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

Consider the Schrödinger operator $-\nabla^2+q$ $$q$,$q=q(x), x \in \mathbf{R}^3$. Let$A(β,α, k)$be the corresponding scattering amplitude,$k^2$be the energy,$α\in S^2$be the incident direction,$β\in S^2$be the direction of scattered wave,$S^2$be the unit sphere in$\mathbf{R}^3$. Assume that$k=k_0 >0$is fixed, and$α=α_0$is fixed. Then the scattering data are$A(β)= A(β,α_0, k_0)=A_q(β)$is a function on$S^2$. The following invers$ \textit{IP: Given an arbitrary $f \in L^2(S^2)$ and an arbitrary small number $$$q \in C_0^{\infty}(D)$, where$D \in \mathbf{R}^3$is an arbitrary fixed domai$ $||A_q(β)-f(β)||_{L^2(S^2)} < ε$?} 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.

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