Bogoliubov quasiparticle dispersion ($E_k=\sqrt{{ϵ}_k({ϵ}_k+2n{V}_k)}$)

The stable quadratic spectrum is $E_k=\sqrt{{ϵ}_k({ϵ}_k+2n{V}_k)}$. If $V_k\to V_0>0$, its low-momentum form is $\hslash |k|\sqrt{n{V}_0/m}$, a linear sound branch. For contact repulsion the healing length separates the phonon and particle regimes. General momentum-dependent interactions can change the higher-energy shape.

A normal mode of linearized condensate dynamics with particle and hole components. Quantizing a suitably normalized mode gives a Bogoliubov quasiparticle. The density and phase components of one mode share a frequency; that fact alone does not establish a twofold excitation degeneracy.