no. 7
Semi-classical formula beyond the Ehrenfest time in quantum chaos. (I) Trace formula
[Formules semi-classiques au-delà du temps d’Erhenfest en chaos quantique. (I) La formule des traces.]
Faure, Frédéric1
1 Institut Fourier 100 rue des Maths, BP74 38402 St Martin d’Heres (France)
Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2525-2599.

On considère une application M, Anosov non linéaire qui conserve l’aire sur le tore T 2 . C’est un des exemples les plus simples d’une dynamique chaotique. On s’intéresse à la dynamique quantique pour les temps longs, générée par un opérateur unitaire M ^. La formule des traces semi-classique habituelle exprime TrM ^ t pour t fini, dans la limite ℏ→0, en termes d’orbites périodiques de M de période t. Des travaux récents atteignent des temps t≪t E /6 où t E =log(1/ℏ)/λ est le temps d’Ehrenfest, et λ est le coefficient de Lyapounov. En utilisant une description uniforme de la dynamique au moyen d’une forme normale semi-classique, nous montrons comment étendre la formule des traces pour des temps plus longs, de la forme t=C.t E , où C est une constante arbitraire, et avec une erreur arbitrairement petite.

We consider a nonlinear area preserving Anosov map M on the torus phase space, which is the simplest example of a fully chaotic dynamics. We are interested in the quantum dynamics for long time, generated by the unitary quantum propagator M ^. The usual semi-classical Trace formula expresses TrM ^ t for finite time t, in the limit ℏ→0, in terms of periodic orbits of M of period t. Recent work reach time t≪t E /6 where t E =log(1/ℏ)/λ is the Ehrenfest time, and λ is the Lyapounov coefficient. Using a semi-classical normal form description of the dynamics uniformly over phase space, we show how to extend the trace formula for longer time of the form t=C.t E where C is any constant, with an arbitrary small error.

DOI : 10.5802/aif.2341
Classification : 81Q50, 37D20
Keywords: Quantum chaos, hyperbolic map, semiclassical trace formula, Ehrenfest time
Mot clés : chaos quantique, application hyperbolique, formule des traces semi-classique, temps d’Ehrenfest

Faure, Frédéric 1

1 Institut Fourier 100 rue des Maths, BP74 38402 St Martin d’Heres (France) 
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Faure, Frédéric. Semi-classical formula beyond the Ehrenfest time in quantum chaos. (I) Trace formula. Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2525-2599. doi : 10.5802/aif.2341. https://aif.centre-mersenne.org/articles/10.5802/aif.2341/

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