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

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