A Birman-Schwinger Principle in Galactic Dynamics

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.

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“…During the writing of this manuscript we have been informed that related results were obtained independently by M. Kunze [36].…”

Section: Outline Of the Papermentioning

confidence: 99%

On the Existence of Linearly Oscillating Galaxies

Hadžić

Rein

Straub

2021

Arch Rational Mech Anal

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“…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].…”

Section: Methodology and Outline Of The Papermentioning

confidence: 99%

“…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.…”

Section: The Periodic Particle Motions and The Period Functionmentioning

confidence: 99%

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

Günther¹,

Rein²,

Straub³

2022

Preprint

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We develop a Birman-Schwinger principle for the spherically symmetric, asymptotically flat Einstein-Vlasov system. It characterizes stability properties of steady states such as the positive definiteness of an Antonov-type operator or the existence of exponentially growing modes in terms of a one-dimensional variational problem for a Hilbert-Schmidt operator. This requires a refined analysis of the operators arising from linearizing the system, which uses action-angle type variables. For the latter, a single-well structure of the effective potential for the particle flow of the steady state is required. This natural property can be verified for a broad class of singularity-free steady states. As a particular example for the application of our Birman-Schwinger principle we consider steady states where a Schwarzschild black hole is surrounded by a shell of Vlasov matter. We prove the existence of such steady states and derive linear stability if the mass of the Vlasov shell is small compared to the mass of the black hole.

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“…In the works [16]- [19] the authors developed a methodology use of an additional continuous variable in general relativistic dynamics, and the subsequent transition to the 7dimensional formalism based on the use of "observer time", weakening the need to introduce conditions on the energy manifold [6], [14] in canonical impulses. In this case, the expression for the total Hilbert-Einstein action and the Hamiltonian equations the movements take on a new form (using the formalism of particle distribution functions), as well as the expression for the energy-momentum tensor in Einstein's equations [20]- [26].…”

Section: Introductionmentioning

confidence: 99%

Cosmological Aspects of the Theory of Equations of the Vlasov–Einstein Type and Their Consequences

Vedenyapin,

Fimin

2023

Equations

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The authors propose a general scheme of derivation from the general relativistic Einstein-Hilbert action for a system of gravitationally interacting charged particles, Hamilton’s dynamics equations and field equations. In accordance with the proposed methodology, new forms of equations of Vlasov type are obtained in the general relativistic case, nonrelativistic and weakly relativistic limits. Expressions are established for the resulting corrections in the equation Poisson, which can contribute to the action of dark matter and dark energy. An effective approach to synchronizing the proper times of different particles of a many-particle system is proposed based on invariance of the form of action. Authors derived (using hydrodynamic substitution) and solved the Euler-type equations leading to the cosmological Friedmann and Milne-McCrea models.

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A Birman-Schwinger Principle in Galactic Dynamics

Cited by 3 publications

References 35 publications

On the Existence of Linearly Oscillating Galaxies

On the Existence of Linearly Oscillating Galaxies

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

Cosmological Aspects of the Theory of Equations of the Vlasov–Einstein Type and Their Consequences

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