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

TL;DR

This paper develops resolvent and propagation estimates for a class of Klein-Gordon equations with potentially non-positive energy, extending analysis to operators with complex spectra and applications to wave equations on various manifolds.

Contribution

It introduces a framework for analyzing Klein-Gordon equations with non-positive energy using Krein space theory, providing new resolvent and propagation estimates.

Findings

Established weighted resolvent boundary value estimates.

Derived propagation estimates for large time behavior.

Applicable to wave and Klein-Gordon equations on diverse manifolds.

Abstract

We study in this paper an abstract class of Klein-Gordon equations: \[ \p_{t}^{2}\phi(t)- 2\i k \p_{t}\phi(t)+ h \phi(t)=0, \] where $\phi: \rr\to \cH$, $\cH$ is a (complex) Hilbert space, and $h$, $k$ are self-adjoint, resp. symmetric operators on $\cH$. 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.