Bouzouina A., Robert D.
Uniform Semi-classical Estimates for the propagation
of Heisenberg Observables
(43K, LaTeX)
ABSTRACT. We prove here that the semi-classical asymptotic expansion
for the propagation of quantum Heisenberg observables,
for $C^\infty$-Hamiltonians growing at most quadratically at infinity,
is uniformly dominated at any order, by an exponential term who's
argument is linear in time. In particular, we recover the Ehrenfest
time for the validity of the semi-classical approximation. This extends
the result proved in \cite{bgp}. Furthermore, if the Hamiltonian and
the initial observables are holomorphic in a complex neighborhood of
the phase space, we prove that the Heisenberg observable is a
semi-classical observable of index Gevrey 2 (3/2 if the Hamiltonian
is purely classical, without lower terms in $\hbar$).