Inverse scattering with non-overdetermined data

Abstract

Let $A(\beta,\alpha,k)$ be the scattering amplitude corresponding to a real-valued potential which vanishes outside of a bounded domain $D\subset \R^3$. The unit vector $\alpha$ 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 $[\nabla^2+k^2-q(x)]u=0$ in $\R^3$. For a suitable class of potentials it is proved that if $A_{q_1}(-\beta,\beta,k)=A_{q_2}(-\beta,\beta,k)$ $\forall \beta\in S^2,$ $\forall k\in (k_0,k_1),$ 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 $A_{q_1}(\beta,\alpha_0,k)=A_{q_2}(\beta,\alpha_0,k)$ $\forall \beta\in S^2_1,$ $\forall k\in (k_0,k_1),$ and $q_1,$ $q_2\in M$,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.