Scattering Structure and Landau Damping for Linearized Vlasov Equations with Inhomogeneous Boltzmannian States

We study the linearized Vlasov–Poisson–Ampère equation for non-constant Boltzmannian states with one region of trapped particles in dimension one and construct the eigenstructure in the context of the scattering theory. This is based on the use of semi-discrete variables (moments in velocity), and it leads to a new Lippmann–Schwinger variational equation. The continuity in quadratic norm of the operator is proved, and the well posedness is proved for a small value of the scaling parameter. It gives a proof of Linear Landau damping for inhomogeneous Boltzmannian states. The linear HMF model is an example.