Exact surface-wave spectrum of a dilute quantum liquid

We consider a dilute gas of bosons with repulsive contact interactions, described on the mean-field level by the Gross-Pitaevskii equation, and bounded by an impenetrable "hard" wall (either rigid or flexible). We solve the Bogoliubov-de Gennes equations for excitations on top of the Bose-Einstein condensate analytically, by using matrix-valued hypergeometric functions. This leads to the exact spectrum of gapless Bogoliubov excitations localized near the boundary. The dispersion relation for the surface excitations represents for small wavenumbers $k$ a ripplon mode with fractional power law dispersion for a flexible wall, and a phonon mode (linear dispersion) for a rigid wall. For both types of excitation we provide, for the first time, the exact dispersion relations of the dilute quantum liquid for all $k$ along the surface, extending to $k \rightarrow \infty$. The small wavelength excitations are shown to be bound to the surface with a maximal binding energy $\Delta= \frac18 (\sqrt{17}-3)^2 mc^2 \simeq 0.158\, mc^2$, which both types of excitation asymptotically approach, where $m$ is mass of bosons and $c$ bulk speed of sound. We demonstrate that this binding energy is close to the experimental value obtained for surface excitations of helium II confined in nanopores, reported in Phys. Rev. B 88, 014521 (2013).