A constructive procedure to recover asymptotics of short-range or long-range potentials

We study an inverse scattering problem for a pair of Hamiltonians (H,H₀) on L²(Rⁿ), where H₀=-Δ and H=H₀+V, V being a short- or long-range potential. By an elementary constructive method, we show that the scattering operator S, which is localized near a fixed energy λ>0, determines the asymptotics of the potential V at infinity, in dimension n?3. This is done by studying the action of the scattering operator on suitable wave packets.