Limiting absorption principle and absence of eigenvalues for massless Klein-Gordon operators on perturbations of the Minkowski spacetime

Abstract

We prove a uniform weighted resolvent estimate for the massless Klein-Gordon operator on a curved spacetime which is sufficiently close to the Minkowski spacetime. This particularly implies the existence and H\"{o}lder continuity of the limiting resolvents at all energies, as well as the absolute continuity, of the massless Klein-Gordon operator. The proof is based on a simple version of Mourre's commutator method and does not rely on microlocal analysis. We also prove the absence of eigenvalues via the Virial theorem under an ellipticity condition on the commutator of the massless Klein-Gordon operator against the generator of a wick-rotated dilation, which is weaker than the smallness condition for the metric perturbation.