Asymptotic Behavior of One-Dimensional Hartree Equations with the Long-Range Interaction
Abstract
We investigate the long-time behavior of small solutions to the one-dimensional Hartree equation with the soft Coulomb interaction. The Hartree equation arises as an effective mean-field model for the evolution of many-body quantum systems, while the soft Coulomb potential provides a regularized one-dimensional approximation of the Coulomb interaction that removes the singularity at the origin while preserving its long-range character. Owing to the slow spatial decay of the interaction kernel, the equation exhibits long-range nonlinear effects. We prove global existence and modified scattering for sufficiently small, exponentially localized initial data. A notable feature is that, in contrast to the logarithmic phase correction arising in the well-known three-dimensional Coulomb Hartree equation, the one-dimensional soft Coulomb interaction produces a leading phase correction of order (log t)2, followed by a lower-order logarithmic correction. The proof is carried out in analytic function spaces. The exponential localization of the initial data is converted into analyticity of the transformed profile, and a method based on a generator function with a time-dependent radius of analyticity is used to compensate for the derivative loss arising from the nonlinear phase equation.
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BibTeX
@misc{yang2026asymptotic,
title = {{Asymptotic Behavior of One-Dimensional Hartree Equations with the Long-Range Interaction}},
author = {Changhun Yang and Chanjin You},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31515},
eprint = {2609.31515},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31515},
}
RIS
TY - UNPB
TI - Asymptotic Behavior of One-Dimensional Hartree Equations with the Long-Range Interaction
AU - Changhun Yang
AU - Chanjin You
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31515
UR - https://arxiv.org/abs/2609.31515
AB - We investigate the long-time behavior of small solutions to the one-dimensional Hartree equation with the soft Coulomb interaction. The Hartree equation arises as an effective mean-field model for the evolution of many-body quantum systems, while the soft Coulomb potential provides a regularized one-dimensional approximation of the Coulomb interaction that removes the singularity at the origin while preserving its long-range character. Owing to the slow spatial decay of the interaction kernel, the equation exhibits long-range nonlinear effects. We prove global existence and modified scattering for sufficiently small, exponentially localized initial data. A notable feature is that, in contrast to the logarithmic phase correction arising in the well-known three-dimensional Coulomb Hartree equation, the one-dimensional soft Coulomb interaction produces a leading phase correction of order (log t)^2, followed by a lower-order logarithmic correction. The proof is carried out in analytic function spaces. The exponential localization of the initial data is converted into analyticity of the transformed profile, and a method based on a generator function with a time-dependent radius of analyticity is used to compensate for the derivative loss arising from the nonlinear phase equation.
ER -