  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.
