01-323 A. Bouzouina, D. Robert

Uniform Semi-classical Estimates for the Propagation of Quantum Observables. (Revised version) (78K, LaTeX) Sep 10, 01

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Abstract. We prove here that the semi-classical asymptotic expansion for the propagation of quantum 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 by Bambusi, Graffi and Paul [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 semi-classical 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.

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