A new integrable system on the sphere and conformally equivariant quantization

Christian Duval (1, 2) , Galliano Valent (3)

1 CPT - E2 Géométrie, Physique et Symétries
2 CPT - Centre de Physique Théorique - UMR 7332
3 LPTHE - Laboratoire de Physique Théorique et Hautes Energies

Taking full advantage of two independent projectively equivalent metrics on the ellipsoid leading to Liouville integrability of the geodesic flow via the well-known Jacobi-Moser system, we disclose a novel integrable system on the sphere $S^n$, namely the ''dual Moser'' system. The latter falls, along with the Jacobi-Moser and Neumann-Uhlenbeck systems, into the category of (locally) Stäckel systems. Moreover, it is proved that quantum integrability of both Neumann-Uhlenbeck and dual Moser systems is insured by means of the conformally equivariant quantization procedure.

Mots clés: Classical integrability; Projectively equivalent metrics; Stäckel systems; Conformally equivariant quantization; Quantum integrability

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Systèmes Solubles et Intégrables [nlin.SI]

Christian Duval, Galliano Valent. A new integrable system on the sphere and conformally equivariant quantization. Journal of Geometry and Physics, 2011, 61 (1), pp.1329-1347. ⟨10.1016/j.geomphys.2011.02.020⟩. ⟨hal-00502566v2⟩