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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| Date: | 2005 |
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| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2005
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/209352 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | 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| Summary: | 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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