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Given a semiclassical distribution $f_h$ microlocalized on a Lagrangianmanifold $\Lambda_0$, $H\in C^\infty( T^\ast {\bf R}^n)$, and $H=E$ a regularenergy 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 theHamilton vector field $v_H$ fails to be transverse to $\Lambda_0$ at somepoints.

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