In the low-temperature regime, the excited states of the system can
be calculated from the ``classical'' frequencies of the linearized
GP equation. Let us look for solutions in the form of small
oscillations of the order parameter around the stationary value.

(1.15)

By keeping terms linear in the complex functions and ,
equation (1.11) becomes

(1.16)

where
.
In a uniform gas, the amplitudes and are plane waves and
the resulting dispersion law takes the Bogoliubov form

(1.17)

where is the wave vector of the excitations and is the density of the gas. For large momenta the spectrum
coincides with the free-particle energy
. At low
momenta equation (1.17) yields the phonon
dispersion , where the sound velocity is given by
the formula