Inverse scattering with non-over-determined data

Let $A(β,α,k)$ be the scattering amplitude corresponding to a real-valued potential which vanishes outside of a bounded domain $D⊂ \R^3$. 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 $[∇^2+k^2-q(x)]u=0$ in $\R^3$. For a suitable class of potentials it is proved that if $A_{q_1}(-β,β,k)=A_{q_2}(-β,β,k)$ $∀ β𝟄 S^2,$ $∀ k𝟄 (k_0,k_1),$ and $q_1,$ $q_2𝟄 M$, then $q_1=q_2$. 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 $A_{q_1}(β,α_0,k)=A_{q_2}(β,α_0,k)$ $∀ β𝟄 S^2_1,$ $∀ k𝟄 (k_0,k_1),$ and $q_1,$ $q_2𝟄 M$,then $q_1=q_2$. Here $S^2_1$ is an arbitrarily small open subset of $S^2$, and $|k_0-k_1|>0$ is arbitrarily small.