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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 \(\lambda_ 2\).

MSC:
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
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