On Inverse Scattering for the N-Body Schrödinger Equation

We consider the generalizedN-body Schrödinger operatorH=-Δ+q(x),q(x)=∑a∈Av_a(x ^a),x∈R^d,x^a∈X^a,R ^d=X^a⊕X_a,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 operatorsŜ_βα(with the same cluster decompositionaforαandβ) and theX-ray transformPI_βα(defined on the set of all lines inX_awith nonsingular directions) of the effective potentialsI_βα(x_a),x_a∈X _a. These results significantly clarify some of those given in the literature. (2)We describe completely KerPand give a method for reconstruction ofI_βα(mod KerP) fromPI_βα. (3)We prove pointwise high energy asymptotics for the two-cluster-two-cluster scattering amplitudesf_βαwith the Fourier transformÎ_βαin 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)=v_a(x^a)+v_b(x^b),x ^a=x₁,x^b=(x₁,...,x_d)=xand eachv_c(x^c) rapidly decreases as x^c→∞,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