Semi-classical behaviour of Schrödinger's dynamics: Revivals of wave packets on hyperbolic trajectory

Article type: Research Article

Authors: Lablée, Olivier

Affiliations: Université Grenoble 1 – CNRS, Institut Fourier, UFR de Mathématiques, UMR 5582, BP 74, 38402 Saint Martin d'Hères, France. E-mail: lablee@ujf-grenoble.fr; URL: http://www-fourier.ujf-grenoble.fr/~lablee/

Abstract: The aim of this paper is to study the semi-classical behaviour of Schrödinger's dynamics for an one-dimensional quantum Hamiltonian with a classical hyperbolic trajectory. As in the regular case (elliptic trajectory), we prove, that for an initial wave packets localized in energy, the dynamics follows the classical motion during short time. This classical motion is periodic and the period Thyp is order of |ln h|. And, for large time, a new period Trev for the quantum dynamics appears: the initial wave packets form again at t=Trev. Moreover, for the time t=p/qTrev a fractional revivals phenomenon of the initial wave packets appears: there is a formation of a finite number of clones of the original wave packet.

Keywords: Schrödinger's dynamics, revivals of wave packets, semi-classical analysis, hyperbolic trajectory, Schrödinger operator with double wells potential

DOI: 10.3233/ASY-2010-1012

Journal: Asymptotic Analysis, vol. 71, no. 1-2, pp. 59-99, 2011