Galilean invariance leaves its imprint on the energy spectrum and eigenstates of quantum particles, bosons, or fermions, confined in a bounded domain. It endows the spectrum with a recurrent structure, which in capillaries or elongated traps of length and cross-section area leads to spectral gaps at wave numbers , where is the number density and is the particle mass. In zero temperature superfluids, in toroidal geometries, it causes the quantization of the flow velocity with the quantum or that of the circulation along the toroid with the known quantum . Adding a “friction” potential, which breaks Galilean invariance, the Hamiltonian can have a superfluid ground state at low flow velocities but not above a critical velocity, which may be different from the velocity of sound. In the limit of infinite and , if is kept fixed, translation invariance is broken, and the center of mass has a periodic distribution, while superfluidity persists at low flow velocities. This conclusion holds for the Lieb-Liniger model.