2005/08/31 by Anatoliy Klimyk
Mathematics · Physics and Astronomy · #Mathematical functions and polynomials #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics #math-ph #math.MP #math.SP #msc:41B15 #msc:81Q10 #msc:81S05
paper · pdf · doi:10.3842/sigma.2005.008
published as SIGMA 1 (2005), 008, 17 pages · Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/
openalex publication_date 2005/10/21 · arxiv created 2005/11/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 selfadjoint 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.