Inverse obstacle scattering with non-over-determined data

TL;DR

This paper proves the first detailed uniqueness theorem for inverse obstacle scattering using minimal, non-over-determined data, establishing that certain fixed and partial scattering measurements uniquely determine the obstacle and boundary conditions.

Contribution

It provides a rigorous proof that minimal scattering data uniquely determine the obstacle's surface and boundary conditions, filling a longstanding gap in inverse scattering theory.

Findings

Uniqueness of obstacle reconstruction from fixed incident wave data.

No closed surfaces of zeros of the scattering solution other than the obstacle boundary.

First detailed proof of such a uniqueness theorem for non-over-determined data.

Abstract

It is proved that the scattering amplitude $A(\beta, \alpha_0, k_0)$, known for all $\beta\in S^2$, where $S^2$ is the unit sphere in $\mathbb{R}^3$, and fixed $\alpha_0\in S^2$ and $k_0>0$, determines uniquely the surface $S$ of the obstacle $D$ and the boundary condition on $S$. The boundary condition on $S$ is assumed to be the Dirichlet, or Neumann, or the impedance one. The uniqueness theorem for the solution of multidimensional inverse scattering problems with non-over-determined data was not known for many decades. A detailed proof of such a theorem is given in this paper for inverse scattering by obstacles for the first time. It follows from our results that the scattering solution vanishing on the boundary $S$ of the obstacle cannot have closed surfaces of zeros in the exterior of the obstacle different from $S$. To have a uniqueness theorem for inverse scattering problems with non-over-determined data is of principal interest because these are the minimal scattering data that allow one to uniquely recover the scatterer.