ar X iv : m at h - ph / 0 50 80 32 v 1 1 5 A ug 2 00 5 Spectra of observables in the q - oscillator and q - analogue of the Fourier transform

Spectra of observables in the q-oscillator and q-analogue of the Fourier transform Abstract. 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 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 at 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.