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Annales scientifiques de l'ENS - Parutions - série 4, 46 (2013)

Parutions < série 4, 46

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 46, fascicule 5 (2013)

Laurent Charles, Álvaro Pelayo, San Vũ Ngoc
Isospectrality for quantum toric integrable systems
Annales scientifiques de l'ENS 46, fascicule 5 (2013), 815-849

Télécharger cet article : Fichier PDF

Résumé :
Isospectralité pour les systèmes intégrables toriques quantiques
Nous donnons une description complète du spectre de tout système intégrable torique quantique, au moyen de l'analyse microlocale des opérateurs de Toeplitz. Ceci résout la question de l'isospectralité pour cette classe de systèmes intégrables : le spectre semi-classique d'un système intégrable quantique torique détermine le système intégrable classique sous-jacent à symplectomorphisme près. Nous donnons aussi une description complète de la théorie spectrale semi-classique des systèmes intégrables toriques quantiques. Ces questions sont classiques en théorie spectrale et remontent aux travaux fondateurs de Colin de Verdière, Guillemin et Sternberg parmi d'autres dans les années 70 et 80.

Mots-clefs : Analyse semi-classique, géométrie symplectique, spectre d'opérateurs, opérateurs de Toeplitz, variété symplectique torique.

Abstract:
We give a full description of the semiclassical spectral theory of quantum toric integrable systems using microlocal analysis for Toeplitz operators. This allows us to settle affirmatively the isospectral problem for quantum toric integrable systems: the semiclassical joint spectrum of the system, given by a sequence of commuting Toeplitz operators on a sequence of Hilbert spaces, determines the classical integrable system given by the symplectic manifold and commuting Hamiltonians. This type of problem belongs to the realm of classical questions in spectral theory going back to pioneer works of Colin de Verdière, Guillemin, Sternberg and others in the 1970s and 1980s.

Keywords: Semi-classical analysis, symplectic geometry, operator spectrum, Toeplitz operators, toric symplectic manifold.

Class. math. : 58J50,,81Q20, 81S10, 47B35, 37J15


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

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