Index and dynamics of quantized contact transformations

Zelditch, Steven

doi:10.5802/aif.1568

Zelditch, Steven

Annales de l'Institut Fourier, Tome 47 (1997) no. 1, pp. 305-363.

Résumé
Abstract

Les transformations de contact quantifiées sont des opérateurs unitaires de Toeplitz de la forme $U_{χ} = Π A χ Π$ sur une variété $(X, α)$ de contact. Ici, $Π : H^{2} (X) \to L^{2} (X)$ est un projecteur de Szegö, $χ$ est une transformation de contact, et $A$ est un opérateur pseudodifférentiel sur $X$ . On peut quantifier une transformation symplectique $χ_{o}$ sur une variété symplectique $(M, ω)$ de cette façon lorsque $χ_{o}$ se relève en une transformation de contact $χ$ sur le fibré “pré-quantique” en cercles $X \to M$ . On montre que les automorphismes symplectiques $χ_{o}$ d’un tore $(M, \sum d x_{i} \land d ξ_{i})$ sont de ce type : le fibré $X$ est alors le quotient du groupe de Heisenberg par son réseau entier $δ$ , le projecteur $Π$ est le noyau de Szegö, et, à une constante près, $Π χ Π$ définit une des lois de transformation de Hermite–Jacobi sur les fonctions thêta. Il en résulte que les applications quantiques du chat (telles qu’elles sont connues dans la littérature physique) ne sont autres que l’action métaplectique du groupe de thêta sur les fonctions thêta. Il résulte aussi que les indices de ces applications symplectiques sont nuls. On donne par ailleurs des résultats généraux sur l’ergodicité quantique des transformations de contact quantifiées, c’est-à-dire, sur les propriétés asymptotiques des valeurs et fonctions propres de $Π A χ Π$ .

Quantized contact transformations are Toeplitz operators over a contact manifold $(X, α)$ of the form $U_{χ} = Π A χ Π$ , where $Π : H^{2} (X) \to L^{2} (X)$ is a Szegö projector, where $χ$ is a contact transformation and where $A$ is a pseudodifferential operator over $X$ . They provide a flexible alternative to the Kähler quantization of symplectic maps, and encompass many of the examples in the physics literature, e.g. quantized cat maps and kicked rotors. The index problem is to determine $ind (U_{χ})$ when the principal symbol is unitary, or equivalently to determine whether $A$ can be chosen so that $U_{χ}$ is unitary. We show that the answer is yes in the case of quantized symplectic torus automorphisms $g$ —by showing that $U_{g}$ duplicates the classical transformation laws on theta functions. Using the Cauchy-Szegö kernel on the Heisenberg group, we calculate the traces on theta functions of each degree $N$ . We also study the quantum dynamics generated by a general q.c.t. $U_{χ}$ , i.e. the quasi-classical asymptotics of the eigenvalues and eigenfunctions under various ergodicity and mixing hypotheses on $χ .$ Our principal results are proofs of equidistribution of eigenfunctions $φ_{N j}$ and weak mixing properties of matrix elements $(B φ_{N i}, φ_{N j})$ for quantizations of mixing symplectic maps.

MR Zbl | 2 citations dans Numdam

DOI : 10.5802/aif.1568

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     title = {Index and dynamics of quantized contact transformations},
     journal = {Annales de l'Institut Fourier},
     pages = {305--363},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {47},
     number = {1},
     year = {1997},
     doi = {10.5802/aif.1568},
     mrnumber = {99a:58082},
     zbl = {0865.47018},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1568/}
}

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Zelditch, Steven. Index and dynamics of quantized contact transformations. Annales de l'Institut Fourier, Tome 47 (1997) no. 1, pp. 305-363. doi : 10.5802/aif.1568. http://www.numdam.org/articles/10.5802/aif.1568/

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