Abstract.
We study modified wave operators for the Hartree equation with a long-range potential \( |x|^{-\nu} \), extending the result in [12] to the whole range of the Dollard type 1/2 < \( \nu \) < 1. We construct the modified wave operators in the whole space of \( (1 + |x|)^{-s}L^2 \). We also have the image, strong continuity and strong asymptotic approximation in the same space. The lower bound \( s > 1 - \nu / 2 \) of the weight is sharp from the scaling argument. Those maps are homeomorphic onto open subsets, which implies in particular asymptotic completeness for small data.
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Submitted 20/08/01, accepted 04/12/01
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Nakanishi, K. Modified Wave Operators for the Hartree Equation with Data, Image and Convergence in the Same Space, II. Ann. Henri Poincaré 3, 503–535 (2002). https://doi.org/10.1007/s00023-002-8626-5
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DOI: https://doi.org/10.1007/s00023-002-8626-5