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

Abstract

We study the time-dependent Hartree--Fock equation in $\R^3$ with physical Coulomb interactions in both direct and exchange channels, with exchange coupling $\eta$, linearized about homogeneous equilibria $\gamma_f=g(-i\nabla)$ having compact momentum support. Nguyen and You showed that, without exchange, two purely imaginary plasmon poles $\pm i\tau_0(|k|)$ 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 $0<k_-\le|k|\le k_+<\kappa_0$, we prove unconditionally for Coulomb exchange that, for small $|\eta|$, 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 $t^{-3/2}$ dispersive decay and endpoint Strichartz estimates. A first-order Ward-type cancellation gives $|\tau_X(r)|\le Cr^2$ 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 $0.7\,\kappa_0$.