  In this note, we consider the wave operator $\square_g$ in the case of
globally hyperbolic, compactly supported perturbations of static spacetimes. We
give an elementary proof of the essential self-adjointness of $\square_g$ and
of uniform microlocal estimates for the resolvent in this setting. This
provides a model for studying Lorentzian spectral zeta functions which is
particularly simple, yet sufficiently general for locally deriving Einstein
equations from a spectral Lagrangian.
