^2$ is the unit sphere in ${\mathbb{R}}^3$, and fixed ${\alpha }_0\in S^2$ 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 theorem for the solution of multidimensional inverse scattering problems with non-over-determined data was not known for many decades. Such a theorem is proved in this paper for inverse scattering by obstacles for the first time.