Intertwining methods in the problem of inverse scattering. (English) Zbl 0618.35029

Melin, Anders

Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1986, Conf. No. 5, 8 p. (1986).

Let \(H_ v\) be the Schrödinger operator with a short range potential v(x), A be the intertwining operator: \(H_ vA=AH_ 0\). By using a fundamental solution of the ultrahyperbolic operator \(\Delta_ x- \Delta_ y\), the author obtains explicit expressions for the intertwining operator A, for wave operators \(W_{\pm}=\lim_{t\to \pm \infty} e^{itH}ve^{-itH}0\), and a formula for the scattering operator \(S=W^*_+W_-.\)
These results are applied to the inverse scattering problems.

Reviewer: S.Tajima

MSC:

35J10	Schrödinger operator, Schrödinger equation
35P25	Scattering theory for PDEs
35R30	Inverse problems for PDEs
47A40	Scattering theory of linear operators

Keywords:

Schrödinger operator; short range potential; intertwining operator; fundamental solution; ultrahyperbolic operator; explicit expressions; scattering operator