The generalized relativistic harmonic oscillator with the Snyder-de Sitter algebra

The Snyder-de Sitter (SdS) algebra is a model of non-commutative space-time admitting three fundamental parameters: the speed of light, the Planck mass and the cosmological constant, and therefore can be seen as an example of triply special relativity. In this paper, we first examine some aspects of the free Dirac equation, notably the Zitterbewegung (ZB) phenomenon, within the (SdS) algebra. Then, we investigate the effect of this algebra on the dynamics of the (1 + 1)-dimensional Generalized Relativistic Harmonic Oscillator (GRHO). Using the momentum representation, the oscillator equation is solved analytically: the exact normalized wave functions are expressed in terms of Gegenbauer polynomials and the corresponding energy equation is obtained. It is shown that the presence of the minimal length and the minimal momentum can drastically affect the spectroscopy of the system. The Dirac oscillator and other important particular cases are also discussed.