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

Vladimir Georgescu (1) , Christian Gérard (2) , Dietrich Häfner (3)

1 AGM - UMR 8088 - Analyse, Géométrie et Modélisation
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay
3 IF - Institut Fourier

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.

Mots clés: Klein-Gordon equations; Krein spaces; resolvent estimates; propagation estimates

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Analyse fonctionnelle [math.FA]; Equations aux dérivées partielles [math.AP]

Vladimir Georgescu, Christian Gérard, Dietrich Häfner. Resolvent and propagation estimates for Klein-Gordon equations with non-positive energy. Journal of Spectral Theory, 2015, 5 (1), pp.113-192. ⟨hal-00802097v2⟩