Spectra of observables and an analogue of the Fourier transform for Biedenharn-Macfarlane
q-oscillator

Abstract:
The position and momentum operators of the Biedenharn-Macfarlane q-oscillator are symmetric but not self-adjoint if q>1. They have one-parameter families of self-adjoint extensions. These extensions are given explicitly. Their spectra and eigenfunctions are derived. Spectrum of each extension is discrete.  Spectra of different extensions do not intersect. An analogue of the Fourier transform is obtained for each extension of the position operator and each extension of the
momentum operator.

Thus, the creation and annihilation operators of the Biedenharn-Macfarlane q-oscillator at q>1 cannot determine a physical system without further more precise definition. Namely, in order to determine a physical system we have to choose appropriate self-adjoint extensions of the position and momentum operators. This means that the Biedenharn-Macfarlane q-oscillator at q>1 in fact determines two-parameter family of q-oscillators. These q-oscillators have different spectra of
the position and momentum operators.