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Modified scattering for the 3D Hartree equation with singular external potentials
Abstract
We establish modified scattering for the three-dimensional Hartree equation with Coulomb interaction and singular external potentials, including repulsive Coulomb and inverse-square potentials. In the defocusing case, the result holds for arbitrary initial data in H0,1(ℝ3)∩ H1,0(ℝ3). In particular, we obtain large-data modified scattering for the flat defocusing Hartree equation. For the focusing problem with repulsive Coulomb potential V(x)=κ|x|−1, we prove radial modified scattering under the sharp mass condition |λ||u0|22≤ 2κ, including the endpoint, and construct standing waves at every larger mass. Our argument obtains strong L2 convergence from rescaled energy and comparison with a frozen-field evolution.
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BibTeX
@misc{xie2026modified,
title = {{Modified scattering for the 3D Hartree equation with singular external potentials}},
author = {Mengyi Xie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32939},
eprint = {2610.32939},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32939},
}
RIS
TY - UNPB
TI - Modified scattering for the 3D Hartree equation with singular external potentials
AU - Mengyi Xie
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32939
UR - https://arxiv.org/abs/2610.32939
AB - We establish modified scattering for the three-dimensional Hartree equation with Coulomb interaction and singular external potentials, including repulsive Coulomb and inverse-square potentials. In the defocusing case, the result holds for arbitrary initial data in H^0,1(ℝ^3)∩ H^1,0(ℝ^3). In particular, we obtain large-data modified scattering for the flat defocusing Hartree equation. For the focusing problem with repulsive Coulomb potential V(x)=κ|x|^−1, we prove radial modified scattering under the sharp mass condition |λ||u_0|_2^2≤ 2κ, including the endpoint, and construct standing waves at every larger mass. Our argument obtains strong L^2 convergence from rescaled energy and comparison with a frozen-field evolution.
ER -