### Abstract

We study the final problem for the nonlinear Schrödinger equation i_{t}u+ 1/2 Δu = λ|u|^{2/n}u, (t,x)∈ R × R^{n} where λ ∈ R,n =1,2,3. If the final data u _{+} ∈ H^{0,α} = {φ ∈ L^{2} :( 1+|x|)^{α} φ ∈ L^{2}}with n/2 < α < min( n, 2,1+2/n) and the norm ∥û_{+}∥L ∞t is sufficiently small, then we prove the existence of the wave operator in L ^{2}. We also construct the modified scattering operator from H ^{0,α} to H ^{0,δ} with n/2 < δ < α.

Original language | English |
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Pages (from-to) | 477-492 |

Number of pages | 16 |

Journal | Communications in Mathematical Physics |

Volume | 267 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2006 Oct 1 |

Externally published | Yes |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

Hayashi, N., & Naumkin, P. I. (2006). Domain and range of the modified wave operator for schrödinger equations with a critical nonlinearity.

*Communications in Mathematical Physics*,*267*(2), 477-492. https://doi.org/10.1007/s00220-006-0057-6