  Let $H(x,p)\sim H_0(x,p)+hH_1(x,p)+\cdots$ be a semi-classical Hamiltonian on
$T^*{\bf R}^n$, and $\Sigma_E=\{H_0(x,p)=E\}$ a non critical energy surface.
Consider $f_h$ a semi-classical distribution (the "source") microlocalized on a
Lagrangian manifold $\Lambda$ which intersects cleanly the flow-out $\Lambda_+$
of the Hamilton vector field $X_{H_0}$ in $\Sigma_E$. Using Maslov 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
kernel). In this report, we elaborate (still at an early stage) on some results
announced in [Doklady Akad. Nauk, Vol. 76, No1, p.1-5, 2017] and provide some
examples, in particular from the theory of wave beams.
