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In this note, we consider the wave operator $\square_g$ in the case ofglobally hyperbolic, compactly supported perturbations of static spacetimes. Wegive an elementary proof of the essential self-adjointness of $\square_g$ andof uniform microlocal estimates for the resolvent in this setting. Thisprovides a model for studying Lorentzian spectral zeta functions which isparticularly simple, yet sufficiently general for locally deriving Einsteinequations from a spectral Lagrangian.

The wave resolvent for compactly supported perturbations of static spacetimes