Semi-classical mass asymptotics on stationary spacetimes

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