On Inverse Scattering for theN-Body Schrödinger Equation

✍ Scribed by Roman G Novikov

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 639 KB
• Volume: 159
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1998.3324

✦ Synopsis

We consider the generalized N-body Schro dinger operator H=&2+q(x), q(x)= :

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 operators S ;: (with the same cluster decomposition a for : and ;) and the X-ray transform PI ;: (defined on the set of all lines in X a with nonsingular directions) of the effective potentials I ;: (x a ), x a # X a . These results significantly clarify some of those given in the literature.

(2) We describe completely Ker P and give a method for reconstruction of I ;: (mod Ker P) from PI ;: .

(3) We prove pointwise high energy asymptotics for the two-cluster-twocluster scattering amplitudes f ;: with the Fourier transform I ;: in the leading term (for the case when the cluster decomposition for : and ; is the same).

(4) We give several additional results for the case (of perturbed stratified medium) when q(x)=v a (x a )+v b (x b ), x a =x 1 , x b =(x 1 , ..., x d )=x and each v c (x c ) rapidly decreases as |x c | Ä , c=a, b.

Our results (including proofs) in some cases significantly simplify methods of high energy inverse scattering for the N-body Schro dinger operator given earlier by Wang and by Enss and Weder. 1998 Academic Press 0. INTRODUCTION The studies of the inverse scattering problem for the N-body Schro dinger equation, which use results of the modern direct scattering theory for this equation, were started in [W2] (and subsequent works [W4, W5]), [EW2] (and subsequent works [EW3, EW4]), [N1] (and subsequent work [N2]), [V].