DISCRETIZATION OF THE PHASE-SPACE FOR A Q-DEFORMED HARMONIC-OSCILLATOR WITH Q A ROOT OF UNITY

The ''position'' and ''momentum'' operators for the q-deformed oscilla tor with q being a root of unity are proved to have discrete eigenvalu es which are roots of deformed Hermite polynomials. The Fourier transf orm connecting the ''position'' and ''momentum'' representations is al so found. The phase space of this oscillator has a lattice structure, which is a non-uniformly distributed grid. Non-equidistant lattice str uctures also occur in the cases of the truncated harmonic oscillator a nd of the q-deformed parafermionic oscillator, while the parafermionic oscillator corresponds to a uniformly distributed grid.