发表机构
Chungbuk National University; Yale University（忠北国立大学; 耶鲁大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究一维软Coulomb相互作用Hartree方程小解的长时间行为，证明全局存在性与修正散射，并发现领先相位修正为$(\log t)^2$阶，方法基于解析函数空间与时间依赖解析半径的生成函数。
AI 中文摘要
我们研究了具有软Coulomb相互作用的一维Hartree方程小解的长时间行为。Hartree方程作为多体量子系统演化的有效平均场模型出现，而软Coulomb势提供了Coulomb相互作用的正则化一维近似，它消除了原点处的奇异性，同时保留了其长程特征。由于相互作用核的空间衰减缓慢，方程表现出长程非线性效应。我们证明了对于足够小且指数局部化的初始数据，全局存在性和修正散射。一个显著特征是，与著名的三维Coulomb Hartree方程中出现的对数相位修正相反，一维软Coulomb相互作用产生阶为$(\log t)^2$的领先相位修正，随后是较低阶的对数修正。证明在解析函数空间中进行。初始数据的指数局部化被转化为变换后轮廓的解析性，并且使用一种基于具有时间依赖解析半径的生成函数的方法来补偿由非线性相位方程引起的导数损失。
英文摘要
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.
Comments27 pages
