  The quasi-normal modes for black holes are the resonances for the scattering
of incoming waves by black holes. Here we consider scattering of massless
uncharged Dirac fields propagating in the outer region of de
Sitter-Reissner-Nordstr{\"o}m black hole, which is spherically symmetric
charged exact solution of the Einstein-Maxwell equations. Using the spherical
symmetry of the equation and restricting to a fixed harmonic the problem is
reduced to a scattering problem for the 1D massless Dirac operator on the line.
The resonances for the problem are related to the resonances for a certain
semiclassical Schr{\"o}dinger operator with exponentially decreasing positive
potential. We give exact relation between the sets of Dirac and Schr{\"o}dinger
resonances. The asymptotic distribution of the resonances is close to the
lattice of pseudopoles associated to the non-degenerate maxima of the
potentials.
  Using the techniques of quantum Birkhoff normal form we give the complete
asymptotic formulas for the resonances. In particular, we calculate the first
three leading terms in the expansion. Moreover, similar results are obtained
for the de Sitter-Schwarzschild quasi-normal modes, thus improving the result
of S\'a Barreto and Zworski from 1997.
