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We analyze the problem of the beam-plasma instability via the analyticaltreatment of the so-called Dyson equation. We first compared the prediction ofthe model constructed by fixing the electric field amplitude with respect to aN-body Hamiltonian numerical simulation. Then, we demonstrate that theshortcomings of such an analytical formulation must be essentially identifiedwith the breaking-down of the self-consistent evolution of the field and theparticle distribution function.

Reanalysis of the beam-plasma instability using the Dyson-like equation formalism