On inverse scattering for the \(N\)-body Schrödinger equation
The paper deals with the inverse scattering problem for many-body Schrödinger operator with local short-range smooth potential. This problem has been studied earlier by \textit{V. Enss} and \textit{R. Weder} [J. Math. Phys., 36, No. 8, 3902-3921 (1995; Zbl 0849.35094)] and by \textit{X. P. Wang} [Inverse Probl. 10, No. 3, 765-784 (1994; Zbl 0803.35169)] who used the high-energy asymptotics of the scattering operator, a Born-type approximation. This asymptotics allows one to calculate some integral transform (the Fourier or \(x\)-ray transform) of the potential. In the paper under review a similar approach is used. High-energy estimates of the resolvent, obtained earlier by \textit{A. Jensen} [Publ. Res. Inst. Math. Sci. 25, No. 1, 155-167 (1989; Zbl 0717.35066)], play an important role in this approach. The results in the paper under review are similar to those of Wang. The author points out that his arguments are simpler. These arguments use some facts which can be found in the literature. For example, formula (2.18) is similar to formula (24) on p. 67 in the book by \textit{A. G. Ramm} [(*) Multidimensional inverse scattering problems. Longman (New York, 1992; Zbl 0746.35056)]; the basic part of Proposition 8.1 is similar to formula b) in Theorem 3 on p. 328 in the paper by \textit{A. G. Ramm} in [Mat. Sb., n. Ser. 66(108), N3, 321-343 (1965; Zbl 0137.07601)]; formula (4.10) is similar to formula (7) on p. 130 in the book (*). In Section 4, the known formula (4.4), and formula (4.5), are derived by applying some asymptotic properties of the Fourier transforms of some functions concentrated on a unit sphere. The author did not mention large literature of the subject, in particular the book by \textit{E. Stein}, [Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals. Princeton Univ. Press, New Jersey, (1993; Zbl 0821.42001) see pp. 361-362 and papers cited there]. In Section 5 inversion of the \(x\)-ray transform is discussed but no references are given to the known results [e.g., \textit{A. G. Ramm}, \textit{A. I. Katsevich}, The Radon transform and local tomography. CRC Press (Boca Raton, 1996; Zbl 0863.44001) pp. 48-53, 54-55] and no comparison with these results are made. From the physical point of view, the solution to inverse scattering problem based on the high-energy asymptotics of the scattering operator is not quite satisfactory by two reasons: first, the high frequency data are very difficult to collect, and second, the Schrödinger equation is no longer a good model when the energy is very high. By these reasons most of the efforts in the study of the inverse scattering problem for two-body problem were concentrated on the inversion methods not based on the high-energy asymptotics.
scientific article; zbMATH DE number 1268054
scientific article