## 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).
[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)].

### MSC:

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

Zbl 0722.00022