Quantization of the 1-D harmonic oscillator with variable mass using the   operators $\hat v$ and $\hat p$
  For the 1-D harmonic oscillator with position depending variable mass, a Hamiltonian and constant of motion are given through a consistent approach. Then, the quantization of this system is carried out using the operator $\hat p$, for the Hamiltonian, and the operator $\hat v$ for the constant of motion. We find that the result of both quantizations brings about different quantum dynamics. 
Gustavo V. L\'opez and Eric M. Reynaga
quant-ph