On the specific heat of a fermionic atomic cloud in the unitary regime

In the unitary regime, when the scattering amplitude greatly exceeds in magnitude the average inter-particle separation, and below the critical temperature thermal properties of an atomic fermionic cloud are governed by the collective modes, specifically the Bogoliubov-Anderson sound modes. The specific heat of an atomic cloud in a elongated trap in particular has a rather compex temperature dependence, which changes from an exponential behavior at very low temperatures ($T\ll\hbar\omega_{||}$), to $\propto T$ for $\hbar\omega_{||}\ll T \ll \hbar\omega_\perp$ and then continuosly to $\propto T^4$ at temperatures just below the critical temperature, when the surface modes play a dominant role. Only the low ($\hbar\omega_{||} \ll T \ll \hbar\omega_\perp$) and high ($\hbar\omega_\perp \ll T < T_c$) temperature power laws are well defined. For the intermediate temperatures one can introduce at most a gradually increasing with temperature exponent.