Microlocal kernel of pseudodifferential operators at an hyperbolic fixed   point
  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. 
J.-F. Bony, S. Fujiie, T. Ramond and M. Zerzeri