Decay of dirac fields in the backgrounds of dilatonic black holes

We find an exact solution to the Dirac equation in the background of the dilatonic black hole in 1 + 1 and 4 + 1 dimensions, and for them we calculate the corresponding quasinormal frequencies. In the two-dimensional case we find that for Weyl spinors the quasinormal modes' (QNM's) frequencies are pure imaginary and have negative sign for all modes, ensuring the stability of the (1 + 1)-dimensional dilatonic black hole. We generalize results to the (4 + 1)-dimensional dilatonic black hole, where the metric is product of a two-dimensional asymptotically flat geometry and a three-sphere with a constant radius. Our conclusion is that the QNMs are pure imaginary, as in the (1 + 1)-dimensional case.