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Discrete coordinate realizations of the \(q\)-oscillator when \(q > 1\). (English) Zbl 1115.81037

Summary: We elaborate on the Macfarlane–Biedenharn \(q\)-oscillator when \(q > 1\). In this case the position operator \(Q = a^{\dagger}+a\) and the momentum operator \(P = i(a^{\dagger}-a)\) are symmetric, but not self-adjoint. For this reason, one cannot specify spectra of \(Q\) and \(P\). Since these operators have one-parameter families of self-adjoint extensions with different spectra, the common definition of such \(q\)-oscillator is not complete. We derive an action of \(Q\) and \(P\) (as well as of the related Hamiltonian) upon functions given on the corresponding coordinate spaces, on which \(Q\) and \(P\) are self-adjoint operators. To each self-adjoint extension of \(Q\) there corresponds an appropriate coordinate space (a spectrum of this self-adjoint extension). Thus, for every fixed \(q > 1\) one obtains a one-parameter family of non-equivalent \(q\)-oscillators in their coordinate spaces.

MSC:

81R12 Groups and algebras in quantum theory and relations with integrable systems
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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