On the denseness of the set of scattering amplitudes

TL;DR

This paper proves that the set of scattering amplitudes for a fixed wave number, known for all incident directions, is dense in the space of square-integrable functions on the sphere, highlighting the richness of scattering data.

Contribution

It establishes the denseness of the set of scattering amplitudes in $L^2(S^2)$ for Dirichlet boundary conditions, under certain spectral conditions, advancing understanding in inverse scattering theory.

Findings

The set of scattering amplitudes is dense in $L^2(S^2)$.

Density holds for all $eta eq$ Dirichlet eigenvalues.

Results apply to obstacles with Dirichlet boundary conditions.

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.