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Galilean invariance leaves its imprint on the energy spectrum and eigenstatesof $N$ quantum particles, bosons or fermions, confined in a bounded domain. Itendows the spectrum with a recurrent structure which in capillaries orelongated traps of length $L$ and cross-section area $s_\perp$ leads tospectral 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 zerotemperature superfluids, in toroidal geometries, it causes the quantization ofthe flow velocity with the quantum $h/(mL)$ or that of the circulation alongthe toroid with the known quantum $h/m$. Adding a "friction" potential whichbreaks Galilean invariance, the Hamiltonian can have a superfluid ground stateat low flow velocities but not above a critical velocity which may be differentfrom 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 centerof mass has a periodic distribution, while superfluidity persists at low flowvelocities. This conclusion holds for the Lieb-Liniger model.

Galilean invariance in confined quantum systems: Implications on spectral gaps, superfluid flow, and periodic order