On the Existence of Linearly Oscillating Galaxies

Titelangaben

Hadžić, Mahir ; Rein, Gerhard ; Straub, Christopher:
On the Existence of Linearly Oscillating Galaxies.
In: Archive for Rational Mechanics and Analysis. Bd. 243 (2022) . - S. 611-696.
ISSN 1432-0673
DOI: https://doi.org/10.1007/s00205-021-01734-4

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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.