Wiener-Hopf factorization in the inverse scattering theory for the \(n\)-D Schrödinger equation. (English) Zbl 0739.35056

Topics in matrix and operator theory, Proc. Workshop, Rotterdam/Neth. 1989, Oper. Theory, Adv. Appl. 50, 1-21 (1991).

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[For the entire collection see Zbl 0722.00022.]
This paper proves the Hölder continuity of the scattering matrix for Schrödinger operators with potentials \(V\) in \(L^ 2_{loc}(\mathbb{R}^ n)\) such that the multiplication by \((1+| x|^ 2)^ sV(x)\) is a bounded operator from \(H^ \alpha\) to \(L^ 2\) for some \(s>1/2\) and some \(\alpha\geq 0\). From this, a Wiener-Hopf factorization of the scattering operator, and the reconstruction of \(V\) from the scattering matrix, are obtained. The result extends earlier work of the authors in three dimensions. See also the authors’ exposition of these results in: Inverse scattering and applications [Proc. AMS-IMS-SIAM Conf., Amherst/MA (USA) 1990, Contemp. Math. 122, 1-11 (1991)].


35P25 Scattering theory for PDEs
35R30 Inverse problems for PDEs
47A40 Scattering theory of linear operators


Zbl 0722.00022