Inverse scattering with non-overdetermined data

TL;DR

This paper proves uniqueness theorems for inverse scattering problems with non-overdetermined data, showing that certain backscattering and limited angular data uniquely determine the potential.

Contribution

It establishes new uniqueness results for inverse scattering with limited data, including backscattering and small angular subsets, for a class of potentials.

Findings

Uniqueness with backscattering data for a class of potentials.

Uniqueness with limited angular data for a class of potentials.

Results hold for potentials vanishing outside a bounded domain.

Abstract

Let $A(\beta,\alpha,k)$ be the scattering amplitude corresponding to a real-valued potential which vanishes outside of a bounded domain $D\subset \R^3$. The unit vector $\alpha$ is the direction of the incident plane wave, the unit vector $\beta$ is the direction of the scattered wave, $k>0$ is the wave number. The governing equation for the waves is $[\nabla^2+k^2-q(x)]u=0$ in $\R^3$. For a suitable class of potentials it is proved that if $A_{q_1}(-\beta,\beta,k)=A_{q_2}(-\beta,\beta,k)$ $\forall \beta\in S^2,$ $\forall k\in (k_0,k_1),$ and $q_1,$ $q_2\in M$, then $q_1=q_2$. This is a uniqueness theorem for the solution to the inverse scattering problem with backscattering data. It is also proved for this class of potentials that if $A_{q_1}(\beta,\alpha_0,k)=A_{q_2}(\beta,\alpha_0,k)$ $\forall \beta\in S^2_1,$ $\forall k\in (k_0,k_1),$ and $q_1,$ $q_2\in M$,then $q_1=q_2$. Here $S^2_1$ is an arbitrarily small open subset of $S^2$, and $|k_0-k_1|>0$ is arbitrarily small.