American Journal of Mathematics

We look at the long-time behavior 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 analyze 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.