On Krein-like theorems for noncanonical Hamiltonian systems with continuous spectra: application to Vlasov-Poisson

TL;DR

This paper extends Krein-like theorems to noncanonical Hamiltonian systems with continuous spectra, specifically analyzing spectral and structural stability of the Vlasov-Poisson system under various perturbations.

Contribution

It establishes conditions for stability and instability of the Vlasov-Poisson system using Krein-like theorems for continuous spectra, including perturbation effects and signature analysis.

Findings

Small perturbations can destabilize stable equilibria.

Signature changes in the spectrum lead to instability.

Embedded discrete modes can be destabilized by infinitesimal perturbations.

Abstract

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 delta f_0' in W^{1,1}(R) such that f_0+delta f_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 previous work. 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 C^n 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.