Application of spectral deformation to the Vlasov–Poisson system. II. Mathematical results

This paper presents a mathematical description of the linearized Vlasov–Poisson operator Lk acting on a family of Banach spaces Xp, related to L p (R), and the application of the method of spectral deformation to this model. It is shown that a type-A analytic family of operators Lk(θ), θ∈C, Lk(0) =Lk can be associated with Lk. By means of this family, the Landau damped modes of the plasma are identified as the spectral resonances of Lk. Existence and uniqueness of solutions to the initial-value problem for the evolution equation ∂ν g =Lk(θ)g is proven. An expansion of any solution to the initial-value problem (with sufficiently smooth initial data) is obtained in terms of the eigenfunctions of Lk(θ) and a spectral integral over the essential spectrum. This is applied to derive an expansion for solutions to the Vlasov equation in which the Landau damped portions of the distribution function are manifestly exhibited. A self-contained discussion of the spectral deformation method and an extension of it to certain closed operators on Banach spaces is also given.