On inverse scattering for the wave equation with a potential term via the Lax-Phillips theory: A simple proof

The inverse scattering problem for the perturbed wave equation (1) □u + V(x)u = 0 in $\text{Ω=R}{}^{\mathrm{n}}$ (n = odd ? 3) is considered. Here the potentials V(x) are real, smooth, with compact support and non-negative. We apply the Lax and Phillips theory, together with some properties of solutions of a Dirichlet problem associated with the operator ?Δ + V(x) to show, in a very simple way, that the scattering operator S(V) associated with (1) determines uniquely the scatterer, provided that a fixed sign condition on the potentials is satisfied. We also show that the map V → S(V) is once-differentiable.