Plasmons for the Hartree equations with Coulomb interaction

TL;DR

This paper proves the existence, decay, and damping of plasmons in the linearized Hartree equations for fermionic gases, providing explicit damping rates and dispersive estimates in a broad setting.

Contribution

It establishes the existence and decay properties of plasmons in the Hartree model, including Landau damping and dispersive estimates for a wide class of steady states.

Findings

Plasmons exist due to long-range interactions.

Landau damping occurs at the threshold frequency.

Density decays rapidly via phase mixing.

Abstract

In this work, we establish the existence and decay of {\em plasmons}, the quantum of Langmuir's oscillatory waves found in plasma physics, for the linearized Hartree equations describing an interacting gas of infinitely many fermions near general translation-invariant steady states, including compactly supported Fermi gases at zero temperature, in the whole space $\RR^d$ for $d\ge 2$. Notably, these plasmons exist precisely due to the long-range pair interaction between the particles. Next, we provide a survival threshold of spatial frequencies, below which the plasmons purely oscillate and disperse like a Klein-Gordon's wave, while at the threshold they are damped by {\em Landau damping}, the classical decaying mechanism due to their resonant interaction with the background fermions. The explicit rate of Landau damping is provided for general radial homogenous equilibria. Above the threshold, the density of the excited fermions is well approximated by that of the free gas dynamics and thus decays rapidly fast for each Fourier mode via {\em phase mixing}. Finally, pointwise bounds on the Green function and dispersive estimates on the density are established.