On $suq(1,1)$-models of quantum oscillator Available to Purchase

Models of the quantum oscillator, based on the discrete series representations of the quantum algebra $suq(1,1)$⁠, are constructed. The position and momentum operators in these models are twisted generators $J2$ and $J1$ for such $suq(1,1)$-representations, respectively. As in the case of the standard harmonic oscillator in quantum mechanics, the position and momentum operators here have continuous simple spectra. These spectra cover a finite interval on the real line, which depends on a value of $q$⁠. Eigenfunctions of these operators are explicitly found. It is shown that the Macfarlane–Biedenharn $q$-oscillator is a limit case of the oscillators under discussion. The $q=1$ limit case, in which spectra of the position and momentum operators cover the whole real line, is also considered in detail.