### 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

Original language | English |
---|---|

Pages (from-to) | 1256-1267 |

Number of pages | 12 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 29 |

Issue number | 5 |

Publication status | Published - 1998 Sep |

Externally published | Yes |

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### Keywords

- Asymptotic behavior
- Hartree equation
- Scattering

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Applied Mathematics

### Cite this

*SIAM Journal on Mathematical Analysis*,

*29*(5), 1256-1267.

**Scattering theory for the Hartree equation.** / Hayashi, Nakao; Naumkin, Pavel I.; Ozawa, Tohru.

Research output: Contribution to journal › Article

*SIAM Journal on Mathematical Analysis*, vol. 29, no. 5, pp. 1256-1267.

}

TY - JOUR

T1 - Scattering theory for the Hartree equation

AU - Hayashi, Nakao

AU - Naumkin, Pavel I.

AU - Ozawa, Tohru

PY - 1998/9

Y1 - 1998/9

N2 - 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

AB - 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

KW - Asymptotic behavior

KW - Hartree equation

KW - Scattering

UR - http://www.scopus.com/inward/record.url?scp=0032349615&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032349615&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0032349615

VL - 29

SP - 1256

EP - 1267

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 5

ER -