We consider a nonlinear area preserving Anosov map 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 . The usual semi-classical Trace formula expresses for finite time , in the limit , in terms of periodic orbits of of period . Recent work reach time where is the Ehrenfest time, and 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 where is any constant, with an arbitrary small error.
On considère une application , Anosov non linéaire qui conserve l’aire sur le tore . 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 . La formule des traces semi-classique habituelle exprime pour fini, dans la limite , en termes d’orbites périodiques de de période . Des travaux récents atteignent des temps où est le temps d’Ehrenfest, et 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 , où est une constante arbitraire, et avec une erreur arbitrairement petite.
Keywords: Quantum chaos, hyperbolic map, semiclassical trace formula, Ehrenfest time
Mot clés : chaos quantique, application hyperbolique, formule des traces semi-classique, temps d’Ehrenfest
@article{AIF_2007__57_7_2525_0, author = {Faure, Fr\'ed\'eric}, title = {Semi-classical formula beyond the {Ehrenfest} time in quantum chaos. {(I)} {Trace} formula}, journal = {Annales de l'Institut Fourier}, pages = {2525--2599}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {57}, number = {7}, year = {2007}, doi = {10.5802/aif.2341}, zbl = {1145.81035}, mrnumber = {2394550}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2341/} }
TY - JOUR AU - Faure, Frédéric TI - Semi-classical formula beyond the Ehrenfest time in quantum chaos. (I) Trace formula JO - Annales de l'Institut Fourier PY - 2007 SP - 2525 EP - 2599 VL - 57 IS - 7 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2341/ DO - 10.5802/aif.2341 LA - en ID - AIF_2007__57_7_2525_0 ER -
%0 Journal Article %A Faure, Frédéric %T Semi-classical formula beyond the Ehrenfest time in quantum chaos. (I) Trace formula %J Annales de l'Institut Fourier %D 2007 %P 2525-2599 %V 57 %N 7 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2341/ %R 10.5802/aif.2341 %G en %F AIF_2007__57_7_2525_0
Faure, Frédéric. Semi-classical formula beyond the Ehrenfest time in quantum chaos. (I) Trace formula. Annales de l'Institut Fourier, Volume 57 (2007) no. 7, pp. 2525-2599. doi : 10.5802/aif.2341. http://www.numdam.org/articles/10.5802/aif.2341/
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