Uniqueness for general backscattering inverse scattering

Establish uniqueness for recovering a compactly supported potential in dimensions n ≥ 2 from backscattering data hat{u}_{\infty}(\omega,k,\omega;V) for all frequencies k > 0 and incident directions \omega, without smallness, genericity, angular-control, or other additional assumptions.

Background

The paper distinguishes the backscattering problem from the fixed-angle problem. In backscattering, the observation direction equals the incident direction, and the available data are the far-field values \hat{u}_{\infty}(\omega,k,\omega;V) over all frequencies and directions. The authors list several partial results for dimensions n ≥ 2, including uniqueness for small or generic potentials, recovery of principal singularities, identification of the zero potential in fixed-angle scattering, and recovery of angularly controlled potentials from backscattering data. The cited statement that the problems remain open in general identifies the absence of a general uniqueness theorem for the backscattering problem; the present paper resolves the fixed-angle problem but does not address general backscattering uniqueness.