Explicitly recovering asymptotics of short range potentials

Inverse scattering for real-valued short range potentials on R-n is studied . It is shown that the scattering matrix at fixed energy is the pull-back o f a pseudo-differential operator and that the symbol of the operator determ ines the asymptotics of the potential. This is done by an explicit construc tion of the Poisson operator for the scattering problem as an oscillatory i ntegral.