Publication : t02/095

Scarred Eigenstates for Quantum Cat Maps of Minimal Periods

Faure F. (LPM2C, Maison des Magistères Jean Perrin, CNRS, Grenoble, FRANCE)
Nonnenmacher S. (CEA, DSM, SPhT (Service de Physique Théorique), F-91191 Gif-sur-Yvette, FRANCE)
De Bièvre S. (UFR de Math.-UMR AGAT Université des Sciences et Technologies de Lille F-59655 Villeneuve d'Ascq, FRANCE)

Abstract:
In this paper we construct a sequence of eigenfunctions of the ``quantum Arnold's cat map'' that, in the semiclassical limit, show a strong scarring phenomenon on the periodic orbits of the dynamics. More precisely, those states have a semiclassical limit measure that is the sum of $1/2$ the normalized Lebesgue measure on the torus plus $1/2$ the normalized Dirac measure concentrated on any a priori given periodic orbit of the dynamics. It is known (the Schnirelman theorem) that ``most'' sequences of eigenfunctions equidistribute on the torus. The sequences we construct therefore provide an example of an exception to this general rule. Our method of construction and proof exploits the existence of special values of $\hbar$ for which the quantum period of the map is relatively ``short'', and a sharp control on the evolution of coherent states up to this time scale. We also provide a pointwise description of these states in phase space, which uncovers their ``hyperbolic'' structure in the vicinity of the fixed points and yields more precise localization estimates.

Année de publication : 2003
Revue : Commun. Math. Phys. 239 449-492 (2003)
Preprint : arXiv:nlin.CD/0207060
PACS : 03.65.Sq, 05.45.Mt, 02.20.Rt
Keywords : Quantum chaos, quantum ergodicity, strong scarring
Langue : Anglais

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