Degeneracy of energy levels in a Maslov-quantized perturbed 1:1 resonant oscillator. A quantum counterpart of a Hamiltonian pitchfork bifurcation

Abstract

It is widely known, in classical theory, that nonlinear Hamiltonian systems depending on several parameters often exhibit bifurcation in their trajectories against the changes in the parameters. A Hamiltonian pitchfork bifurcation in the oscillator's periodic trajectories is a typical example. In this article, to seek a quantum counterpart of that bifurcation, the Maslov quantization is applied to a perturbed 1:1 resonant oscillator in the Birkhoff-Gustavson normal form with two parameters. By using a geometric method the degeneracy of energy levels is found to be taken as a quantum counterpart to that bifurcation. The bifurcation set for the Hamiltonian pitchfork bifurcation in classical theory is viewed as the classical limit of a 'bifurcation set' for the degeneracy of energy levels in quantum theory.