Born–Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator

We apply the Born–Jordan and Weyl quantization formulas for polynomials in canonical coordinates to the constants of motion of some examples of the superintegrable 2D anisotropic harmonic oscillator. Our aim is to study the behaviour of the algebra of the constants of motion after the different quantization procedures. In the examples considered, we have that the Weyl formula always preserves the original superintegrable structure of the system, while the Born–Jordan formula, when producing different operators than the Weyl's one, does not.