Isospectrality for quantum toric integrable systems

Charles, Laurent; Pelayo, Álvaro; Vũ Ngọc, San

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 5, pp. 815-849.

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.

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.

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