Asymptotic solutions: glancing trajectories, Lagrangian singularities, and Bessel cylinders

We find a normal form for the simplest Lagrangian singularity appearing as the projection onto the space-energy of a solution to the Hamilton-Jacobi equation. Using this normal form we approximate asymptotic solutions of an equation $(\widehat{H}-E_0)u_h=f_h$ with a semiclassical distribution $f_h$ microlocalized on a Lagrangian manifold ${\mathrm{\Lambda }}_0$ when $E_0$ is a critical value of the restriction $H{\mathrm{\mid }}_{{\mathrm{\Lambda }}_0}$.