[Approximation semiclassique et géométrie non commutative]
Nous considérons lʼévolution semiclassique à temps long pour lʼéquation de Schrödinger linéaire. Nous montrons que, dans le cas dʼune dynamique sous-jacente chaotique, le symbole principal dʼune observable est propagé, jusquʼà des temps de lʼordre de , par le flot classique sous-jacent, à condition de considérer un calcul symbolique de type Toeplitz que nous précisons et pour lequel le symbole appartient à lʼalgèbre non commutative du feuilletage (fort) instable de la dynamique classique correspondante.
We consider the long time semiclassical evolution for the linear Schrödinger equation. We show that, in the case of chaotic underlying classical dynamics and for times up to , the symbol of a propagated observable by the corresponding von Neumann–Heisenberg equation is, in a sense made precise below, precisely obtained by the push-forward of the symbol of the observable at time . The corresponding definition of the symbol calls upon a kind of Toeplitz quantization framework, and the symbol itself is an element of the noncommutative algebra of the (strong) unstable foliation of the underlying dynamics.
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@article{CRMATH_2011__349_21-22_1177_0,
author = {Paul, Thierry},
title = {Semiclassical approximation and noncommutative geometry},
journal = {Comptes Rendus. Math\'ematique},
pages = {1177--1182},
publisher = {Elsevier},
volume = {349},
number = {21-22},
year = {2011},
doi = {10.1016/j.crma.2011.10.011},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2011.10.011/}
}
TY - JOUR AU - Paul, Thierry TI - Semiclassical approximation and noncommutative geometry JO - Comptes Rendus. Mathématique PY - 2011 SP - 1177 EP - 1182 VL - 349 IS - 21-22 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2011.10.011/ DO - 10.1016/j.crma.2011.10.011 LA - en ID - CRMATH_2011__349_21-22_1177_0 ER -
%0 Journal Article %A Paul, Thierry %T Semiclassical approximation and noncommutative geometry %J Comptes Rendus. Mathématique %D 2011 %P 1177-1182 %V 349 %N 21-22 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2011.10.011/ %R 10.1016/j.crma.2011.10.011 %G en %F CRMATH_2011__349_21-22_1177_0
Paul, Thierry. Semiclassical approximation and noncommutative geometry. Comptes Rendus. Mathématique, Tome 349 (2011) no. 21-22, pp. 1177-1182. doi : 10.1016/j.crma.2011.10.011. http://www.numdam.org/articles/10.1016/j.crma.2011.10.011/
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