Semi-classical mass asymptotics on stationary spacetimes

We study the spectrum of a timelike Killing vector field acting as a differential operator on the solution space of the Klein–Gordon equation on a globally hyperbolic stationary spacetime with compact Cauchy hypersurface . We endow with a natural inner product, so that is a self-adjoint operator on with discrete spectrum . In earlier work, we proved a Weyl law for the number of eigenvalues in an interval for fixed mass . In this sequel, we prove a Weyl law along ‘ladders’ such that as . More precisely, we given an asymptotic formula as for the counting function for . The asymptotics are determined from the dynamics of the Killing flow on the hypersurface in the space of mass 1 geodesics where . The method is to treat as a semi-classical parameter and employ techniques of homogeneous quantization.