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

Abstract

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 $\hslash$. Here the quantum evolution $U_{\hslash ,ϵ}(t)$ is generated by the Schrödinger operator defined by the maximal action in $L^2({\mathbb{R}}^n)$ of $-{\hslash }^2\Delta +\mathcal{Q}+ϵV$; $\mathcal{Q}$ is a positive definite quadratic form on ${\mathbb{R}}^n$; 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 $ϵ$, the time required for an exchange of energy between the unperturbed oscillator modes is exponentially long time independently of $\hslash$.