  We look at the long-time behaviour of solutions to a semi-classical
Schr\"odinger equation on the torus. We consider time scales which go to
infinity when the semi-classical parameter goes to zero and we associate with
each time-scale the set of semi-classical measures associated with all possible
choices of initial data. On each classical invariant torus, the structure of
semi-classical measures is described in terms of two-microlocal measures,
obeying explicit propagation laws. We apply this construction in two
directions. We first analyse the regularity of semi-classical measures, and we
emphasize the existence of a threshold : for time-scales below this threshold,
the set of semi-classical measures contains measures which are singular with
respect to Lebesgue measure in the "position" variable, while at (and beyond)
the threshold, all the semi-classical measures are absolutely continuous in the
"position" variable, reflecting the dispersive properties of the equation.
Second, the techniques of two- microlocal analysis introduced in the paper are
used to prove semiclassical observability estimates. The results apply as well
to general quantum completely integrable systems.
