Calogero-Moser粒子由连续多孤子在全时间区间上的一致逼近

中文摘要 AI 辅助

我们证明经典有理Calogero-Moser系统的每个解，在从粒子位置中减去共同的线性漂移后，可以被聚焦连续Calogero-Moser方程的有理多孤子解在全时间上一致逼近。该构造将多孤子逆谱矩阵识别为Moser的时间依赖厄米位置矩阵的秩一耗散扰动。粒子间距的一致下界随后给出多孤子极点的定量控制。它们的实部和实速度以$\u03b5^2$阶误差逼近平移后的粒子位置和速度，而极点高度为正且总和恰好等于$\u03b5$。利用多孤子密度的精确Poisson核表示，我们获得在界-Lipschitz距离下以$O(\u03b5(1+|\u006cog\u03b5|))$的速率收敛到原子粒子测度，且关于时间一致。我们还推导了固定$\u03b5$下极点高度的大时间分布，其中一个显著分支保持极限高度，其余高度随时间二次衰减。显式的两粒子公式说明了该逼近并表明实位置误差估计是尖锐的。

英文摘要

We prove that every solution of the classical rational Calogero-Moser system can be approximated, after subtracting a common linear drift from the particle positions, by rational multisoliton solutions of the focusing continuum Calogero-Moser equation, uniformly for all time. The construction identifies the multisoliton inverse spectral matrix as a rank-one dissipative perturbation of Moser's time-dependent Hermitian position matrix. A uniform lower bound on the particle separation then yields quantitative control of the multisoliton poles. Their real parts and real velocities approximate the shifted particle positions and velocities with errors of order $\varepsilon^2$, while the pole heights are positive and sum exactly to $\varepsilon$. Using the exact Poisson-kernel representation of the multisoliton density, we obtain convergence to the atomic particle measure in bounded-Lipschitz distance at rate $O(\varepsilon(1+|\log\varepsilon|))$, uniformly in time. We also derive the large-time distribution of pole height at fixed $\varepsilon$, with one distinguished branch retaining the limiting height and the remaining heights decaying quadratically in time. Explicit two-particle formulas illustrate the approximation and show that the real-position error estimate is sharp.

发表机构

• Georgetown University（乔治城大学）

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