Abstract:
We discuss the existence of oscillating solutions close to stable steady states of the gravitational Vlasov-Poisson system, which is the fundamental system of equations used in astrophysics to describe the evolution of galaxies, on the linear level. Since this corresponds to the existence of positive eigenvalues of the operator governing the linearised dynamics, we first explicitly characterise the essential spectrum of this operator by introducing action-angle type variables on phase space. In particular, we show that the essential spectrum possesses a gap at the origin. We then derive a sharp criterion for the existence of eigenvalues inside this “principal gap” by applying a Birman-Schwinger type principle. After giving some (conditional) examples of equilibria where the criterion is satisfied, we conclude by discussing the implications on Landau damping. The talk is based on joint work with Mahir Hadžić and Gerhard Rein.