  The zero-temperature properties of a dilute two-component Fermi gas in the
BCS-BEC crossover are investigated. On the basis of a generalization of the
Hylleraas-Undheim method, we construct rigorous upper bounds to the collective
frequencies for the radial and the axial breathing mode of the Fermi gas under
harmonic confinement in the framework of the hydrodynamic theory. The bounds
are compared to experimental data for trapped vapors of Li6 atoms.
