34 Pages
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QUANTUM REVIVALS IN TWO DEGREES OF FREEDOM INTEGRABLE SYSTEMS THE TORUS CASE

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Niveau: Supérieur, Licence, Bac+2
QUANTUM REVIVALS IN TWO DEGREES OF FREEDOM INTEGRABLE SYSTEMS : THE TORUS CASE OLIVIER LABLEE ABSTRACT. The paper deals with the semi-classical behaviour of quantum dy- namics for a semi-classical completely integrable system with two degrees of free- dom near Liouville regular torus. The phenomomenon of wave packet revivals is demonstrated in this article. The framework of this paper is semi-classical analy- sis (limit : h ? 0). For the proofs we use standard tools of real analysis, Fourier analysis and basic analytic number theory. 1. INTRODUCTION 1.1. Motivation. In quantum physics, on a Riemannian manifold (M, g) the evo- lution of an initial state ?0 ? L2(M) is given by the famous Schrodinger equation : ih ∂?(t) ∂t = Ph?(t); ?(0) = ?0. Here h > 0 is the semi-classical parameter and the operator Ph : D (Ph) ? L2 (M) ? L2 (M) is h-pseudo-differential operator (for example Ph = ? h 2 2 ∆g + V). In the case of dimension 1 or for completely integrable systems, we can describe the semi-classical eigenvalues of the Hamiltonian Ph and by linearity we can write the solutions of the Schrodinger equation. Nevertheless, the behaviour of the so- lutions when the times t evolves in larges times scales remains quite mysterious.

  • initial state

  • classical completely

  • larges times

  • semi-classical analysis

  • ?0 ?

  • harmonic oscillators

  • quantum dynamics

  • integrable system


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