Ripplon dispersion and finite-range effects in the quantum-liquid surface

We derive within the linear-response theory for inhomogeneous Bose liquids the dispersion relation of a surface excitation (ripplon). It is shown that, in an infinite half space, the ripplon dispersion relation is r(q?)q?3/2, where q? is the momentum parallel to the surface. This dispersion relation holds only if the system is not in an external field. For the special case of a Skyrme interaction with a gradient coupling term, we relate the coefficient of proportionality to the surface energy and thus reproduce the hydrodynamic prediction from linear-response theory. The ripplon-dispersion relation and the shape of the collective excitations is evaluated and discussed for a nonlocal, phenomenological Skyrme interaction and a microscopic interaction based on variational wave functions.