  We consider two classes of steady states of the three-dimensional,
gravitational Vlasov-Poisson system: the spherically symmetric Antonov-stable
steady states (including the polytropes and the King model) and their plane
symmetric analogues. We completely describe the essential spectrum of the
self-adjoint operator governing the linearized dynamics in the neighborhood of
these steady states. We also show that for the steady states under
consideration, there exists a gap in the spectrum. We then use a version of the
Birman-Schwinger principle first used by Mathur to derive a general criterion
for the existence of an eigenvalue inside the first gap of the essential
spectrum, which corresponds to linear oscillations about the steady state. It
follows in particular that no linear Landau damping can occur in the
neighborhood of steady states satisfying our criterion. Verification of this
criterion requires a good understanding of the so-called period function
associated with each steady state. In the plane symmetric case we verify the
criterion rigorously, while in the spherically symmetric case we do so under a
natural monotonicity assumption for the associated period function. Our results
explain the pulsating behavior triggered by perturbing such steady states,
which has been observed numerically.
