The Hawking effect predicts that, in a space- time describing the collapse of a spherically symmetric star to a Schwarzschild black hole, a static observer at infinity sees the Unruh state as a thermal state at Hawking temperature. The first mathematical proof of the Hawking effect, in the original setting of Hawking, is due to Bachelot. His work on Klein-Gordon fields has been extended to Dirac fields, in the first place by Bachelot himself, and by Melnyk after that. Those works, placed in the setup of a spherically symmetric star, have been completed by Häfner, who gave a rigorous proof of the Hawking effect for Dirac fields, outside a star collapsing to a Kerr black hole. The aim of this thesis is to study the Hawking effect not for a model of free quantum fields, in which case the problems can be reduced to studies on linear partial differential equations, but for a model of interacting Dirac fields. The interaction will be considered as a static, compactly-supported interaction, living outside the star. We choose to study a toy model in a 1+1 dimensional space-time. Using the fact that the problem is spherically symetric, one can, at least in the free case, reduce the real problem to this toy model. We study the behavior of Dirac fermions fields in various situations : first, for an observable following the star's collapse ; then, for a static observable ; finally, for a time-dependent interaction, fixed close to the star's boundary. In each of those cases, we show the existence of the Hawking Effect and give the corresponding limit state.
