2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/145077Using the free-space translation representation (modified Radon transform) of Lax and Phillips in odd dimensions, it is shown that the generalized backscattering transform (so outgoing angle $ω=Sθ$ in terms of the incoming angle with $S$ orthogonal and $\Id-S$ invertible) may be further restricted to give an entire, globally Fredholm, operator on appropriate Sobolev spaces of potentials with compact support. As a corollary we show that the modified backscattering map is a local isomorphism near elements of a generic set of potentials.Minor changes, typos corrected, references addedAnalysis of PDEstext