Plasmon poles for the linearized Coulomb--Hartree--Fock equation
Abstract
We study the time-dependent Hartree--Fock equation in R3 with physical Coulomb interactions in both direct and exchange channels, with exchange coupling η, linearized about homogeneous equilibria γf=g(−i∇) having compact momentum support. Nguyen and You showed that, without exchange, two purely imaginary plasmon poles ± iτ0(|k|) exist below a survival threshold κ0>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-≤|k|≤ k+<κ0, 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 τX 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 |τX(r)|≤ Cr2 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 C9 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 κ0.
Related
- Unconditional scattering below the ground state for the three-dimensional focusing energy-critical generalized Hartree equation
- A Gauge-Covariant Semiclassical Limit from the magnetic Hartree--Fock equation to the magnetic Vlasov--Poisson equation
- Global well-posedness and scattering for mass-critical Hartree equation
- Ground state uniqueness and quantitative stability for the mass-critical Hartree equation
- Classification of minimal-mass blowup solutions to the mass-critical Hartree equation at the L2-critical level
BibTeX
@misc{dan2026plasmon,
title = {{Plasmon poles for the linearized Coulomb--Hartree--Fock equation}},
author = {Yuya Dan},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29671},
eprint = {2608.29671},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29671},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Plasmon poles for the linearized Coulomb--Hartree--Fock equation
AU - Yuya Dan
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29671
UR - https://arxiv.org/abs/2608.29671
AB - 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∇) having compact momentum support. Nguyen and You showed that, without exchange, two purely imaginary plasmon poles ± 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_-≤|k|≤ 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)|≤ 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.
ER -