Zero sound and first sound in a disk-shaped normal Fermi gas

We study the zero sound and the first sound in a dilute and ultracold disk-shaped normal Fermi gas with a strong harmonic confinement along the axial direction and uniform in the two planar directions. Working at zero temperature we calculate the chemical potential mu of the fermionic fluid as a function of the uniform planar density rho and find that mu changes its slope in correspondence to the filling of harmonic axial modes (shell effects). Within the linear response theory, and under the random phase approximation, we calculate the velocity c(s)(0) of the zero sound. We find that also c(s)(0) changes its slope in correspondence of the filling of the harmonic axial modes and that this effect depends on the Fermi-Fermi scattering length a(F). In the collisional regime, we calculate the velocity c(s) of first sound showing that c(s) displays jumps at critical densities fixed by the scattering length a(F). Finally, we discuss the experimental achievability of these zero sound and first sound waves with ultracold alkali-metal atoms.