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<p>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.</p>

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On Absence of Embedded Eigenvalues and Stability of BGK Waves

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月3日

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  • M. Hadžić,
  • M. Moreno

摘要

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.