Dario Bambusi, Mirko Degli Esposti, Sandro Graffi
Uniform convergence of the Lie-Dyson expansion with respect to the Planck constant
(30K, Latex 2e)

ABSTRACT.  We prove that the Lie-Dyson expansion for the Heisenberg observables has a nonzero convergence radius in the variable $\epsilon t$ which does not depend on the Planck constant $\hbar$. Here the quantum evolution $U_{\hbar,\epsilon}(t)$ is generated by the Schr\"odinger operator defined by the maximal action in $L^2(\R^n)$ of $-\hbar^2\Delta+\Q+\epsilon V$; $\Q$ is a positive definite quadratic form on $\R^n$; 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 $\epsilon$, the time required for an exchange of energy between the unperturbed oscillator modes is exponentially long independently of $\hbar$