Uniform convergence of the Lie-Dyson expansion with respect to the Planck constant

We prove that the Lie-Dyson expansion for the Heisenberg observables has a nonzero convergence radius in the variable εt which does not depend on the Planck constant h. Here the quantum evolution Uh,ε(t) is generated by the Schrödinger operator defined by the maximal action in L2 (Rn) of -h δ + Q + ε V; Q is a positive definite quadratic form on Rn; the observables and V belong to a suitable class of pseudodifferential operators with analytic symbols. It is furthermore proved that, up to an error of order e, the time required for an exchange of energy between the unperturbed oscillator modes is exponentially long independently of h.