Collective modes of trapped gases at the BEC-BCS crossover

The collective mode frequencies in isotropic and deformed traps are calculated for general polytropic equation of states, P proportional n(gamma+1), and expressed in terms of gamma and the trap geometry. For molecular and standard Bose-Einstein condensates and Fermi gases near Feshbach resonances, the effective power gamma approximately 0.5-1.3 is calculated from Jastrow type wave-function ansatz, and from the crossover model of Leggett. The resulting mode frequencies are calculated for these phases around the BEC-BCS crossover.