Semiclassical localization in phase space

We study semiclassical properties of quantum systems whose classical limit is neither chaotic nor integrable. The phase space of the corresponding classical systems possess plenty of invariant sets, and we construct approximate projection operators associated with open invariant sets as Anti-Wick quantizations of their characteristic functions. Under an additional condition on the invariant set, called stable invariance, the construction yields approximate projection operators whose commutator with the Hamilton operator is bounded by any power of Planks constant. Finally, we discuss some applications to time evolution and quasimodes