Abstract We consider space-periodic and inhomogeneous steady states of the one-dimensional electrostatic Vlasov–Poisson system. We prove that there exists a large class of fixed background ion densities and spatial periods, so that the corresponding linearised operator around such Bernstein–Greene–Kruskal (BGK)-type 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-type 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.
Hadžić et al. (2026) studied this question.