We study the spectrum $\{\lambda_j(m)\}_{j=1}^{\infty}$ 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 $(\Box_g + m^2) u = 0$ 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 $m \to \infty$ for the number $$N_{\nu, C}(m)= \# \{j \mid \frac{\lambda_j(m)}{m} \in [\nu - \frac{C}{m}, \nu + \frac{C}{m} ]\}$$ 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 $\langle \dot{\gamma}, Z \rangle= \nu$.

中文翻译：

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静止时空中的半经典质量渐近

我们研究了类时间杀死向量场 $Z$ 的频谱 $\{\lambda_j(m)\}_{j=1}^{\infty}$ 在 Hilbert 解的空间上充当微分算子 $D_Z$大规模的 Klein-Gordon 方程 $(\Box_g + m^2) u = 0$ 在全局双曲平稳时空 $(M, g)$ 上，具有紧凑的柯西超曲面。反质量 $m^{-1}$ 在形式上类似于 Schr\"odinger 方程中的普朗克常数，并且我们将 Weyl 渐近性定义为 $m \to \infty$ 对于数字 $$N_{\nu, C} (m)= \# \{j \mid \frac{\lambda_j(m)}{m} \in [\nu - \frac{C}{m}, \nu + \frac{C}{m} ] \}$$ 对于给定的 $C > 0$。半经典质量渐近性由质量 $1$ 测地线 $\gamma$ 空间中超曲面上的杀伤流 $e^{tZ} $ 的动力学控制其中 $\langle \dot{\gamma}, Z \rangle= \nu$。