Essential Self-Adjointness of Klein-Gordon Type Operators on Asymptotically Static, Cauchy-Compact Spacetimes

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