ABSTRACT:

We will present several recent results concerning the link between quantum and classical mechanics, in the regime of small values of the Planck constant. We will first show how the knowledge of the spectrum of quantum Hamiltonians, suitably localized, permit the reconstruction of classical Hamiltonians and Birkhoff normal forms nears periodic trajectories. In particular we will give a pure quantum construction of the normal from containing the classical one. Secondly we will consider long, Planck constant dependent, time semiclassical evolution. We will show how different scales of time, all of them diverging at the classical limit, present different behaviours, some classical, some not, as for example ubiquity. We will finish by discussing a link between sensitivity to initial conditions in classical mechanics and quantum intrinsic randomness.