Inverse scattering with non-overdetermined data

Let A(β,α,k) be the scattering amplitude corresponding to a real-valued potential which vanishes outside of a bounded domain D⊂R³. The unit vector α is the direction of the incident plane wave, the unit vector β is the direction of the scattered wave, k>0 is the wave number. The governing equation for the waves is [∇²+k²−q(x)]u=0 in R³. For a suitable class M of potentials it is proved that if Aq₁(−β,β,k)=Aq₂(−β,β,k),∀β∈S², ∀k∈(k₀,k₁), and q₁, q₂∈M, then q₁=q₂. This is a uniqueness theorem for the solution to the inverse scattering problem with backscattering data. It is also proved for this class of potentials that if , ∀k∈(k₀,k₁), and q₁, q₂∈M, then q₁=q₂. Here is an arbitrarily small open subset of S², and |k₀−k₁|>0 is arbitrarily small.