  We study the microlocal kernel of h-pseudodifferential operators P(x,hD)-z,
where z belongs to some neighborhood of size O(h) of a critical value of its
principal symbol. We suppose that this critical value corresponds to a
hyperbolic fixed point of the associated Hamiltonian flow.
  First we describe propagation of singularities at such a hyperbolic fixed
point, both in the analytic and in the smooth category. In both cases, we show
that the null solution is the only element of this microlocal kernel which
vanishes on the stable incoming manifold, but for energies z in some discrete
set.
  For energies z out of this set, we build the element of the microlocal kernel
with prescribed data on the incoming manifold. We describe completely the
operator which associate the value of this null solution on the outgoing
manifold to the initial data on the incoming one. In particular it appears to
be a semiclassical Fourier integral operator associated to some natural
canonical relation.
