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

### Sandro Graffi

Università di Bologna, Italy### Dario Bambusi

Università degli Studi di Milano, Italy### Mirko Degli Espositi

Università di Bologna, Italy

## Abstract

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