Semiclassical approximation and noncommutative geometry

Thierry Paul

doi:10.1016/j.crma.2011.10.011

Partial Differential Equations/Mathematical Physics

Semiclassical approximation and noncommutative geometry
[Approximation semiclassique et géométrie non commutative]

Présenté par : Yves Meyer

Thierry Paul ¹

¹ CNRS and CMLS École polytechnique, 91128 Palaiseau cedex, France

Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1177-1182.

Abstract (VO)
Résumé

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 : 2011-08-28
Accepté le : 2011-10-12
Publié le : 2011-10-26

DOI : 10.1016/j.crma.2011.10.011

Affiliations des auteurs :

Thierry Paul ¹

¹ CNRS and CMLS École polytechnique, 91128 Palaiseau cedex, France

@article{CRMATH_2011__349_21-22_1177_0,
     author = {Thierry Paul},
     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},
}

TY  - JOUR
AU  - Thierry Paul
TI  - Semiclassical approximation and noncommutative geometry
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 1177
EP  - 1182
VL  - 349
IS  - 21-22
PB  - Elsevier
DO  - 10.1016/j.crma.2011.10.011
LA  - en
ID  - CRMATH_2011__349_21-22_1177_0
ER  -

%0 Journal Article
%A Thierry Paul
%T Semiclassical approximation and noncommutative geometry
%J Comptes Rendus. Mathématique
%D 2011
%P 1177-1182
%V 349
%N 21-22
%I Elsevier
%R 10.1016/j.crma.2011.10.011
%G en
%F CRMATH_2011__349_21-22_1177_0

Thierry Paul. Semiclassical approximation and noncommutative geometry. Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1177-1182. doi : 10.1016/j.crma.2011.10.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.10.011/

Bibliographie
Cité par

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

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

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

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

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

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

Thierry Paul Species of spaces, Theoretical physics, wavelets, analysis, genomics. An indisciplinary tribute to Alex Grossmann, Cham: Springer, 2023, pp. 307-386 | DOI:10.1007/978-3-030-45847-8_16 | Zbl:1555.81092
Thierry Paul Mathematical Entities without Objects. On Realism in Mathematics and a Possible Mathematization of (Non)Platonism: Does Platonism Dissolve in Mathematics?, European Review, Volume 29 (2021) no. 2, p. 253 | DOI:10.1017/s1062798720000393
Thierry Paul Symbolic calculus for singular curve operators, Schrödinger operators, spectral analysis and number theory. In memory of Erik Balslev, Cham: Springer, 2021, pp. 195-221 | DOI:10.1007/978-3-030-68490-7_10 | Zbl:1471.81085
Fabricio Macià High-frequency dynamics for the Schrödinger equation, with applications to dispersion and observability, Nonlinear optical and atomic systems. At the interface of physics and mathematics. Based on lecture notes given at the 2013 Painlevé-CEMPI-PhLAM thematic semester., Cham: Springer; Lille: Centre Européen pour les Mathématiques, la Physiques et leurs Interactions (CEMPI), 2015, pp. 275-335 | DOI:10.1007/978-3-319-19015-0_4 | Zbl:1337.35126

Cité par 4 documents. Sources : Crossref, zbMATH

Commentaires - Politique

Il n'y a aucun commentaire pour cet article. Soyez le premier à écrire un commentaire !

Publier un nouveau commentaire:

Publier une nouvelle réponse: