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, H-0) on L-2(R-n), where H-0 = -Delta and H = H-0 + 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 greater than or equal to 3. This is done by studying the action of the scattering operator on suitable wave packets. (C) 2004 Elsevier Inc. All rights reserved.