线性化库仑-哈特里-福克方程的等离子体极点
Plasmon poles for the linearized Coulomb--Hartree--Fock equation
- Matsuyama University（松山大学）
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AI总结：
该研究针对线性化库仑-哈特里-福克方程，证明小交换耦合下等离子体极点持续存在，推导其偏移公式并通过数值计算验证了相关分析结果。
AI中文摘要：
我们研究三维空间中具有物理库仑相互作用（包含直接和交换通道）的时间相关哈特里-福克方程，其交换耦合为η，围绕具有紧凑动量支撑的均匀平衡态γ_f = g(-i∇)进行线性化。Nguyen和You证明，在无交换作用时，在存活阈值κ₀>0以下存在两个纯虚等离子体极点±iτ₀(|k|)。现有非线性哈特里-福克理论要求直接相互作用为短程且交换核光滑且小，不包含本文的库仑情形。在每个紧凑带0<k₋≤|k|≤k₊<κ₀上，我们无条件证明，对于库仑交换，当|η|较小时，极点持续存在、保持简单且为纯虚数，并实解析依赖于η。其一阶偏移τ_X可分解为显式静态自能和动态顶点修正。每个极点产生一个精确的无阻尼模式，具有闭式单位密度本征函数，嵌入纤维生成元的本质谱中。因果密度格林函数分解为显式无阻尼正弦波和余项，其拉普拉斯变换在极点附近全纯，而等离子体通道满足均匀t^(-3/2)色散衰减和端点Strichartz估计。一阶沃德型抵消给出|τ_X(r)|≤Cr²（r降至0），因此交换在一阶下不改变等离子体间隙，且我们获得刚度修正的闭式公式。对C⁹光滑费米球的数值计算支持该分析：在计算范围内，自能和顶点项的抵消在1.2%以内，且交换在接近0.7κ₀处对色散的软化作用最强。
英文摘要：
We study the time-dependent Hartree--Fock equation in $\R^3$ with physical Coulomb interactions in both direct and exchange channels, with exchange coupling $η$, linearized about homogeneous equilibria $γ_f=g(-i\nabla)$ having compact momentum support. Nguyen and You showed that, without exchange, two purely imaginary plasmon poles $\pm iτ_0(|k|)$ exist below a survival threshold $κ_0>0$. Existing nonlinear Hartree--Fock theory requires a short-range direct interaction and a smooth small exchange kernel, excluding the present Coulomb setting. On every compact band $0<k_-\le|k|\le k_+<κ_0$, we prove unconditionally for Coulomb exchange that, for small $|η|$, the poles persist, remain simple and purely imaginary, and depend real-analytically on $η$. Their first-order shift $τ_X$ splits into explicit static self-energy and dynamic vertex corrections. Each pole yields an exact undamped mode with a closed-form unit-density eigenfunction, embedded in the fiber generator's essential spectrum. The causal density Green function splits into an explicit undamped sine wave and a remainder whose Laplace transform is holomorphic near the poles, while the plasmon channel satisfies uniform $t^{-3/2}$ dispersive decay and endpoint Strichartz estimates. A first-order Ward-type cancellation gives $|τ_X(r)|\le Cr^2$ down to $r=0$; hence exchange leaves the plasma gap unchanged to first order, and we obtain a closed formula for the stiffness correction. Numerics for a $C^9$ smoothed Fermi ball support the analysis: self-energy and vertex terms cancel within $1.2\%$ on the computed range, and exchange softens the dispersion most strongly near $0.7\,κ_0$.
