### Abstract

We consider the scattering problem for the Hartree type equation in ℝ^{n} with n≧2: {Mathematical expression} where {Mathematical expression} and * denotes the convolution in ℝ^{n}. We prove the existence of wave operators in H^{0, k} = {ψ∈L^{2}(ℝ^{n});|x|^{kψ∈}L^{2}(ℝ^{n})} for any positive integer k under the assumption 1<γ_{1}, γ_{2}<2. This is an optimal result in the sense that the existence of wave operators breaks down if min (γ_{1}, γ_{2}≢1. The case where 1<γ_{1}, γ_{2} = 2 is also treated according to the sign of λ_{2}.

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

Pages (from-to) | 259-275 |

Number of pages | 17 |

Journal | Communications in Mathematical Physics |

Volume | 146 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1992 May |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*146*(2), 259-275. https://doi.org/10.1007/BF02102628

**Nonlinear scattering with nonlocal interaction.** / Nawa, Hayato; Ozawa, Tohru.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 146, no. 2, pp. 259-275. https://doi.org/10.1007/BF02102628

}

TY - JOUR

T1 - Nonlinear scattering with nonlocal interaction

AU - Nawa, Hayato

AU - Ozawa, Tohru

PY - 1992/5

Y1 - 1992/5

N2 - We consider the scattering problem for the Hartree type equation in ℝn with n≧2: {Mathematical expression} where {Mathematical expression} and * denotes the convolution in ℝn. We prove the existence of wave operators in H0, k = {ψ∈L2(ℝn);|x|kψ∈L2(ℝn)} for any positive integer k under the assumption 1<γ1, γ2<2. This is an optimal result in the sense that the existence of wave operators breaks down if min (γ1, γ2≢1. The case where 1<γ1, γ2 = 2 is also treated according to the sign of λ2.

AB - We consider the scattering problem for the Hartree type equation in ℝn with n≧2: {Mathematical expression} where {Mathematical expression} and * denotes the convolution in ℝn. We prove the existence of wave operators in H0, k = {ψ∈L2(ℝn);|x|kψ∈L2(ℝn)} for any positive integer k under the assumption 1<γ1, γ2<2. This is an optimal result in the sense that the existence of wave operators breaks down if min (γ1, γ2≢1. The case where 1<γ1, γ2 = 2 is also treated according to the sign of λ2.

UR - http://www.scopus.com/inward/record.url?scp=0001178744&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001178744&partnerID=8YFLogxK

U2 - 10.1007/BF02102628

DO - 10.1007/BF02102628

M3 - Article

VL - 146

SP - 259

EP - 275

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -