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 XH
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.