Collective excitations of a one-dimensional Fermi gas under harmonic confinement

We study the spectrum of collective excitations of a spin-polarized Fermi gas confined in a one-dimensional harmonic trap at zero temperature. In the collisionless regime we evaluate exactly the dynamic structure factor, while in the collisional regime we solve analytically the linearized equations of hydrodynamics in the Thomas-Fermi approximation. We also verify the validity of the Thomas-Fermi theory by solving numerically a time-dependent nonlinear Schroedinger equation with a fifth-order interaction term. We find that in both the collisionless and the collisional regime the excitation frequencies of the Fermi gas are multiples of the trap frequency, analogously to the case of the one-dimensional homogeneous Fermi fluid where the velocities of zero and first sound coincide. Due to boson-fermion dynamical mapping our results for the spectrum apply as well to a one-dimensional Bose gas with hard-core point-like interactions ("Tonks gas").