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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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[1] Bateman H., Erdelyi A. (1954) Tables of Integral Transforms. McGraw-Hill Book Co., N.Y. pp. 343
[2] Bergh J., Löfström J. (1976) Interpolation spaces. An introduction. Springer-Verlag, Berlin-N.Y., pp. 207 · Zbl 0344.46071
[3] Carles R. (2001) Geometric optics and Long range scattering for one dimensional nonlinear Schrödinger equations. Commun. Math. Phys. 220, 41–67 · Zbl 1029.35211 · doi:10.1007/s002200100438
[4] Friedman A. (1969) Partial Differential Equations. Holt-Rinehart and Winston, New York · Zbl 0224.35002
[5] Ginibre J., Ozawa T. (1993) Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension n 2. Commun. Math. Phys. 151, 619–645 · Zbl 0776.35070 · doi:10.1007/BF02097031
[6] Ginibre J., Ozawa T., Velo G. (1994) On the existence of the wave operators for a class of nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor. 60(2): 211–239 · Zbl 0808.35136
[7] Hayashi N., Naumkin P.I. (1998) Asymptotics in large time of solutions to nonlinear Schrödinger and Hartree equations. Amer. J. Math. 120, 369–389 · Zbl 0917.35128 · doi:10.1353/ajm.1998.0011
[8] Hayashi, N., Tsutsumi, Y. Remarks on the scattering problem for nonlinear Schrödinger equations. In: Knowles, I.W., Saito, Y. (eds.) Differential Equations and Mathematical Physics, Lecture Note in Mathematics. Berlin-Heidelberg-New York: Springer-Verlag, 1285, 1986, pp. 162–168
[9] Ozawa T. (1991) Long range scattering for nonlinear Schrödinger equations in one space dimension. Commun. Math. Phys. 139, 479–493 · Zbl 0742.35043 · doi:10.1007/BF02101876
[10] Shimomura A., Tonegawa S. (2004) Long range scattering for nonlinear Schrödinger equations in one and two space dimensions. Differ. Int. Eqs. 17, 127–150 · Zbl 1164.35325
[11] Stein, E.M. Singular Integrals and Differentiability Properties of Functions. 30, Princeton, NJ: Princeton Univ. Press, 1970 · Zbl 0207.13501
[12] Tsutsumi Y. (1987) L 2-solutions for nonlinear Schrödinger equations and nonlinear groups. Funkcialaj Ekvacioj 30, 115–125 · Zbl 0638.35021
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