The Dirac system on the Anti-de Sitter Universe

We investigate the global solutions of the Dirac equation on the Anti-de-Sitter Universe. Since this space is not globally hyperbolic, the Cauchy problem is not, {\it a priori}, well-posed. Nevertheless we can prove that there exists unitary dynamics, but its uniqueness crucially depends on the ratio beween the mass $M$ of the field and the cosmological constant $\mathrm{\Lambda }>0$ : it appears a critical value, $\mathrm{\Lambda }\mathrm{/}12$, which plays a role similar to the Breitenlohner-Freedman bound for the scalar fields. When $M^2\ge \mathrm{\Lambda }\mathrm{/}12$ there exists a unique unitary dynamics. In opposite, for the light fermions satisfying $M^2<\mathrm{\Lambda }\mathrm{/}12$, we construct several asymptotic conditions at infinity, such that the problem becomes well-posed. In all the cases, the spectrum of the hamiltonian is discrete. We also prove a result of equipartition of the energy.