Resolvent and propagation estimates for Klein–Gordon equations with non-positive energy

Abstract

We study in this paper an abstract class of Klein-Gordon equations:

where $\varphi :\mathbb{R}\to \mathcal{H}$, $\mathcal{H}$ is a (complex) Hilbert space, and $h$, $k$ are self-adjoint, resp. symmetric operators on $\mathcal{H}$.

We consider their generators $H$ (resp. $K$) in the two natural spaces of Cauchy data, the energy (resp. charge) spaces. We do not assume that the dynamics generated by $H$ or $K$ has any positive conserved quantity, in particular these operators may have complex spectrum. Assuming conditions on $h$ and $k$ which allow to use the theory of selfadjoint operators on Krein spaces, we prove weighted estimates on the boundary values of the resolvents of $H$, $K$ on the real axis. From these resolvent estimates we obtain corresponding propagation estimates on the behavior of the dynamics for large times.

Examples include wave or Klein-Gordon equations on asymptotically euclidean or asymptotically hyperbolic manifolds, minimally coupled with an external electro-magnetic field decaying at infinity.