  If X is a contact Anosov vector field on a smooth compact manifold M and V is
a smooth function on M, it is known that the differential operator A=-X+V has
some discrete spectrum called Ruelle-Pollicott resonances in specific Sobolev
spaces. We show that for |Im(z)| large the eigenvalues of A are restricted to
vertical bands and in the gaps between the bands, the resolvent of A is bounded
uniformly with respect to |Im(z)|. In each isolated band the density of
eigenvalues is given by the Weyl law. In the first band, most of the
eigenvalues concentrate of the vertical line Re(z)=< D >, the space average of
the function D(x)=V(x)-1/2 div(X)/E_u where Eu is the unstable distribution.
This band spectrum gives an asymptotic expansion for dynamical correlation
functions.
