  We analyse the structure of semiclassical and microlocal Wigner measures for
solutions to the linear Schr\"{o}dinger equation on the disk, with Dirichlet
boundary conditions.
  Our approach links the propagation of singularities beyond geometric optics
with the completely integrable nature of the billiard in the disk. We prove a
"structure theorem", expressing the restriction of the Wigner measures on each
invariant torus in terms of {\em second-microlocal measures}. They are obtained
by performing a finer localization in phase space around each of these tori, at
the limit of the uncertainty principle, and are shown to propagate according to
Heisenberg equations on the circle.
  Our construction yields as corollaries (a) that the disintegration of the
Wigner measures is absolutely continuous in the angular variable, which is an
expression of the dispersive properties of the equation; (b) an observability
inequality, saying that the $L^2$-norm of a solution on any open subset
intersecting the boundary (resp. the $L^2$-norm of the Neumann trace on any
nonempty open set of the boundary) controls its full $L^2$-norm (resp.
$H^1$-norm). These results show in particular that the energy of solutions
cannot concentrate on periodic trajectories of the billiard flow other than the
boundary.
