Harmonic oscillator in the context of the extended uncertainty principle

At large-scale distances where space-time is curved due to gravity, a nonzero minimal uncertainty in the momentum emerges. The presence of minimal uncertainty in momentum allows a modification to the quantum uncertainty principle, which is known as the Extended Uncertainty Principle (EUP). In this work, we handle the harmonic oscillator problem in the EUP scenario and obtain analytical exact solutions in classical and semiclassical domains. In the classical context, we establish the equations of motion of the oscillator and show that the EUP-corrected frequency is depending on the energy and deformation parameter. In the semiclassical domain, we derive the energy eigenvalue levels and demonstrate that the energy spectrum depends on $n^{2}$, as the feature of hard confinement. Finally, we investigate the impact of the EUP on the harmonic oscillator's thermodynamic properties by using the EUP-corrected partition functions in classical limit in the (A)dS backgrounds.