Hydrodynamic action and Bose spectrum of superfluid Fermi systems

A “hydrodynamic action” functional for a superfluid Fermi gas is constructed in the path integral formalism. The integral over the Fermi fields is transformed into an integral over the Bose field that describes the Cooper pairs (and also over the field of the electric potential in the model with Coulomb interaction). An investigation is made of the Bose spectrum determined by the hydrodynamic action. For $T\sim T_c$ two branches of the spectrum are obtained. For $T\ll T_c$, the dispersion is calculated and stability with respect to decays of Bogolyubov sound is proved. The branch $E\approx 2\Delta$ is investigated; this remains when a Coulomb interaction is added.