Nous cherchons des solutions globales de l'équation de Dirac dans l'univers Anti-de Sitter. Comme cet espace n'est pas globalement hyperbolique, le problème de Cauchy n'est pas, a priori, bien posé. Nous montrons que c'est toutefois le cas quand la masse du champ est grande par rapport à la constante cosmologique. En revanche, pour les faibles masses, nous construisons diverses conditions asymptotiques à l'infini, rendant le problème bien posé. Dans tous les cas, l'hamiltonien a un spectre discret. On établit également un résultat d'équipartition de l'énergie.
We investigate global solutions of the Dirac equation on the Anti-de-Sitter Universe. Since this space is not globally hyperbolic, the Cauchy problem is not, a priori, well-posed. Nevertheless, this is the case when the mass of the field is large compared to the cosmological constant. In opposite, for the light fermions, 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 get a result of equipartition of the energy.
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@article{CRMATH_2007__345_8_435_0, author = {Bachelot, Alain}, title = {The {Dirac} equation on the {Anti-de-Sitter} {Universe}}, journal = {Comptes Rendus. Math\'ematique}, pages = {435--440}, publisher = {Elsevier}, volume = {345}, number = {8}, year = {2007}, doi = {10.1016/j.crma.2007.09.011}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2007.09.011/} }
TY - JOUR AU - Bachelot, Alain TI - The Dirac equation on the Anti-de-Sitter Universe JO - Comptes Rendus. Mathématique PY - 2007 SP - 435 EP - 440 VL - 345 IS - 8 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2007.09.011/ DO - 10.1016/j.crma.2007.09.011 LA - en ID - CRMATH_2007__345_8_435_0 ER -
Bachelot, Alain. The Dirac equation on the Anti-de-Sitter Universe. Comptes Rendus. Mathématique, Tome 345 (2007) no. 8, pp. 435-440. doi : 10.1016/j.crma.2007.09.011. http://www.numdam.org/articles/10.1016/j.crma.2007.09.011/
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