Title :

Scarred eigenstates for quantum cat maps of minimal periods

Author(s) :

Faure, Frédéric [Auteur]
Institut Fourier [IF ]
Nonnenmacher, Stéphane [Auteur]
Service de Physique Théorique [SPhT]
De Bievre, Stephan [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Journal title :

Communications in Mathematical Physics

Pages :

449-492

Publisher :

Springer Verlag

Publication date :

2003

ISSN :

0010-3616

HAL domain(s) :

Mathématiques [math]/Physique mathématique [math-ph]
Physique [physics]/Physique mathématique [math-ph]
Science non linéaire [physics]/Dynamique Chaotique [nlin.CD]

English abstract : [en]

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 ...
Show more >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 Planck's constant 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.Show less >

Language :

Anglais

Popular science :

Non

Comment :

LaTeX, 49 pages, includes 10 figures. I added section 6.6. To be published in Commun. Math. Phys

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL