Fermi resonance monodromy

B. Two-Dimensional Quantum Monodromy in Fermi Resonance Polyads... 

The tacit assumption above is that the monodromy matrix is defined with respect to the primitive unit cell, with sides (5v, 8fe) = (0,1) and (1, 0), because the twist angle that determines the monodromy is given by A9 = — (Sv/Sfe)j.. However, situations can arise where other choices are more convenient. For example, the energy levels within a given Fermi resonance polyad are labeled by a counting number v = 0,1,... and an angular momentum that takes only even or only odd values. Thus the convenient elementary cell has sides (8v, 8L) = (0,2) and (1, 0), and the natural basis, say, y, is related to the primitive basis, x, by... 

Analysis of the prototypical resonant swing spring model [11-13] shows that Fermi resonance with conserved angular momentum is an intrinsically three-dimensional phenomenon. The form of the 3x3 monodromy matrix was given. 

The standard effective spectroscopic Fermi resonant Hamiltonian allows more complicated types of behavior. The full three-dimensional aspects of the monodromy remain to be worked out, but it was shown, with the help of the Xiao—KeUman [28, 29] catastrophe map, that four main dynamical regimes apply, and that successive polyads of a given molecule may pass from one regime to another. 

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