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.
A. Bouzouina. D. Robert. "Uniform semiclassical estimates for the propagation of quantum observables." Duke Math. J. 111 (2) 223 - 252, 1 February 2002. https://doi.org/10.1215/S0012-7094-02-11122-3