Lagrangian intersections and glancing points: typical transitions of phase in semiclassical approximations

Given a semiclassical distribution $f_h$ microlocalized on a Lagrangian manifold ${\mathrm{\Lambda }}_0$, $H\in C^{\mathrm{\infty }}(T^{\ast }{\mathbf{R}}^n)$, and $H=E$ a regular energy surface, we find asymptotic solutions of the PDE $(H(x,\widehat{p})-E)u_h(x,E)=f_h(x)$ in terms of the Maslov canonical operator, when the Hamilton vector field $v_H$ fails to be transverse to ${\mathrm{\Lambda }}_0$ at some points.