JournalsrlmVol. 18 , No. 2DOI 10.4171/rlm/487

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

Abstract

We prove that the Lie-Dyson expansion for the Heisenberg observables has a nonzero convergence radius in the variable \ept\ep t which does not depend on the Planck constant \hbar. Here the quantum evolution U,\ep(t)U_{\hbar,\ep}(t) is generated by the \Sc\ operator defined by the maximal action in L2(Rn)L^2(\R^n) of 2Δ+\Q+\epV-\hbar^2\Delta+\Q+\ep V; \Q\Q is a positive definite quadratic form on Rn\R^n; the observables and VV belong to a suitable class of pseudodifferential operators with analytic symbols. It is furthermore proved that, up to an error of order \ep\ep, the time required for an exchange of energy between the unperturbed oscillator modes is exponentially long time independently of \hbar.