Semi-classical mass asymptotics on stationary spacetimes

We study the spectrum $\{{\lambda }_j(m){\}}_{j=1}^{\mathrm{\infty }}$ of a timelike Killing vector field $Z$ acting as a differential operator $D_Z$ on the Hilbert space of solutions of the massive Klein-Gordon equation $({\mathrm{\square }}_g+m^2)u=0$ on a globally hyperbolic stationary spacetime $(M,g)$ with compact Cauchy hypersurface. The inverse mass $m^{-1}$ is formally like the Planck constant in a Schr"odinger equation, and we give Weyl asymptotics as $m\to \mathrm{\infty }$ for the number $$N_{\nu ,C}(m)=\mathrm{\#}\{j\mid \frac{{\lambda }_j(m)}{m}\in [\nu -\frac{C}{m},\nu +\frac{C}{m}]\}$$ for a given $C>0$. The semi-classical mass asymptotics are governed by the dynamics of the Killing flow $e^{tZ}$ on the hypersurface in the space of mass $1$ geodesics $\gamma$ where $\langle \dot{\gamma}, Z \rangle= \nu$.