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

This paper presents a mathematical description of the linearized Vlasov-Poisson operator L_k acting on a family of Banach spaces X _p, 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 L_k (θ), θ∈ℂ, L_k (0) = L_k can be associated with L_k. By means of this family, the Landau damped modes of the plasma are identified as the spectral resonances of L_k. Existence and uniqueness of solutions to the initial-value problem for the evolution equation ∂_νg = L _k (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 L_k (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.