Inverse scattering with non-overdetermined data

Let $A(\beta ,\alpha ,k)$ be the scattering amplitude corresponding to a real-valued potential which vanishes outside of a bounded domain $D\subset \R^3$\mathrm{.}Theunitvector$\alpha$ is the direction of the incident plane wave, the unit vector $\beta$ is the direction of the scattered wave, $k>0$ is the wave number. The governing equation for the waves is $[{\mathrm{\nabla }}^2+k^2-q(x)]u=0$ in ${\mathbb{R}}^3$. For a suitable class of potentials it is proved that if $A_{q_1}(-\beta ,\beta ,k)=A_{q_2}(-\beta ,\beta ,k)$ $\mathrm{\forall }\beta \in S^2,$ $\mathrm{\forall }k\in (k_0,k_1),$ and $q_1,$ $q_2\in M$, 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 $A_{q_1}(\beta ,{\alpha }_0,k)=A_{q_2}(\beta ,{\alpha }_0,k)$ $\mathrm{\forall }\beta \in S_1^2,$ $\mathrm{\forall }k\in (k_0,k_1),$ and $q_1,$ $q_2\in M$,then $q_1=q_2$. Here $S_1^2$ is an arbitrarily small open subset of $S^2$, and $\mathrm{\mid }{k}_0-k_1\mathrm{\mid }>0$ is arbitrarily small.