We show asymptotic completeness for linear massive Dirac fields on the Schwarzschild-Anti-de Sitter spacetime. The proof is based on a Mourre estimate. We also construct an asymptotic velocity for this field.

We prove the existence of exponentially accurate quasimodes using the square of the Dirac operator on the Schwarzschild-Anti-de Sitter spacetime and the Agmon estimates. We then deduce a logarithmic lower bound for the local energy decay of the Dirac propagator. IntroductionThere has been a lot of works concerning the stability of black holes spacetimes in mathematical general relativity. Recently, some significant results have been obtained in this direction. Concerning the Schwarzschild solution, M. Dafermos, G. Holzegel and I. Rodnianski [9] obtained linear stability to gravitational perturbation by means of vector field methods. F. Finster and J. Smoller [16] obtained linear stability of the non-extreme Kerr black hole using an integral representation for the solution of the Teukolsky equation. Concerning the full non-linear stability problem, P. Hintz and A. Vasy [22] obtained the non-linear stability of the Kerr-de Sitter family of black hole by means of a general framework which uses microlocal techniques. In particular, the latest uses a precise understanding of the resonances for their problem. Resonances allow to obtain a precise decay rate for the fields and at which frequency this decay happens. This has been a subject of advanced study in the last decades. In mathematical general relativity, the analysis of resonances, which are also called quasi-normal modes, begins with the work of Bachelot and Motet-Bachelot [5] for the Schwarzschild black hole. This study was then pursued by A. Sá Barreto and M. Zworski [38] for spherically symmetric black holes where resonances are presented as poles of the meromorphic continuation of the resolvent. J-F Bony and D. Häfner [6] gave a full expansion of the solution of the wave equation in terms of resonant states which gives an exponential decay of the local energy for the Schwarzschild-de Sitter solution. R. Melrose, A. Sá Barreto and A. Vasy [36] extended this result to more general manifolds and initial data. This was also extended to the case of rotating black hole with positive cosmological constant by S. Dyatlov [12], [13] and then by A. Vasy [41] who develops more general and flexible techniques. The method developed by S. Dyatlov was also extended to the framework of Dirac operators for the Kerr-Newman-de Sitter black hole by A. Iantchenko [29] to define resonances . This result generalizes papers by the same author concerning resonances in the de Sitter-Reissner-Nordström spacetime where expansion of the solution in term of resonant states was also given [28], [30]. Concerning resonances in Anti-de Sitter black holes, C. Warnick [42] defines resonances in asymptotically Anti-de Sitter black holes. Quasi-normal modes were also defined for Kerr-Anti-de Sitter black holes by O. Gannot [19]. In the case of Schwarzschild-Anti-de Sitter black hole, the same author proves the existence of resonances using the black box method.In this section, we present the Schwarzschild Anti-de Sitter space-time and give the coordinate system that we will work with in the r...

We study the resolvent of the massive Dirac operator in the Schwarzschild-Anti-de Sitter space-time. After separation of variables, we use standard one dimensional techniques to obtain an explicit formula. We then make use of this formula to extend the resolvent meromorphically accross the real axis.