This research was stimulated by the recent studies of damping solutions in dynamically stable spherical stellar systems. Using the simplest model of the homogeneous stellar medium, we discuss non-trivial features of stellar systems. Taking them into account will make it possible to correctly interpret the results obtained earlier and will help to set up decisive numerical experiments in the future. In particular, we compare the initial value problem versus the eigenvalue problem. It turns out that in the unstable regime, the Landau-damped waves can be represented as a superposition of van Kampen modes plus a discrete damped mode, usually ignored in the stability study. This mode is a solution complex conjugate to the unstable Jeans mode. In contrast, the Landau-damped waves are not genuine modes: in modes, eigenfunctions depend on time as exp (−iωt), while the waves do not have eigenfunctions on the real $v$-axis at all. However, ‘eigenfunctions’ on the complex $v$-contours do exist. Deviations from the Landau damping are common and can be due to singularities or cut-off of the initial perturbation above some fixed value in the velocity space.

中文翻译：

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恒星系统中的阻尼扰动：真实模式和朗道阻尼波

最近对动态稳定球形恒星系统中阻尼解决方案的研究激发了这项研究。使用最简单的均质恒星介质模型，我们讨论恒星系统的重要特征。将它们考虑在内将有可能正确解释早期获得的结果，并将有助于在未来建立决定性的数值实验。特别是，我们比较了初始值问题和特征值问题。事实证明，在不稳定状态下，朗道阻尼波可以表示为 van Kampen 模式加上离散阻尼模式的叠加，通常在稳定性研究中被忽略。此模式是不稳定 Jeans 模式的解复数共轭。相比之下，朗道阻尼波不是真正的模式：在模式中，特征函数依赖于时间，如 exp (−iωt)，而波在真实的 $v$ 轴上根本没有特征函数。然而，复杂的 $v$-contours 上的“特征函数”确实存在。朗道阻尼的偏差很常见，可能是由于速度空间中高于某个固定值的初始扰动的奇异性或截止。