Inverse obstacle scattering with non-over-determined data

TL;DR

This paper proves a unique determination of obstacle surfaces and boundary conditions from non-over-determined scattering data, solving a long-standing problem in inverse scattering theory.

Contribution

It establishes the first uniqueness theorem for inverse obstacle scattering with limited data, advancing theoretical understanding in the field.

Findings

Proves unique determination of obstacle surface and boundary condition from fixed incident data.

First to establish a uniqueness theorem for inverse scattering with non-over-determined data.

Addresses a problem unsolved for decades in multidimensional inverse scattering theory.

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. Such a theorem is proved in this paper for inverse scattering by obstacles for the first time.