SIGMA 1 (2005), 008, 17 pages      math-ph/0508032      https://doi.org/10.3842/SIGMA.2005.008

Spectra of Observables in the q-Oscillator and q-Analogue of the Fourier Transform

Anatoliy U. Klimyk
Bogolyubov Institute for Theoretical Physics, 14-b Metrologichna Str., 03143 Kyiv, Ukraine

Received August 26, 2005, in final form October 19, 2005; Published online October 21, 2005

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.

Key words: Biedenharn-Macfarlane q-oscillator; position operator; momentum operator; spectra; continuous q^-1-Hermite polynomials; Fourier transform.

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