 06324 Dario Bambusi, Mirko Degli Esposti, Sandro Graffi
 Uniform convergence of the LieDyson expansion with respect to the Planck constant
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Nov 9, 06

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Abstract. We prove that the LieDyson 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 $\hbar$
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