The Rotation Number for Quantum Integrable Systems

Monique Dauge (1) , Michael A. Hall (2) , San Vu Ngoc (3, 1)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 USC - University of Southern California
3 IUF - Institut universitaire de France

For a two degree of freedom quantum integrable system, a new spectral quantity is defined, the quantum rotation number. In the semiclassical limit, the quantum rotation number can be detected on a joint spectrum and is shown to converge to the well-known classical rotation number. The proof requires not only semiclassical analysis (including Bohr-Sommerfeld quantization rules) but also a detailed study on how quantum labels can be assigned to the joint spectrum in a smooth way. This leads to the definition and analysis of asymptotic lattices. The general results are applied to the semitoric case where formulas become particularly natural.

Mots clés: good labelling; asymptotic lattice; semiclassical analysis; pseudo-differential operators; rotation number; semitoric systems; quantization; Liouville integrable systems; inverse problem; symplectic invariants; lattice detection

Domaines: Théorie spectrale [math.SP]; Géométrie symplectique [math.SG]; Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]

Monique Dauge, Michael A. Hall, San Vu Ngoc. The Rotation Number for Quantum Integrable Systems. 2019. ⟨hal-02104892v2⟩