On the existence of linearly oscillating galaxies

TL;DR

This paper analyzes the spectral properties of steady states in the gravitational Vlasov-Poisson system, establishing conditions for linear oscillations and explaining pulsating behaviors observed in simulations.

Contribution

It provides a complete spectral description of certain steady states and introduces a criterion for the existence of oscillatory eigenmodes, linking spectral gaps to pulsating galaxy models.

Findings

Existence of spectral gaps in the linearized operator.

A criterion for eigenvalues indicating oscillations.

Verification of the criterion in symmetric cases.

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.