  Galilean invariance leaves its imprint on the energy spectrum and eigenstates
of $N$ 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 $L$ and cross-section area $s_\perp$ leads to
spectral gaps $n^2h^2s_\perp\rho/(2mL)$ at wavenumbers $2n\pi s_\perp\rho$,
where $\rho$ is the number density and $m$ is the particle mass. In zero
temperature superfluids, in toroidal geometries, it causes the quantization of
the flow velocity with the quantum $h/(mL)$ or that of the circulation along
the toroid with the known quantum $h/m$. 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 $N$ and $L$, if
$N/L=s_\perp\rho$ is kept fixed, translation invariance is broken, the center
of mass has a periodic distribution, while superfluidity persists at low flow
velocities. This conclusion holds for the Lieb-Liniger model.
