[Propagation quantique pour les opérateurs de Berezin–Toeplitz]
We describe the asymptotic behaviour of the quantum propagator generated by a Berezin–Toeplitz operator with real-valued principal symbol. We also give precise asymptotics for smoothed spectral projectors associated with the operator in the autonomous case; this leads us to introducing quantum states associated with immersed Lagrangian submanifolds. These descriptions involve geometric quantities of two origins, coming from lifts of the Hamiltonian flow to the prequantum bundle and the canonical bundle respectively. The latter are the main contribution of this article and are connected to the Maslov indices appearing in trace formulas, as will be explained in a forthcoming paper.
Nous décrivons le comportement asymptotique du propagateur quantique d’un opérateur de Berezin–Toeplitz de symbole principal réel. Nous donnons également une asymptotique précise pour les projecteurs spectraux régularisés associés à l’opérateur dans le cas autonome ; cela nous mène à introduire des états quantiques associés à des sous-variétés lagrangiennes immergées. Ces descriptions mettent en jeu des quantités géométriques de deux origines, provenant de relevés du flot hamiltonien au fibré préquantifiant et au fibré canonique, respectivement. Ces dernières sont la contribution principale de l’article et sont reliées aux indices de Maslov apparaissant dans les formules de traces, comme nous l’expliquerons dans un futur article.
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Keywords: Berezin–Toeplitz operators, Schrödinger equation, Lagrangian states, geometric quantization, semiclassical limit
Mots-clés : opérateurs de Berezin–Toeplitz, équation de Schrödinger, états lagrangiens, quantification géométrique, limite semiclassique
Laurent Charles 1 ; Yohann Le Floch 2
CC-BY 4.0
Laurent Charles; Yohann Le Floch. Quantum propagation for Berezin–Toeplitz operators. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 35 (2026) no. 3, pp. 543-586. doi: 10.5802/afst.1857
@article{AFST_2026_6_35_3_543_0,
author = {Laurent Charles and Yohann Le Floch},
title = {Quantum propagation for {Berezin{\textendash}Toeplitz} operators},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {543--586},
year = {2026},
publisher = {Universit\'e de Toulouse, Toulouse},
volume = {Ser. 6, 35},
number = {3},
doi = {10.5802/afst.1857},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1857/}
}
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%0 Journal Article %A Laurent Charles %A Yohann Le Floch %T Quantum propagation for Berezin–Toeplitz operators %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2026 %P 543-586 %V 35 %N 3 %I Université de Toulouse, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1857/ %R 10.5802/afst.1857 %G en %F AFST_2026_6_35_3_543_0
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