We prove here that the semiclassical asymptotic expansion for the
propagation of quantum observables, for C\sp ∞-Hamiltonians
growing at most quadratically at infinity, is uniformly dominated at
any order by an exponential term whose argument is linear in time. In
particular, we recover the Ehrenfest time for the validity of the
semiclassical approximation. This extends the result proved in
[BGP]. Furthermore, if the Hamiltonian and the initial observables are
holomorphic in a complex neighborhood of the phase space, we prove
that the quantum observable is an analytic semiclassical
observable. Other results about the large time behavior of observables
with emphasis on the classical dynamic are also given. In particular,
precise Gevrey estimates are established for classically integrable
systems.