# 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 $\hbar$.

## Cite this article

Sandro Graffi, Dario Bambusi, Mirko Degli Espositi, Uniform convergence of the Lie–Dyson expansion with respect to the Planck constant. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 18 (2007), no. 2, pp. 153–162

DOI 10.4171/RLM/487