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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 $$\frac{n}{2} < \delta < \alpha$$.

##### MSC:
 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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