On absence of embedded eigenvalues and stability of BGK waves

TL;DR

This paper proves the absence of embedded eigenvalues in certain BGK wave equilibria of the Vlasov-Poisson system, leading to insights on Landau damping and spectral stability using advanced analytical techniques.

Contribution

It introduces a novel energy-based method to exclude embedded eigenvalues in BGK equilibria with trapped particles, applicable to complex spectral problems.

Findings

No embedded eigenvalues inside the essential spectrum for a large class of BGK waves.

Establishment of a nonquantitative Landau damping result for specific equilibria.

Development of a robust analytical approach for spectral problems with elliptic and hyperbolic critical points.

Abstract

We consider space-periodic and inhomogeneous steady states of the one-dimensional electrostatic Vlasov-Poisson system, known as the Bernstein-Greene-Kruskal (BGK) waves. We prove that there exists a large class of fixed background ion densities and spatial periods, so that the corresponding linearised operator around the associated BGK-equilibria has no embedded eigenvalues inside the essential spectrum. As a consequence we conclude a nonquantitative version of Landau damping around a subclass of such equilibria with monotone dependence on particle energy. The BGK equilibria under investigation feature trapped electrons which lead to presence of both elliptic and hyperbolic critical points in the characteristic phase-space diagram. They also feature a small parameter, which roughly speaking governs the size of the trapped zone - also referred to as electron hole. Our argument uses action-angle variables and a careful analysis of the associated period function. To exclude embedded eigenvalues we develop an energy-based approach which deals with resonant interactions between the energy (action)-space and the angle frequencies; their singular structure and summability properties are the key technical challenge. Our approach is robust and applicable to other spectral problems featuring elliptic and hyperbolic critical points.