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

Abstract

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 selfadjoint 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.

Completeness of rank one perturbations of normal operators with lacunary spectrum

“…This Jacobi matrix defines a symmetric operator A 0 with indices (1,1). In [22,Theorem 1], a parametrization of all selfadjoint extensions of A 0 is given and their spectra are calculated. Any self-adjoint extension of A 0 is lacunary.…”

On Orthogonality Relations for Dual Discrete q-Ultraspherical Polynomials

“…It is well-known (see [3]) that q −1 -Hermite polynomials are closely related to the BiedenharnMacfarlane q-oscillator. Thus, the polynomials, considered in this paper, is also related to this q-oscillator.…”

“…In this paper we deal with the so-called q −1 -Hermite polynomials (which are closely related to the Biedenharn-Macfarlane oscillator; see [3]) and with the dual discrete q-ultraspherical polynomials. The q −1 -Hermite orthogonal polynomials were discovered by Askey [4] and the discrete q-ultraspherical polynomials and their duals were introduced in [5].…”

“…This means that they have a one-parameter family of orthogonality relations (these orthogonalities are important for studying the Biedenharn-Macfarlane oscillator when q > 1; see [3]). These orthogonality relations were explicitly derived in [6].…”

Application of Fibonacci oscillators in the Debye model

“…The study of quantum groups and quantum algebras has attracted great interest in recent years and stimulated intense research in various fields of physics [1,2,3] taking into account a range of applications covering astrophysics and condensed matter, for instance, black holes and high-temperature superconductors [4].…”