On the denseness of the set of scattering amplitudes

Abstract

It is proved that the set of scattering amplitudes $\{A(\beta, \alpha, k)\}_{\forall \alpha \in S^2}$, known for all $\beta\in S^2$, where $S^2$ is the unit sphere in $\mathbb{R}^3$, $k>0$ is fixed, $k^2$ is not a Dirichlet eigenvalue of the Laplacian in $D$, is dense in $L^2(S^2)$. Here $A(\beta, \alpha, k)$ is the scattering amplitude corresponding to an obstacle $D$, where $D\subset \mathbb{R}^3$ is a bounded domain with a boundary $S$. The boundary condition on $S$ is the Dirichlet condition.