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 is contained in 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), for all β element of S2, for all k element of (k0,k1), and q1, q2 element of 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 Aq1(β,α0,k)=Aq2(β,α0,k), for all β element of S12, for all k element of (k0,k1), and q1, q2 element of M, then q1=q2. Here S12 is an arbitrarily small open subset of S2, and |k0-k1|>0 is arbitrarily small.