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

Details 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

2005-08-15

arxiv.org/abs/math-ph/0508032v2

SIGMA 1 (2005), 008, 17 pages

Physics

Mathematical Physics

Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

Scientific paper

10.3842/SIGMA.2005.008

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.

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