Abstract
We consider a family of isolated inhomogeneous steady states of the gravitational Vlasov-Poisson system with a point mass at the centre. These are parametrised by the polytropic index k > 1/2 , so that the phase space density of the steady state is C1 at the vacuum boundary if and only if k > 1 . We prove the following sharp dichotomy result: if k > 1 , the linear perturbations Landau damp and if 1/2 < k ≤ 1 they do not. The above dichotomy is a new phenomenon and highlights the importance of steady state regularity at the vacuum boundary in the discussion of the long-time behaviour of the perturbations. Our proof of (nonquantitative) gravitational relaxation around steady states with k > 1 is the first such result for the gravitational Vlasov-Poisson system. The key novelty of this work is the proof that no embedded eigenvalues exist in the essential spectrum of the linearised system.