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Astérisque - Titles - 375 (2015)

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Prequantum transfer operator for symplectic Anosov diffeomorphism
Frédéric Faure and Masato Tsujii
Astérisque 375 (2015), ix+222 pages
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Résumé :
Opérateur de transfert préquantique pour un difféomorphisme symplectique Anosov
On définit la prequantification d'un difféomorphisme symplectique et Anosov f:MM comme étant une extension U(1) de f qui préserve une connexion dont la courbure est la forme symplectique sur M. On étudie les propriétés spectrales de l'opérateur de transfert associé avec un potentiel VC^(M). On l'appelle l'opérateur de transfert préquantique. C'est un modèle pour les opérateurs associés au flot géodésique sur les variétés de courbure négative (ou les flots Anosov de contact). On restreint l'opérateur de transfert au mode de Fourier N par rapport à l'action de U(1) et on étudie ses propriétés spectrales dans la limite N, en considérant l'opérateur de transfert comme un opérateur intégral de Fourier et en utilisant l'analyse semi-classique. Le résultat principal, avec des conditions de pincements, montre que le spectre a une structure en bandes, c'est à dire qu'il est contenu dans des anneaux séparés et concentriques à l'origine. On montre qu'avec le potentiel spécial (et seulement Hölder continu) V_0=12|Df|_E_u|, où E_u est l'espace instable, la bande la plus externe est le cercle unité et est séparé des autres bandes par un gap uniforme en N. Pour cela on utilise une extension de l'opérateur de transfert au fibré de Grassmann. En utilisant la formule des traces de Atiyah-Bott, on établit une formule des traces de Gutzwiller avec un reste décroissant exponentiellement vite en temps longs. Pour un potentiel V général, et pour N, la plupart des valeurs propres de la bande externe se concentrent et s'équidistribuent sur le cercle de rayon (V-V_0_M)._M signifie la moyenne sur M. Le nombre de valeurs propres sur la bandes externe satisfait la loi de Weyl c'est à dire N^dVol(M) à l'ordre dominant, avec d=12dimM. On développe un calcul semi-classique associé à l'opérateur préquantique en définissant une quantification des observables Op_N() comme étant la projection de l'opérateur multiplication par sur l'espace spectral de la bande extérieure. On obtient une formule de transport de type Egorov qui est exacte. Les fonctions de corrélations définies par l'opérateur de transfert sont gouvernées en temps long par l'opérateur restreint à la bande externe que l'on appelle opérateur quantique. On interprète ces résultats d'un point de vue physique comme l'émergence de la dynamique quantique dans les fonctions de corrélations classiques en temps longs. On compare ces résultats avec la quantification géométrique (standard) en chaos quantique.

Mots-clefs : Opérateurs de transfert, résonances de Ruelle, décroissance des corrélations, analyse semi-classique.

Abstract:
We define the prequantization of a symplectic Anosov diffeomorphism f:MM as a U(1) extension of the diffeomorphism f preserving a connection related to the symplectic structure on M. We study the spectral properties of the associated transfer operator with a given potential VC^(M), called prequantum transfer operator. This is a model of transfer operators for geodesic flows on negatively curved manifolds (or contact Anosov flows). We restrict the prequantum transfer operator to the N-th Fourier mode with respect to the U(1) action and investigate the spectral property in the limit N, regarding the transfer operator as a Fourier integral operator and using semi-classical analysis. In the main result, under some pinching conditions, we show a ``band structure'' of the spectrum, that is, the spectrum is contained in a few separated annuli and a disk concentric at the origin. We show that, with the special (Hölder continuous) potential V_0=12|Df|_E_u|, where E_u is the unstable subspace, the outermost annulus is the unit circle and separated from the other parts. For this, we use an extension of the transfer operator to the Grassmanian bundle. Using Atiyah-Bott trace formula, we establish the Gutzwiller trace formula with exponentially small reminder for large time. We show also that, for a potential V such that the outermost annulus is separated from the other parts, most of the eigenvalues in the outermost annulus concentrate on a circle of radius (V-V_0) where . denotes the spatial average on M. The number of the eigenvalues in the outermost annulus satisfies a Weyl law, that is, N^dVol(M) in the leading order with d=12dimM. We develop a semiclassical calculus associated to the prequantum operator by defining quantization of observables Op_N() as the spectral projection of multiplication operator by  to this outer annulus. We obtain that the semiclassical Egorov formula of quantum transport is exact. The correlation functions defined by the classical transfer operator are governed for large time by the restriction to the outer annulus that we call the quantum operator. We interpret these results from a physical point of view as the emergence of quantum dynamics in the classical correlation functions for large time. We compare these results with standard quantization (geometric quantization) in quantum chaos.

Class. math. : 37D20, 37D35, 37C30, 81Q20, 81Q50


ISBN : 978-2-85629-823-7
ISSN : 0303-1179
Publié avec le concours de : Centre National de la Recherche Scientifique

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