Semi-classical Green functions

Let H(x, p) ∼ H 0 (x, p) + hH 1 (x, p) + … be a semi-classical Hamiltonian on T ∗ R n , and Σ E = {H 0 (x, p) = E} be a non-critical energy surface. Consider f h , a semi-classical distribution (the "source") microlocalized on a Lagrangian manifold Λ which intersects cleanly the flow-out Λ + of the Hamilton vector field ${X_{{H_0}}}$ in Σ. Using Maslov’s canonical operator, we look for a semi-classical distribution u h satisfying the limiting absorption principle and H w (x, hD x )u h = f h (semi-classical Green function). In this report, we elaborate (still at an early stage) on some results announced in [1] and provide some examples, in particular, from the theory of wave beams.