We report on an exact, first principles result which shows that when the superfluid of He-4 is spatially confined, gaps are created in the phonon excitation energy and that the phonon density of states (PDOS) presents sharp peaks positioned at the gap edges. This finding complements our knowledge of the physics occurring in a spatially confined quantum system, for the superfluid of He-4 in particular, this has been shown to result in the critical velocity of a flow motion. Here, we first derive an exact expression of the PDOS of a confined superfluid, and then discuss related aspects regarding the probing of the detailed structure of the peaked PDOS using inelastic neutron scattering.