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

Presented by: 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
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.

Received: 2011-08-28
Accepted: 2011-10-12
Published online: 2011-10-26

DOI: 10.1016/j.crma.2011.10.011

Author's affiliations:

Thierry Paul ¹

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

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     title = {Semiclassical approximation and noncommutative geometry},
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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/

References
Cited by

[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

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