Super Algebra and Harmonic Oscillator in Anti de Sitter Space

The harmonic oscillator in anti de Sitter space(AdS) is discussed. We con- sider the harmonic oscillator potential and then time independent Schrodinger equation in AdS space. Then, we apply the supersymmetric Quantum Mechanics approach to solve our dierential equation. In this paper, we have solved Schrodinger equation for harmonic oscillator in AdS spacetime by supersymmetry approach. The shape invari- ance, charge conjugation and other properties of supersymmetric Quantum Mechanics for our equation are discussed. We investigate the dynamical symmetry into denite group. We also obtain the Casimir operator and thermodynamics properties of har- monic oscillator in Anti de sitter space. We have discussed the corresponding algebra for the N = 2 Supersymmetry. The energy spectrum of the harmonic oscillator is similar to the at space but the constant terms of it are dierent, This constant terms depend on the geometrical parameter of the background. Finally, we have obtained the various thermodynamics quantity of the system.