  Following our previous paper, Bergeron et al, Smooth quantum dynamics of the
mixmaster universe, Phys. Rev. D 92, 061302(R) (2015), concerning the
quantization of the vacuum Bianchi IX model and the Born-Huang-Oppenheimer
framework, we present a further analysis of the dynamical properties of the
model. Consistently with the deep quantum regime, we implement the harmonic
approximation of the anisotropy potential. We thus obtain manageable dynamical
equations. We study the quantum anisotropic oscillations during the bouncing
phase of the universe. Neglecting the backreaction from transitions between
quantum anisotropy states we obtain analytical results. In particular, we
identify a parameter which is associated with dynamical properties of the
quantum model and describes a sort of phase transition. Once the parameter
exceeds its critical value, the Born-Huang-Oppenheimer approximation breaks
down. The application of the present result to a simple model of the Universe
indicates that the parameter indeed exceeds its critical value and that there
takes place a huge production of anisotropy at the bounce. This in turn must
lead to a sustained phase of accelerated expansion, an inflationary phase. The
quantitative inclusion of backreaction shall be examined in a follow-up paper
based on the vibronic approach.
