Deformation Dependence of Breathing Oscillations in Bose - Fermi Mixtures at Zero Temperature

We study the breathing oscillations in bose-fermi mixtures in the axially-symmetric deformed trap of prolate, spherical and oblate shapes, and clarify the deformation dependence of the frequencies and the characteristics of collective oscillations. The collective oscillations of the mixtures in deformed traps are calculated in the scaling method. In largely-deformed prolate and oblate limits and spherical limit, we obtain the analytical expressions of the collective frequencies. The full calculation shows that the collective oscillations become consistent with the analytically-obtained frequencies when the system is deformed into both prolate and oblate regions. The complicated changes of oscillation characters are shown to occur in the transcendental regions around the spherically-deformed region. We find that these critical changes of oscillation characters are explained by the level crossing behaviors of the intrinsic oscillation modes. The approximate expressions are obtained for the level crossing points that determine the transcendental regions. We also compare the results of the scaling methods with those of the dynamical approach.