Collective excitations of a trapped degenerate Fermi gas

We evaluate the small-amplitude excitations of a spin-polarized vapour of Fermi atoms confined inside a harmonic trap. The dispersion law $\omega ={\omega }_f[l+4n(n+l+2)\mathrm{/}3{]}^{1\mathrm{/}2}$ is obtained for the vapour in the collisional regime inside a spherical trap of frequency ${\omega }_f$, with $n$ the number of radial nodes and $l$ the orbital angular momentum. The low-energy excitations are also treated in the case of an axially symmetric harmonic confinement. The collisionless regime is discussed with main reference to a Landau-Boltzmann equation for the Wigner distribution function: this equation is solved within a variational approach allowing an account for non-linearities. A comparative discussion of the eigenmodes of oscillation for confined Fermi and Bose vapours is presented in an Appendix.