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.

A Birman-Schwinger Principle in Galactic Dynamics

“…Since ω 1 is continuous on the compact set D and T 1 is non-zero, certainly (b) follows from (a), but it is also possible to give a proof using the explicit period relation (6.12). Similar results have been obtained in [16]. ✷…”

“…These are the (somewhat extended) lecture notes for four lectures delivered at the spring school during the thematic programme "Mathematical Perspectives of Gravitation beyond the Vacuum Regime" at ESI Vienna in February 2022. The main reference for the lectures is [25], which has some overlap with [16], although we wanted to emphasize the action-angle variables approach and put a main focus on the Birman-Schwinger principle, as is done in [25]. Since the lectures have been aimed at newcomers, some parts of them cover basic background material.…”

Collisionless equilibria in general relativity: stable configurations beyond the first binding energy maximum

“…For small values of z c -corresponding to less relativistic settings-we observe a stable oscillation of the solution around the original equilibrium. For the spherically symmetric Vlasov-Poisson system, which is the non-relativistic limit of the Einstein-Vlasov system [28], similar oscillations have been observed numerically [23] and have recently been proven to exist on the linear level for certain equilibria in [15].…”

A Birman-Schwinger Principle in General Relativity: Linearly Stable Shells of Collisionless Matter Surrounding a Black Hole

“…Crucial parts of our investigation are based on action-angle type variables, which we introduce in Section 3.3. While action-angle variables are a classical tool in Hamiltonian mechanics [9,32,38], they have been used recently to derive a Birman-Schwinger principle in Newtonian galactic dynamics [25,31] and to analyze phase mixing [51].…”

“…As in the non-relativistic setting [25,31], the radial period function T and its properties are crucial to derive a Birman-Schwinger principle. We next show that this function is bounded and bounded away from zero on the steady state support for sufficiently small shells around a Schwarzschild black hole.…”