Deformed su(1,1) Algebra as a Model for Quantum Oscillators

Abstract

The Lie algebra $\mathfrak{su}(1,1)$ can be deformed by a reflection operator, in such a way that the positive discrete series representations of $\mathfrak{su}(1,1)$ can be extended to representations of this deformed algebra $\mathfrak{su}(1,1{)}_{\gamma }$. Just as the positive discrete series representations of $\mathfrak{su}(1,1)$ can be used to model a quantum oscillator with Meixner-Pollaczek polynomials as wave functions, the corresponding representations of $\mathfrak{su}(1,1{)}_{\gamma }$ can be utilized to construct models of a quantum oscillator. In this case, the wave functions are expressed in terms of continuous dual Hahn polynomials. We study some properties of these wave functions, and illustrate some features in plots. We also discuss some interesting limits and special cases of the obtained oscillator models.

References in corpus (2)

• Supersymmetric Quantum Mechanics with Reflections
• From $s{l}_q(2)$ to a Parabosonic Hopf Algebra

Cited by in corpus (5)

• The algebra of dual -1 Hahn polynomials and the Clebsch-Gordan problem of sl_{-1}(2)
• The oscillator model for the Lie superalgebra sh(2|2) and Charlier polynomials
• Non-Hermitian Oscillator and R-deformed Heisenberg Algebra
• Where do bosons actually belong?
• Mass-deformed ABJ and ABJM theory, Meixner-Pollaczek polynomials, and $su(1,1)$ oscillators