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 H_m:={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 H_m with a natural inner product, so that [Formula presented] is a self-adjoint operator on H_m with discrete spectrum {λ_j(m)}. In earlier work, we proved a Weyl law for the number of eigenvalues λ_j(m) in an interval for fixed mass m. In this sequel, we prove a Weyl law along ‘ladders’ {(m,λ_j(m)):m∈R₊} such that [Formula presented] as m→∞. More precisely, we given an asymptotic formula as m→∞ for the counting function [Formula presented] for C>0. The asymptotics are determined from the dynamics of the Killing flow e^tZ on the hypersurface N_1,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.