Spectra of Observables in the $q$-Oscillator and $q$-Analogue of the Fourier Transform

Spectra of the position and momentum operators of the Biedenharn–Macfarlane $q$-oscillator (with the main relation $aa^+-qa^+a=1$) are studied when $q>1$. These operators are symmetric but not self-adjoint. They have a one-parameter family of self-adjoint extensions. These extensions are derived explicitly. Their spectra and eigenfunctions are given. Spectra of different extensions do not intersect. The results show that the creation and annihilation operators $a^+$ and $a$ of the $q$-oscillator for $q>1$ cannot determine a physical system without further more precise definition. In order to determine a physical system we have to choose appropriate self-adjoint extensions of the position and momentum operators.