  In this article we study limits of Wigner distributions (the so-called
semiclassical measures) corresponding to sequences of solutions to the
semiclassical Schroedinger equation at times scales $\alpha_{h}$ tending to
infinity as the semiclassical parameter $h$ tends to zero (when $\alpha
_{h}=1/h$ this is equivalent to consider solutions to the non-semiclassical
Schreodinger equation). Some general results are presented, among which a weak
version of Egorov's theorem that holds in this setting. A complete
characterization is given for the Euclidean space and Zoll manifolds (that is,
manifolds with periodic geodesic flow) via averaging formulae relating the
semiclassical measures corresponding to the evolution to those of the initial
states. The case of the flat torus is also addressed; it is shown that
non-classical behavior may occur when energy concentrates on resonant
frequencies. Moreover, we present an example showing that the semiclassical
measures associated to a sequence of states no longer determines those of their
evolutions. Finally, some results concerning the equation with a potential are
presented.
