On Inverse Scattering for theN-Body Schrِdinger Equation

We consider the generalizedN-body Schrödinger operatorH=−Δ+q(x), q(x)=∑a∈A va(xa), x∈Rd, xa∈Xa, Rd=Xa⊕Xa,with short-range regular interactions. We obtain, in particular, the following results. (1) We give new formulas and proofs relating high energy asymptotics (in a weak sense) of the scattering operatorsŜβα(with the same cluster decompositionaforαandβ) and theX-ray transformPIβα(defined on the set of all lines inXawith nonsingular directions) of the effective potentialsIβα(xa),xa∈Xa. These results significantly clarify some of those given in the literature. (2) We describe completely Ker Pand give a method for reconstruction ofIβα(mod Ker P) fromPIβα. (3) We prove pointwise high energy asymptotics for the two-cluster-two-cluster scattering amplitudesfβαwith the Fourier transformÎβαin the leading term (for the case when the cluster decomposition forαandβis the same). (4) We give several additional results for the case (of perturbed stratified medium) whenq(x)=va(xa)+vb(xb),xa=x1,xb=(x1, …, xd)=xand eachvc(xc) rapidly decreases as |xc|→∞,c=a, b. Our results (including proofs) in some cases significantly simplify methods of high energy inverse scattering for theN-body Schrödinger operator given earlier by Wang and by Enss and Weder