The dynamics of strongly interacting trapped dilute Fermi gases (dilute in the sense that the range of interatomic potential is small compared with inter-particle spacing ) is investigated in a single-equation approach to the time-dependent density-functional theory. Our results are in good agreement with recent experimental data in the BCS-BEC crossover regime.It is also shown that the calculated corrections to the hydrodynamic approximation may be important even for systems with a rather large number of atoms.

A time-dependent Kohn-Sham (KS) like theory is presented for N bosons in three and lower-dimensional traps. We derive coupled equations, which allow one to calculate the energies of elementary excitations. A rigorous proof is given to show that the KS like equation correctly describes properties of the one-dimensional condensate of impenetrable bosons in a general time-dependent harmonic trap in the large N limit.

Dynamics of strongly interacting trapped dilute Fermi gases is investigated at zero temperature. As an example of application we consider the expansion of the cloud of fermions initially confined in an anisotropic harmonic trap, and study the equation of state dependence of the radii of the trapped cloud and the collective oscillations in the vicinity of a Feshbach resonance.

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 6 Li atoms.

We investigate the stability of the Bose-Einstein condensate (BEC) for the case of atoms with negative scattering lengths at zero temperature using the Ginzburg-Pitaevskii-Gross (GPG) stationary theory. We have found a new exact equation for determining the upper bound of the critical numbers N cr of atoms for a metastable state to exist. Our calculated value of N cr for Bose-Einstein condensation of lithium atoms based on our new equation is in agreement with those observed in a recent experiment. PACS number(s): 03.75. Fi, 03.65.Db, 05.30.Jp, 67.90.+z