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 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.