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.
\Bibitem{Kli05}
\by Anatoliy U. Klimyk
\paper Spectra of Observables in the $q$-Oscillator and $q$-Analogue of the Fourier Transform
\jour SIGMA
\yr 2005
\vol 1
\papernumber 008
\totalpages 17
\mathnet{http://mi.mathnet.ru/sigma8}
\crossref{https://doi.org/10.3842/SIGMA.2005.008}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2173595}
\zmath{https://zbmath.org/?q=an:1098.81035}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000207064600008}
Linking options:
https://www.mathnet.ru/eng/sigma8
https://www.mathnet.ru/eng/sigma/v1/p8
This publication is cited in the following 6 articles:
Baranov A.D., Yakubovich D.V., “Completeness of Rank One Perturbations of Normal Operators With Lacunary Spectrum”, J. Spectr. Theory, 8:1 (2018), 1–32
Setare M.R. Majari P., “(2+1)-Dimensional F-Deformed Dirac Oscillator as F-Deformed Ajc Model”, Eur. Phys. J. Plus, 132:11 (2017), 458
Marinho A.A., Brito F.A., Chesman C., “Thermal and Electrical Properties of a Solid Through Fibonacci Oscillators”, Physica A, 443 (2016), 324–332
Marinho A.A.A., Brito F.A., Chesman C., “Application of Fibonacci Oscillators in the Debye Model”, 27Th International Conference on Low Temperature Physics (Lt27), Pts 1-5, Journal of Physics Conference Series, 568, ed. Calzetta E., IOP Publishing Ltd, 2014, 012009
Marinho A.A., Brito F.A., Chesman C., “Thermal Properties of a Solid Through Q-Deformed Algebra”, Physica A, 391:12 (2012), 3424–3434
Valentyna A. Groza, Ivan I. Kachuryk, “On Orthogonality Relations for Dual Discrete $q$-Ultraspherical Polynomials”, SIGMA, 2 (2006), 034, 8 pp.
Citation in format AMSBIB
Contact us: