no. 21-22
Partial Differential Equations/Mathematical Physics
Semiclassical approximation and noncommutative geometry
[Approximation semiclassique et géométrie non commutative]
Présenté par : Yves Meyer
Paul, Thierry1
1 CNRS and CMLS École polytechnique, 91128 Palaiseau cedex, France
Comptes Rendus. Mathématique, Tome 349 (2011) no. 21-22, pp. 1177-1182

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 ℏ−2+ϵ,ϵ>0, 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 t=0. 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.

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 ℏ−2+ϵ,ϵ>0, 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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.10.011

Paul, Thierry 1

1 CNRS and CMLS École polytechnique, 91128 Palaiseau cedex, France 
@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},
     year = {2011},
     publisher = {Elsevier},
     volume = {349},
     number = {21-22},
     doi = {10.1016/j.crma.2011.10.011},
     language = {en},
     url = {https://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  - https://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 https://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

[1] Bambusi, D.; Graffi, S.; Paul, T. Long time semiclassical approximation of quantum flows: A proof of the Ehrenfest time, Asymptot. Anal., Volume 21 (1999), pp. 149-160

[2] Bouzouina, A.; Robert, D. Uniform semiclassical estimates for the propagation of quantum observables, Duke Math. J., Volume 111 (2002), pp. 223-252

[3] Connes, A. Noncommutative Geometry, Academic Press, Inc., 1994

[4] Connes, A. Sur la thérie non commutative de lʼintégration, Lectures Notes in Math., vol. 725, Springer, Berlin, 1979

[5] Hasselblatt, B.; Katok, A. Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and Its Applications, vol. 54, Cambridge University Press, 1995

[6] T. Paul, in preparation.

[7] Paul, T.; Uribe, A. The semi-classical trace formula and propagation of wave packets, J. Funct. Anal., Volume 132 (1995), pp. 192-249

[8] Schubert, R. Semiclassical wave propagation for large times | arXiv

Cité par Sources :