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 \(\frac{n}{2} < \delta < \alpha\).
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Hayashi, N., Naumkin, P.I. Domain and Range of the Modified Wave Operator for Schrödinger Equations with a Critical Nonlinearity. Commun. Math. Phys. 267, 477–492 (2006). https://doi.org/10.1007/s00220-006-0057-6
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DOI: https://doi.org/10.1007/s00220-006-0057-6