A differential geometric approach to a quantum counterpart
of bifurcations
in a certain class of 1:1 resonant Hamiltonian
systems
Y. Uwano
Department of Applied Mathematics and Physics, Kyoto University,
Kyoto 606-01, Japan
In quantizing nonlinear Hamiltonian systems exibiting bifurcations, a question arises as to what interesting phenomena come out in the quantized systems as quantum counterparts of the bifurcations. It is shown in [1] that a degeneracy of energy levels is observed as a quantum counterpart of a Hamiltonian pitchfork bifurcation of periodic trajectries. As a generalization of [1], this talk is focused on a quantum counterpart of a bifurcation taking place in a certain class of 1:1 resonant Hamiltonian systems in the Birkhoff-Gustavson normal form with three parameters. The torus quantization method is applied to the Hamiltonian systems in a differential geometric setting: A degeneracy of energy levels is found to be taken as a quantum counterpart to that bifurcation. The bifurcation set for the bifurcation in classical theory is viewed as the classical limit of a bifurcation set' for the degeneracy of energy levels in quantum theory.