Deformation dependence of breathing oscillations in Bose-Fermi mixtures at zero temperature

We study the breathing oscillations in Bose-Fermi mixtures in axially symmetric deformed traps of prolate, spherical and oblate shapes, and clarify the deformation dependence of the frequencies and the characteristics of the collective oscillations. The collective oscillations of the mixtures in the deformed traps are calculated using a scaling method. We obtain for the greatly deformed prolate and oblate limits and the spherical limit analytical expressions for the collective frequencies. The full calculation shows that the collective oscillations become consistent with the frequencies obtained analytically when the system is deformed into prolate or oblate regions. Complicated changes of the oscillation characteristics are shown to occur in the transcendental regions around the spherically deformed region. We find that these critical changes in oscillation characteristics are explained by the level crossing behaviors of the intrinsic oscillation modes. Approximate expressions are obtained for the level crossing points that determine the transcendental regions. We also compare the results from the scaling method with those from the dynamical approach.