On the existence of linearly oscillating galaxies

Abstract

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.

104 pages; some errors have been corrected

References in corpus (3)

• A numerical investigation of the stability of steady states and critical phenomena for the spherically symmetric Einstein-Vlasov system
• A non-variational approach to nonlinear stability in stellar dynamics applied to the King model
• On the transport operators arising from linearizing the Vlasov-Poisson or Einstein-Vlasov system about isotropic steady states

Cited by in corpus (4)

• Collisionless equilibria in general relativity: stable configurations beyond the first binding energy maximum
• Damping versus oscillations for a gravitational Vlasov-Poisson system
• A Birman-Schwinger Principle in General Relativity: Linearly Stable Shells of Collisionless Matter Surrounding a Black Hole
• A Birman-Schwinger principle in galactic dynamics