Some properties of the scattering amplitude and the inverse scattering problem

Abstract

an asymptotic formula (0.3) lim #(P(K)x k " * f > = -2\Q(y) \y-z \ ~2 dy for #< 3 , where P(k)=-2k-\S(k)-I) with the S-matrix S for (see 2), * M ()=exp(ikz)<=L\S 2 } and ( . ) S 2 i s the inner product of L 2 (S 2 ).(0.3) is valid for any short-range potential Q(y) and the uniqueness of the inverse scattering problem will be shown by the use of (0,3).(0.3) can be also used to reconstruct the potential Q(y) by its 5-matrix S(k).We shall discuss it elsewhere. In this paper we restrict ourselves to the three dimensional case for the sake of simplicity.But all the results can be easily extended to the higher dimensional case.It will be discussed elsewhere, too.