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Nonlinear scattering with nonlocal interaction. (English) Zbl 0748.35046
Summary: We consider the scattering problem for the Hartree type equation in \(\mathbb{R}^ n\) with \(n\geq 2\) \[ i\partial u/\partial t+\Delta u/2=(V*| u|^ 2)u, \] where \(V(x)=\sum^ 2_{j=1}\lambda_ j| x|^{-\gamma_ j}\), \((\lambda_ 1,\lambda_ 2)\neq (0,0)\), \(\lambda_ j\in\mathbb{R}\), \(\gamma_ j>0\), and * denotes the convolution in \(\mathbb{R}^ n\). We prove the existence of wave operators in \(H^{0,k}=\{\psi\in L^ 2(\mathbb{R}^ n)\); \(| x|^ k\psi\in L^ 2(\mathbb{R}^ n)\}\) for any positive integer \(k\) under the assumption \(1<\gamma_ 1\), \(\gamma_ 2<2\). This is an optimal result in the sense that the existence of wave operators breaks down if \(\min(\gamma_ 1,\gamma_ 2)\leq 1\). The case where \(1<\gamma_ 1<\gamma_ 2=2\) is also treated according to the sign of \(\lambda_ 2\).

35Q55 NLS equations (nonlinear Schrödinger equations)
35P25 Scattering theory for PDEs
35B40 Asymptotic behavior of solutions to PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
Full Text: DOI
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