Uniqueness theorems for inverse obstacle scattering in Lipschitz domains
  An inverse problem of finding an obstacle and the boundary condition on its surface from the fixed-energy scattering data is studied. A new method is developed for a proof of the uniqueness results. The method does not use the discreteness of the spectrum of the corresponding Laplacian in a bounded domain. Proof of the uniqueness results is based on the fact that the Hilbert space of square integrable functions is separable. 
A.G.Ramm