  Energy spectra of quasi-one-dimensional quantum rings with a few electrons
are studied using several different theoretical methods. Discrete Hubbard
models and continuum models are shown to give similar results governed by the
special features of the one-dimensionality. The energy spectrum of the
many-body system can be described with a rotation-vibration spectrum of a
'Wigner molecule' of 'localized' electrons, combined with the spin-state
determined from an effective antiferromagnetic Heisenberg Hamiltonian. The
persistent current as a function of magnetic flux through the ring shows
periodic oscillations arising from the 'rigid rotation' of the electron ring.
For polarized electrons the periodicity of the oscillations is always the flux
quantum Phi_0. For nonpolarized electrons the periodicity depends on the
strength of the effective Heisenberg coupling and changes from \Phi_0 first to
Phi_0/2 and eventually to Phi_0/N when the ring gets narrower.
