Inverse obstacle scattering with non-over-determined data - Ramm, Alexander G. (Global Journal of Mathematical Analysis, 2018)
t is proved that the scattering amplitude \(A(β, α_0, k_0)\), known for all \(βS^2\), where \(S^2\) is the unit sphere in \(R^3\), and fixed \(α_0S^2\) and \(k_0>0\), determines uniquely the surface \(S\) of the obstacleÂ \(D\) and the boundary condition on \(S\). The boundary condition onÂ \(S\) 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.A detailed proof of such a theorem is given in this paper for inverse scattering by obstacles for the first time. It follows from our results that the scattering solution vanishing on the boundaryÂ \(S\) of the obstacle cannot have closed surfaces of zeros in the exterior of the obstacle different from \(S\). To have a uniqueness theorem for inverse scattering problems with non-over-determined data is of principal interest because these are the minimal scattering data that allow one to uniquely recover the scatterer.