Quantization Of Nonintegrable Systems By Lyapunov Exponents And Periodic Orbits

Highly excited molecular vibration is nonintegrable or even chaotic due to its strong nonlinearity. The quantization of a nonintegralbe system is a very frontier topic of physics which remains unresolved up to now. By the coset method of Lie algebra (group), we studied the problem of quantization of the nonintegrable systems. The main contents are as follows.1. Formal quantum numbers as retrieved via diabatic correlation to the zero-order states where all inter-mode couplings are switched off can be assigned to the eigenstates of the molecular highly excited vibration. Formal quantum numbers embody the important characteristics of the highly excited vibrational states, and are shown as approximate constants of motion. These formal quantum numbers can be employed to classify and assign the highly excited vibrational states. Furthermore, they are helpful to the quantization of nonintegrable systems by utilizing periodic orbits.2. The quantization of the system of one electron in four sites and the vibration of H2O with Fermi resonance were studied by the Lyapunov analysis. The quantization condition is that the average Lyapunov exponents show local minima as a function of the classical energy. This emphasizes that the quantized levels request the least global chaoticity. This is a fresh idea that treats quantization from the viewpoint of chaoticity and has not been reported before to the best of our knowledge. 3. We studied the periodic motions due to the nonlinear resonances of D-C and C-N stretches of DCN molecule. The nonlinearity due to these resonances plays an important role in the dynamics of highly excited molecular vibrational states. These periodic motions bear peculiar characters not possessed by the normal modes due to linearity. The characteristics of the periodic motions due to nonlinearity were found related to the resonance forms.4. Quantization of the non-integrable (chaotic) systems of DCN vibration and Henon-Heiles potential were analyzed by the action integrals of periodic orbits. Our results show that: the number of periodic orbits of a nonintegrable systemdecreases and the system becomes more chaotic as the perturbation on the integrable part is increased. These survived periodic orbits though scarce in a chaotic system, form the invariant skeleton of the phase space and possess important characters relating to the (approximate) constants of motion and the quantal levels. Hence, we can employ the traditional action integral method by the remnant periodic orbits to explore the quantization condition for a nonintegrable (and chaotic) system. We found that the action integrals correspond to the constants of motion of the quantal levels. This point emphasizes the importance of periodic orbits. On the other hand, we note that the least global chaoticity throughout the whole phase space via the chaotic orbits can also lead to proper quantization for the nonintegrable systems. These works shed on the speculation of the intrinsic connection between periodic and chaotic orbits in a nonintegrable system.