On Krein-Like Theorems for Noncanonical Hamiltonian Systems with Continuous Spectra: Application to Vlasov-Poisson

The notions of spectral stability and the spectrum for the Vlasov-Poisson system linearized about homogeneous equilibria, f 0(v), are reviewed. Structural stability is reviewed and applied to perturbations of the linearized Vlasov operator through perturbations of f 0. We prove that for each f 0 there is an arbitrarily small δf′0 in such that f 0+δf 0 is unstable. When f 0 is perturbed by an area preserving rearrangement, f 0 will always be stable if the continuous spectrum is only of positive signature, where the signature of the continuous spectrum is defined as in Morrison and Pfirsch (1992) and Morrison (2000). If there is a signature change, then there is a rearrangement of f 0 that is unstable and arbitrarily close to f 0 with f′0 in W.1,1 This result is analogous to Krein's theorem for the continuous spectrum. We prove that if a discrete mode embedded in the continuous spectrum is surrounded by the opposite signature there is an infinitesimal perturbation in Cn norm that makes f 0 unstable. If f 0 is stable we prove that the signature of every discrete mode is the opposite of the continuum surrounding it.