Damped perturbations in stellar systems: Genuine modes and Landau-damped
  waves
  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 nontrivial 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 {\it 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 $\exp (-{\rm i} \omega t)$, 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.
