  We theoretically investigate breathing oscillations of weakly-interacting
degenerate Fermi gases in highly-anisotropic harmonic oscillator traps. If the
traps are not highly anisotropic, the fermions behave as three-dimensional (3D)
gases and exhibit the coupled breathing oscillations as studied in a previous
paper [T. Maruyama and T. Nishimura, Phys. Rev. A 75 (2007) 033611]; Otherwise
the fermions exhibit quasi-low-dimensional (QLD) properties derived from
specific structures in their single-particle spectrum, called QLD structures.
In the present paper, we focus on effects of the QLD structures on the
breathing oscillations of the two-component fermions with symmetric population
densities. Here we develop the semi-classical Thomas-Fermi approximation
extended to the highly-anisotropic systems and obtain the collective
frequencies in the sum-rule-scaling method and perturbation theory. As a
result, we reveal that the effects of the QLD structures can not be seen in the
transverse modes in the first-order perturbation and appear only in the
longitudinal modes with hierarchies reflecting the QLD structures. We also
demonstrate time-evolution of the oscillations in the present framework.
