Inverse scattering with the data at fixed energy and fixed incident direction

Consider the Schr\"odinger operator $-\nabla^2+q$ $$q$,$q=q(x), x \in \mathbf{R}^3$. Let$A(\beta,\alpha, k)$be the corresponding scattering amplitude,$k^2$be the energy,$\alpha \in S^2$be the incident direction,$\beta \in S^2$be the direction of scattered wave,$S^2$be the unit sphere in$\mathbf{R}^3$. Assume that$k=k_0 >0$is fixed, and$\alpha=\alpha_0$is fixed. Then the scattering data are$A(\beta)= A(\beta,\alpha_0, k_0)=A_q(\beta)$is a function on$S^2$. The following invers$ \textit{IP: Given an arbitrary $f \in L^2(S^2)$ and an arbitrary small number $$ $q \in C_0^{\infty}(D)$, where $D \in \mathbf{R}^3$ is an arbitrary fixed domai$$||A_q(\beta)-f(\beta)||_{L^2(S^2)} < \epsilon?} A positive answer to this question is given. A method for constructing such a q$is proposed. There are infinitely many such$q$, not necessarily real-valued.