Semi-classical mass asymptotics on stationary spacetimes

We study the spectrum of a timelike Killing vector field Z acting as a differential operator on the solution space Hm:={u∣(□g+m2)u=0} of the Klein–Gordon equation on a globally hyperbolic stationary spacetime (M,g) with compact Cauchy hypersurface Σ. We endow Hm with a natural inner product, so that DZ=1i∇Z is a self-adjoint operator on Hm with discrete spectrum {λj(m)}. In earlier work, we proved a Weyl law for the number of eigenvalues λj(m) in an interval for fixed mass m. In this sequel, we prove a Weyl law along ‘ladders’ {(m,λj(m)):m∈R+} such that λj(m)m≃ν as m→∞. More precisely, we given an asymptotic formula as m→∞ for the counting function Nν,C(m):=#{j∣λj(m)m∈[ν−Cm,ν+Cm]} for C>0. The asymptotics are determined from the dynamics of the Killing flow etZ on the hypersurface N1,ν in the space N1 of mass 1 geodesics γ where 〈γ̇,Z〉=ν. The method is to treat m as a semi-classical parameter h−1 and employ techniques of homogeneous quantization.