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- Title
Domain and Range of the Modified Wave Operator for Schrödinger Equations with a Critical Nonlinearity.
- Authors
Hayashi, Nakao; Naumkin, Pavel I.
- Abstract
We study the final problem for the nonlinear Schrödinger equation where $$\lambda \in{\bf R},n=1,2,3$$ . If the final data $$u_{+}\in {\bf H}^{0,\alpha }=\left\{ \phi \in {\bf L}^{2}:\left( 1+\left\vert x\right\vert \right) ^{\alpha }\phi \in {\bf L}^{2}\right\} $$ with $$\frac{ n}{2} < \alpha < \min \left( n,2,1+\frac{2}{n}\right) $$ and the norm $$\Vert \widehat{u_{+}}\Vert _{{\bf L}^{\infty }}$$ is sufficiently small, then we prove the existence of the wave operator in L 2. We also construct the modified scattering operator from H 0, α to H 0, δ with $$\frac{n}{2} < \delta < \alpha$$ .
- Subjects
SCHRODINGER equation; FOURIER transforms; SOBOLEV spaces; CAUCHY problem; MATHEMATICAL physics
- Publication
Communications in Mathematical Physics, 2006, Vol 267, Issue 2, p477
- ISSN
0010-3616
- Publication type
Article
- DOI
10.1007/s00220-006-0057-6