Plasmon poles for the linearized Coulomb--Hartree--Fock equation

We study the time-dependent Hartree–Fock equation in ${\mathbb{R}}^3$ with physical Coulomb interactions in both direct and exchange channels, with exchange coupling $\eta$, linearized about homogeneous equilibria ${\gamma }_f=g(-i\mathrm{\nabla })$ having compact momentum support. Nguyen and You showed that, without exchange, two purely imaginary plasmon poles $\pm i{\tau }_0(\mathrm{\mid }k\mathrm{\mid })$ exist below a survival threshold ${\kappa }_0>0$. 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 $0<k_{-}\le \mathrm{\mid }k\mathrm{\mid }\le k_{+}<{\kappa }_0$, we prove unconditionally for Coulomb exchange that, for small $\mathrm{\mid }\eta \mathrm{\mid }$, the poles persist, remain simple and purely imaginary, and depend real-analytically on $\eta$. Their first-order shift ${\tau }_X$ 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 $t^{-3\mathrm{/}2}$ dispersive decay and endpoint Strichartz estimates. A first-order Ward-type cancellation gives $\mathrm{\mid }{\tau }_X(r)\mathrm{\mid }\le C{r}^2$ 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 $1.2\mathrm{\%}$ on the computed range, and exchange softens the dispersion most strongly near $0.7{\kappa }_0$.