Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point

We study the microlocal kernel of h-pseudodifferential operators Oph(p) − z, where z belongs to some neighborhood of size O(h) of a critical value of its principal symbol p0(x, ξ ). We suppose that this critical value corresponds to a hyperbolic fixed point of the Hamiltonian flow Hp0 . First we describe propagation of singularities at such a hyperbolic fixed point, both in the analytic and in the C∞ 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 given 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. © 2007 Elsevier Inc. All rights reserved.