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Scattering problem for nonlinear Schrödinger equations. (English) Zbl 0612.35104

Es seien \[ \| v\|^ 2_{\Sigma}:=\| v\|^ 2_{L^ 2({\mathbb{R}}^ n)}+\| \nabla v\|^ 2_{L^ 2({\mathbb{R}}^ n)}+\| xv\|^ 2_{L^ 2({\mathbb{R}}^ n)}, \] und \(\Sigma\) der Hilbertraum mit der Norm \(\| v\|_{\Sigma}\). Der Verf. untersucht das asymptotische Verhalten für \(t\to \pm \infty\) der nichtlinearen Schrödingergleichung \[ i\partial u/\partial t=-\Delta u+| u|^{p-1}u \] mit \(t\in {\mathbb{R}}\), \(x\in {\mathbb{R}}^ n\) und \(u(0,x)=u_ 0(x)\in \Sigma\). Er zeigt die Existenz der Wellenoperatoren und ihre asymptotische Vollständigkeit für \[ \gamma (n)<p<(n+2)/(n- 2)\quad mit\quad \gamma (n)=(n+2+\sqrt{n^ 2+12n+4})/(2n). \] Die Ergebnisse sind Erweiterungen von Resultaten, die Ginibre und Velo erhalten haben. Benutzt werden ein pseudokonformes Erhaltungsgesetz, die Strichartz-Abschätzung und eine Transformation \(u(t,x)=(it)^{- n/2}e^{ix^ 2/2t}v(1/t,x/t)\).
Reviewer: R.Leis

MSC:

35P25 Scattering theory for PDEs
35J60 Nonlinear elliptic equations
47A40 Scattering theory of linear operators
35J10 Schrödinger operator, Schrödinger equation
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References:

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