Spectra of Observables in the q-Oscillator and q-Analogue of the Fourier Transform
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 ex...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2005 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2005
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/209352 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Spectra of Observables in the q-Oscillator and q-Analogue of the Fourier Transform / A.U. Klimyk // Symmetry, Integrability and Geometry: Methods and Applications. — 2005. — Т. 1. — Бібліогр.: 22 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862647221642592256 |
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| author | Klimyk, A.U. |
| author_facet | Klimyk, A.U. |
| citation_txt | Spectra of Observables in the q-Oscillator and q-Analogue of the Fourier Transform / A.U. Klimyk // Symmetry, Integrability and Geometry: Methods and Applications. — 2005. — Т. 1. — Бібліогр.: 22 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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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| first_indexed | 2025-12-01T12:18:04Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-209352 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-01T12:18:04Z |
| publishDate | 2005 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Klimyk, A.U. 2025-11-19T12:29:02Z 2005 Spectra of Observables in the q-Oscillator and q-Analogue of the Fourier Transform / A.U. Klimyk // Symmetry, Integrability and Geometry: Methods and Applications. — 2005. — Т. 1. — Бібліогр.: 22 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 47B15; 81Q10; 81S05 https://nasplib.isofts.kiev.ua/handle/123456789/209352 https://doi.org/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. This research was partially supported by Grant 10.01/015 of the State Foundation of Fundamental Research of Ukraine. Discussions with N. Atakishiyev, I. Burban, and O. Gavrylyk are gratefully acknowledged. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Spectra of Observables in the q-Oscillator and q-Analogue of the Fourier Transform Article published earlier |
| spellingShingle | Spectra of Observables in the q-Oscillator and q-Analogue of the Fourier Transform Klimyk, A.U. |
| title | Spectra of Observables in the q-Oscillator and q-Analogue of the Fourier Transform |
| title_full | Spectra of Observables in the q-Oscillator and q-Analogue of the Fourier Transform |
| title_fullStr | Spectra of Observables in the q-Oscillator and q-Analogue of the Fourier Transform |
| title_full_unstemmed | Spectra of Observables in the q-Oscillator and q-Analogue of the Fourier Transform |
| title_short | Spectra of Observables in the q-Oscillator and q-Analogue of the Fourier Transform |
| title_sort | spectra of observables in the q-oscillator and q-analogue of the fourier transform |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209352 |
| work_keys_str_mv | AT klimykau spectraofobservablesintheqoscillatorandqanalogueofthefouriertransform |