We prove quantitative decay rates for the linearised gravitational potential around compactly supported equilibria of the Vlasov-Poisson system featuring an attractive point mass potential at the origin.  Such equilibria feature stably trapped particles with a stationary elliptic point of the characteristic flow. This presents a severe obstacle to any kind of dispersion. The problem is further complicated by the presence of an infinite-dimensional kernel. To handle these issues we combine tools from dynamical systems, Hamiltonian geometry, and scattering theory. 

 

Our theorem can be viewed as a first quantitative proof of (linear) gravitational Landau damping. It gives in particular the first complete treatment of linear stability of the vacuum solution around the point mass - an essential ingredient for nonlinear stability.  Joint work with Matthew Schrecker.