## Abstract

We study the scattering problem for the Hartree equation i∂_{t}u = - 1/2Δu +f(\u\^{2})u, (t,x) ∈ R x R^{n}, with initial data u(0,x) = U_{0}(X), x ∈ R^{n}, where f(|u|^{2}) = V * |u|^{2}, V(x) = λ|x|^{-1}, λ ∈ R, n ≥ 2. We prove that for any U_{0} ∈ H^{0,γ}∩ H^{γ,0}, with 1/2 < γ < n/2, such that the value ∈ = ∥u_{0}∥0,γ + ∥u_{0}∥γ,0 is sufficiently small, there exist unique u± 6 H'0 n /f0''7 with | < CT < 7 such that for all |t| > 1 ||u(t) - exp (Ti/(|u± 2) (y ) log |i|) l/(t)U± L^{2} < C'eltrμ+7, where JJL = min(l, γ), O < v < min(l, -1γf-), <f denotes the Fourier transform of if, U(t) is the free Schrödinger evolution group, and Hm<s is the weighted Sobolev space defined by Hm<s = {<fe S'; IMIm1, = Ii(I + N2)s/2(l - Ar/VllL2 < oo}.

Original language | English |
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Pages (from-to) | 1256-1267 |

Number of pages | 12 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 29 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1998 Sept |

## Keywords

- Asymptotic behavior
- Hartree equation
- Scattering