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

### Vladimir Georgescu

Université de Cergy-Pontoise, France### Christian Gérard

Université de Paris 11, Orsay, France### Dietrich Häfner

Université de Grenoble I, Saint-Martin-d'Hères, France

## Abstract

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

where $ϕ:R→H$, $H$ is a (complex) Hilbert space, and $h$, $k$ are self-adjoint, resp. symmetric operators on $H$.

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.

## Cite this article

Vladimir Georgescu, Christian Gérard, Dietrich Häfner, Resolvent and propagation estimates for Klein–Gordon equations with non-positive energy. J. Spectr. Theory 5 (2015), no. 1, pp. 113–192

DOI 10.4171/JST/93