On suq(1,1)-models of quantum oscillator

Abstract

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.