Essential self-adjointness for the Klein-Gordon type operators on asymptotically static spacetime

Let $X=\mathbb{R}\times M$ be the spacetime, where $M$ is a closed manifold equipped with a Riemannian metric $g$, and we consider a symmetric Klein-Gordon type operator $P$ on $X$, which is asymptotically converges to ${\mathrm{\partial }}_t^2-{\mathrm{△}}_g$ as $\mathrm{\mid }t\mathrm{\mid }\to \mathrm{\infty }$, where ${\mathrm{△}}_g$ is the Laplace-Beltrami operator on $M$. We prove the essential self-adjointness of $P$ on $C_0^{\mathrm{\infty }}(X)$. The idea of the proof is closely related to a recent paper by the authors on the essential self-adjointness for Klein-Gordon operators on asymptotically flat spaces.