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We look at the long-time behaviour of solutions to a semi-classicalSchr\"odinger equation on the torus. We consider time scales which go toinfinity when the semi-classical parameter goes to zero and we associate witheach time-scale the set of semi-classical measures associated with all possiblechoices of initial data. On each classical invariant torus, the structure ofsemi-classical measures is described in terms of two-microlocal measures,obeying explicit propagation laws. We apply this construction in twodirections. We first analyse the regularity of semi-classical measures, and weemphasize the existence of a threshold : for time-scales below this threshold,the set of semi-classical measures contains measures which are singular withrespect 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 areused to prove semiclassical observability estimates. The results apply as wellto general quantum completely integrable systems.

Semiclassical Completely Integrable Systems : Long-Time Dynamics And Observability Via Two-Microlocal Wigner Measures