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

Abstract

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.