Qualitative analysis of the classical and quantum Manakov top (Q2473443)

The paper gives a heuristic, physically motivated (molecular physics), analysis of links between the classical and quantum monodromy transformations for a special integrable version of the Euler top, introduced by \textit{S. V. Manakov} [Funct. Anal. Appl. 10 (1976), 328--329 (1977Funkts. Anal. Prilozh. 10, No. 4, 93--94 (1976; Zbl 0343.70003)]. Its classical phase-space is compact, hence the quantum problem is an example of a finite quantum system, whose properties are analyzed by means of a geometrical representation of the joint spectrum of commuting quantum observables. That is interpreted as a spectral lattice with defects. The classical monodromy map, arising due to the focus-focus singularity (pinched torus), in the quantum version [c.f. also \textit{S. Vũ Ngoc}, Commun. Math. Phys. 203, No. 2, 465--479 (1999; Zbl 0981.35015)] induces a transformation of an elementary cell of a quantum spectral lattice via a propagation (analogue of a parallel transport) along paths which cross singular strata. This effect relies on the existence of local action-angle variables, whereas a nontrivial monodromy demonstrates the absence of global action-angle variables [c.f. \textit{J. J. Duistermaat}, Commun. Pure Appl. Math. 33, 687--706 (1980; Zbl 0439.58014)].