99-409 Bouzouina A., Robert D.

Uniform Semi-classical Estimates for the propagation of Heisenberg Observables (43K, LaTeX) Oct 27, 99

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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 $\hbar$).

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