Generalized semiconfined harmonic oscillator model with a position-dependent effective mass

By using a point canonical transformation starting from the constant-mass Schr\"odinger equation for isotonic potential, it is shown that semiconfined harmonic oscillator model with position-dependent mass in BenDaniel-Duke setting and same spectrum as standard can be easily constructed extended to shifted oscillator, which could result presence of uniform gravitational field. A further generalization proposed by considering $m$-dependent $0<m<2$ deriving associated potential. This results family potential pairs, original pair belongs corresponds $m=1$. Finally, would general von Roos kinetic energy operator presented examples Zhu-Kroemer Mustafa-Mazharimousavi settings are briefly discussed.