Damping versus oscillations for a gravitational Vlasov-Poisson system

We consider a family of isolated inhomogeneous steady states to the gravitational Vlasov-Poisson system with a point mass at the centre. They are parametrised by the polytropic index $k>1/2$, so that the phase space density of the steady state is $C^1$ 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 $1/2< k\le1$ 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 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 step in the proof is to show that no embedded eigenvalues exist in the essential spectrum of the linearised system.