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 L^2(Rn), where H0=-(delta) 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 (lambda)>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.