%0 Journal Article
%T Spectra of observables in the q-oscillator and q-analogue of the Fourier transform
%A Anatoliy Klimyk
%J Mathematics
%D 2005
%I arXiv
%R 10.3842/SIGMA.2005.008
%X 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.
%U http://arxiv.org/abs/math-ph/0508032v2