Forward and backward propagators applied to direct and inverse scattering of the scalar field in the resonance region

Complete families of linearly independent solutions to the scalar Helmholtz equation introduced by Vekua (1953) and further studied by Kleinman et al. (1984), Ramm (1986), provide a unified view over a class of direct and inverse scattering problems and have an impact over solution algorithms. They are shown to improve the understanding of two known direct scattering algorithms and to suggest a new solution method for an imaging problem: the reconstruction of an impenetrable obstacle in the resonance region from knowledge of the incident field and the scattering amplitude and from prior information.