  In this article, we are concerned with long-time behaviour of solutions to a
semi-classical Schr\"odinger-type 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. 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.
