Galaxy dynamics, gravitational Vlasov-Poisson system, Landau damping, and scattering theory

Abstract

We consider the gravitational Vlasov-Poisson system linearized around steady states that are extensively used to study the dynamics of galaxies, or of clusters of galaxies. Namely, polytropes and King steady states. We develop a complete stationary scattering theory for the selfadjoint, strictly positive, Antonov operator that governs the plane-symmetric linearized dynamics. We identify the absolutely continuous spectrum of the Antonov operator. Moreover, we prove that the part of the singular spectrum of the Antonov operator that is embedded in its absolutely continuous spectrum is contained in a closed set of measure zero, that we characterize. We construct the generalized Fourier maps, and we prove that the wave operators exist and are complete. Moreover, we obtain stationary formulae for the wave operators, and we prove that Birman's invariance principle holds. Using these results we obtain a precise description of the dynamics of the stars in the galaxies, or of the galaxies in the clusters of galaxies, for large times. Namely, we prove that the distribution function of the solutions to the linearized gravitational Vlasov-Poisson system with initial data in the absolutely continuous subspace of the Antonov operator are asymptotic, for large times, to the solutions to the unperturbed linearized gravitational Vlasov-Poisson system. This implies that they are asymptotic to the trajectories of the solutions to Newton's equation with the gravitational potential of the steady state, in the sense that they are transported along these trajectories. Moreover, for these initial states the gravitational Landau damping holds. Namely, we prove that the gravitational force and its time derivative, as well as the gravitational potential and its time derivative, tend to zero for large times.