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Domain and range of the modified wave operator for Schrödinger equations with a critical nonlinearity. (English) Zbl 1113.81121
Summary: We study the final problem for the nonlinear Schrödinger equation \[ i{\partial}_{t}u+\frac{1}{2}\Delta u=\lambda|u|^{\frac{2}{n}}u,\quad (t,x)\in \mathbb R\times \mathbb R^{n}, \] where \(\lambda \in\mathbb R\), \(n=1,2,3\). If the final data \(u_{+}\in {\mathcal H}^{0,\alpha}=\{\phi \in {\mathcal L}^{2}: (1+|x|)^{\alpha}\phi \in {\mathcal L}^{2}\}\) with \(\frac{n}{2} < \alpha < \min (n,2,1+\frac{2}{n})\) and the norm \(\| \widehat{u_+}\|_{{\mathbf L}^{\infty}}\) is sufficiently small, then we prove the existence of the wave operator in \({\mathcal L} ^{2}\). We also construct the modified scattering operator from \({\mathcal H} ^{0, \alpha}\) to \({\mathcal H} ^{0, \delta}\) with \(\frac{n}{2} < \delta < \alpha\).

81U20 \(S\)-matrix theory, etc. in quantum theory
81U05 \(2\)-body potential quantum scattering theory
35Q55 NLS equations (nonlinear Schrödinger equations)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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