Comptes Rendus
no. 2
Partial differential equations/Mathematical physics
Wave packets and the quadratic Monge–Kantorovich distance in quantum mechanics

Presented by: Jean-Michel Coron

François Golse1; Thierry Paul1
1 CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau cedex, France
Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 177-197.

In this paper, we extend the upper and lower bounds for the “pseudo-distance” on quantum densities analogous to the quadratic Monge–Kantorovich(–Vasershtein) distance introduced in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016) 165–205] to positive quantizations defined in terms of the family of phase space translates of a density operator, not necessarily of rank 1 as in the case of the Töplitz quantization. As a corollary, we prove that the uniform as ħ→0 convergence rate for the mean-field limit of the N-particle Heisenberg equation holds for a much wider class of initial data than in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016) 165–205]. We also discuss the relevance of the pseudo-distance compared to the Schatten norms for the purpose of metrizing the set of quantum density operators in the semiclassical regime.

Nous considérons dans ce texte la « pseudo-distance » entre densités quantiques introduite dans [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016) 165–205], analogue à la distance quadratique de Monge–Kantorovich(–Vasershtein). Nous en étendons les bornes inférieures et supérieures aux quantifications positives définies en termes de la famille des espaces de phase translatés d'un opérateur de densité, pas nécessairement de rang 1 comme dans le cas de la quantification de Töplitz. Comme corollaire, nous démontrons que le taux de convergence uniforme, lorsque ħ tend vers 0, de la limite de champ moyen de l'équation de Heisenberg à N particules vaut pour une classe beaucoup plus large de données initiales que dans [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016) 165–205]. Nous discutons également la pertinence de la pseudo-distance, comparée aux normes de Schatten, dans le but de métriser l'ensemble des opérateurs de densité quantique en régime semi-classique.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2017.12.007

François Golse 1; Thierry Paul 1

1 CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau cedex, France 
@article{CRMATH_2018__356_2_177_0,
     author = {Fran\c{c}ois Golse and Thierry Paul},
     title = {Wave packets and the quadratic {Monge{\textendash}Kantorovich} distance in quantum mechanics},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {177--197},
     publisher = {Elsevier},
     volume = {356},
     number = {2},
     year = {2018},
     doi = {10.1016/j.crma.2017.12.007},
     language = {en},
}
TY  - JOUR
AU  - François Golse
AU  - Thierry Paul
TI  - Wave packets and the quadratic Monge–Kantorovich distance in quantum mechanics
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 177
EP  - 197
VL  - 356
IS  - 2
PB  - Elsevier
DO  - 10.1016/j.crma.2017.12.007
LA  - en
ID  - CRMATH_2018__356_2_177_0
ER  - 
%0 Journal Article
%A François Golse
%A Thierry Paul
%T Wave packets and the quadratic Monge–Kantorovich distance in quantum mechanics
%J Comptes Rendus. Mathématique
%D 2018
%P 177-197
%V 356
%N 2
%I Elsevier
%R 10.1016/j.crma.2017.12.007
%G en
%F CRMATH_2018__356_2_177_0
François Golse; Thierry Paul. Wave packets and the quadratic Monge–Kantorovich distance in quantum mechanics. Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 177-197. doi : 10.1016/j.crma.2017.12.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.12.007/

[1] C. Bardos; L. Erdös; F. Golse; N. Mauser; H.-T. Yau Derivation of the Schrödinger–Poisson equation from the quantum N-body problem, C. R. Acad. Sci. Paris, Ser. I, Volume 334 (2002), pp. 515-520

[2] C. Bardos; F. Golse; N. Mauser Weak coupling limit of the N particle Schrödinger equation, Methods Appl. Anal., Volume 7 (2000), pp. 275-293

[3] C. Cohen-Tannoudji; B. Diu; F. Laloë Quantum Mechanics, vol. 1, John Wiley, New York, 1991

[4] L. Erdös; H.-T. Yau Derivation of the nonlinear Schrödinger equation from a many body Coulomb system, Adv. Theor. Math. Phys., Volume 5 (2001), pp. 1169-1205

[5] F. Golse; C. Mouhot; T. Paul On the mean field and classical limits of quantum mechanics, Commun. Math. Phys., Volume 343 (2016), pp. 165-205

[6] C. Gomez; M. Hauray Rigorous derivation of Lindblad equations from quantum jump processes in 1D | arXiv

[7] A. Grossmann; J. Morlet; T. Paul Transforms associated to square integrable representations I, J. Math. Phys., Volume 26 (1985), pp. 2473-2479

[8] L.D. Landau; E.M. Lifshitz Quantum Mechanics. Nonrelativistic Theory, Pergamon Press Ltd., 1977

[9] P.-L. Lions; T. Paul Sur les mesures de Wigner, Rev. Mat. Iberoam., Volume 9 (1993), pp. 553-618

[10] P. Pickl A simple derivation of mean-field limits for quantum systems, Lett. Math. Phys., Volume 97 (2011), pp. 151-164

[11] M. Reed; B. Simon Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-Adjointness, Academic Press, 1975

[12] M. Reed; B. Simon Methods of Modern Mathematical Physics. I: Functional Analysis, Academic Press, 1980

[13] I. Rodnianski; B. Schlein Quantum fluctuations and rate of convergence towards mean-field dynamics, Commun. Math. Phys., Volume 291 (2009), pp. 31-61

[14] E. Schrödinger Der stetige Übergang von der Mikro- zur Makromechanik, Naturwissenschaften, Volume 14 (1926), pp. 664-666

[15] H. Spohn Kinetic equations from Hamiltonian dynamics, Rev. Mod. Phys., Volume 52 (1980), pp. 600-640

[16] C. Villani Topics in Optimal Transportation, American Math. Soc., Providence, RI, USA, 2003

Cited by Sources:

Comments - Policy