Supersymmetric Quantum Mechanics Approach to Unified Treatment of the Non-Relativistic Harmonic Oscillator and the Klein-Gordon Oscillator Subject to a Uniform Magnetic Field in the Framework of Snyder-de Sitter Algebra

In this work, using the method of supersymmetric quantum mechanics (SUSY QM) and the concept of shape invariance, we determine the energy levels and wavefunctions for a two-dimensional Klein-Gordon oscillator (KGO) and a non-relativistic isotropic harmonic oscillator (NRHO) of a charged particle subject to a constant magnetic field within the framework of the Snyder-de Sitter (SdS) model. Both cases are solved simultaneously by introducing a Hamiltonian that depends on a real parameter, enabling an easy transition between the two cases. By transforming the corresponding equations into a position-dependent Schrödinger-like equation, the problem is solved exactly in position space. The wavefunctions are explicitly derived using recursion relations involving Jacobi polynomials.