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Scattering theory in the weighted \(L^ 2({\mathbb{R}}^ n)\) spaces for some Schrödinger equations. (English) Zbl 0659.35078

We study the scattering problem for the following Schrödinger equation: \[ (**)\quad i\partial_ tu+(1/2)\Delta u=V_ 1u+(V_ 2*| u|^ 2)u,\quad (t,x)\in {\mathbb{R}}\times {\mathbb{R}}^ n,\quad u(0,x)=\phi (x),\quad x\in {\mathbb{R}}^ n, \] where \(V_ 1=V_ 1(x)=\lambda_ 1| x|^{-\gamma_ 1}\) \((\lambda_ 1\geq 0\), \(1<\gamma_ 1<\min (2,n/2))\), \(V_ 2=V_ 2(x)=\sum^{3}_{k=2}\lambda_ k| x|^{-\gamma_ k},\) \((\lambda_ k\geq 0\), \(1<\gamma_ k<\min (2,n))\), * denotes the convolution in \({\mathbb{R}}^ n\), \[ H^{m,s}=\{v\in L^ 2({\mathbb{R}}^ n);\quad \| v\|_{m,s}=\| (1+| x|^ 2)^{s/2}(I-\Delta)^{m/2} v\|_{L^ 2}<\infty \},\quad m,s\in {\mathbb{R}}. \] We show that (1) if \(\phi \in H^{0,1}\), all solutions of (**) are asymptotically free in \(L^ 2({\mathbb{R}}^ n)\), (2) if \(n\geq 4\), \((3/2)\leq \gamma_ 1,\gamma_ 2,\gamma_ 3<2\), \(\phi \in H^{0,2}\), all solutions of (**) are asymptotically free in \(H^{0,1}\), (3) If \(\lambda_ 1=0\), \(n\geq 3\), \((4/3)<\gamma_ 2,\gamma_ 3<2\), \(s\in {\mathbb{N}}\), \(\phi \in H^{0,2}\), the wave operators and the scattering operator are well defined in \(H^{0,s}\) and homeomorphisms from \(H^{0,s}\) to \(H^{0,s}\).

MSC:

35P25 Scattering theory for PDEs
35J10 Schrödinger operator, Schrödinger equation
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References:

[1] J.E. Barab , Nonexistence of asymptotic free solutions for a nonlinear Schrödinger equation . J. Math. Phys. , t. 25 , 1984 , p. 3270 - 3273 . MR 761850 | Zbl 0554.35123 · Zbl 0554.35123 · doi:10.1063/1.526074
[2] J. Bergh and J. Löfström , Interpolation Spaces . Berlin , Heidelberg , New York , Springer , 1976 . MR 482275 | Zbl 0344.46071 · Zbl 0344.46071
[3] A. Friedman , Partial Differential Equations . Holt - Rinehart and Winston , New York , 1969 . MR 445088 | Zbl 0224.35002 · Zbl 0224.35002
[4] J. Ginibre and G. Velo , On a class of nonlinear Schrödinger equations I, II . J. Funct. Anal. , t. 32 , 1979 , p. 1 - 32 , 33 - 71 ; III. Ann. Inst. Henri Poincaré, Physique Théorique , t. 28 , 1978 , p. 287 - 316 . Numdam | MR 533219 | Zbl 0396.35028 · Zbl 0396.35028 · doi:10.1016/0022-1236(79)90076-4
[5] J. Ginibre and G. Velo , Scattering theory in the energy space for a class of nonlinear Schrödinger equations . J. Math. pures et appl. , t. 64 , 1985 , p. 363 - 401 . MR 839728 | Zbl 0535.35069 · Zbl 0535.35069
[6] J. Ginibre and G. Velo , Private communication .
[7] N. Hayashi and Y. Tsutsumi , Scattering theory for Hartree type equations . Ann. Inst. Henri Poincaré, Physique Théorique , t. 46 , 1987 , p. 187 - 213 . Numdam | MR 887147 | Zbl 0634.35059 · Zbl 0634.35059
[8] N. Hayashi and Y. Tsutsumi , Remarks on the scattering problem for nonlinear Schrödinger equations , to appear in the Proceedings of UAB conference on Differential Equations and Mathematical Physics , Springer-Verlag , New York , 1986 . MR 921265 | Zbl 0633.35059 · Zbl 0633.35059
[9] N. Hayashi and T. Ozawa , Time decay of solutions to the Cauchy problem for time-dependent Schrödinger-Hartree equations . Commun. Math. Phys. , t. 110 , 1987 , p. 467 - 478 . Article | MR 891948 | Zbl 0648.35078 · Zbl 0648.35078 · doi:10.1007/BF01212423
[10] N. Hayashi and T. Ozawa , Smoothing effectfor some Schrödinger equations , preprint RIMS-583, 1987 . MR 1012208
[11] N. Hayashi and T. Ozawa , Time decay for some Schrödinger equations , preprint RIMS-554, 1987 . MR 987581
[12] T. Kato , On nonlinear Schrödinger equations . Ann. Inst. Henri Poincaré, Physique Théorique , t. 46 , 1987 , p. 113 - 129 . Numdam | MR 877998 | Zbl 0632.35038 · Zbl 0632.35038
[13] E.M. Stein , Singular Integral and Differentiability Properties of Functions . Princeton Univ. Press , Princeton Math. Series 30 , 1970 . MR 290095 | Zbl 0207.13501 · Zbl 0207.13501
[14] W.A. Strauss , Decay and asymptotic for \square u = F(u) . J. Funct. Anal. , t. 2 , 1968 , p. 409 - 457 . Zbl 0182.13602 · Zbl 0182.13602 · doi:10.1016/0022-1236(68)90004-9
[15] W.A. Strauss , Nonlinear scattering theory at low energy: Sequel . J. Funct. Anal. , t. 43 , 1981 , p. 281 - 293 . MR 636702 | Zbl 0494.35068 · Zbl 0494.35068 · doi:10.1016/0022-1236(81)90019-7
[16] Y. Tsutsumi , Scattering problem for nonlinear Schrödinger equations . Ann. Inst. Henri Poincaré, Physique Théorique , t. 43 , 1985 , p. 321 - 347 . Numdam | MR 824843 | Zbl 0612.35104 · Zbl 0612.35104
[17] Y. Tsutsumi and K. Yajima , The asymptotic behavior of nonlinear Schrödinger equations . Bull. (New Series), Amer. Math. Soc. , t. 11 , 1984 , p. 186 - 188 . Article | MR 741737 | Zbl 0555.35028 · Zbl 0555.35028 · doi:10.1090/S0273-0979-1984-15263-7
[18] K. Yajima , Existence of solutions for Schrödinger evolution equations . Commun. Math. Phys. , t. 110 , 1987 , p. 415 - 426 . Article | MR 891945 | Zbl 0638.35036 · Zbl 0638.35036 · doi:10.1007/BF01212420
[19] J. Ginibre , A remark on some papers by N. Hayashi and T. Ozawa , preprint Or say, 1987 . · Zbl 0692.35076
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