Scattering theory for the Hartree equation

Nakao Hayashi, Pavel I. Naumkin, Tohru Ozawa

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27 Citations (Scopus)


We study the scattering problem for the Hartree equation i∂tu = - 1/2Δu +f(\u\2)u, (t,x) ∈ R x Rn, with initial data u(0,x) = U0(X), x ∈ Rn, where f(|u|2) = V * |u|2, V(x) = λ|x|-1, λ ∈ R, n ≥ 2. We prove that for any U0 ∈ H0,γ∩ Hγ,0, with 1/2 < γ < n/2, such that the value ∈ = ∥u0∥0,γ + ∥u0∥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± L2 < 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 languageEnglish
Pages (from-to)1256-1267
Number of pages12
JournalSIAM Journal on Mathematical Analysis
Issue number5
Publication statusPublished - 1998 Sept


  • Asymptotic behavior
  • Hartree equation
  • Scattering


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