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By using a point canonical transformation starting from the constant-massSchr\"odinger equation for the isotonic potential, it is shown that asemiconfined harmonic oscillator model with a position-dependent mass in theBenDaniel-Duke setting and the same spectrum as the standard harmonicoscillator can be easily constructed and extended to a semiconfined shiftedharmonic oscillator, which could result from the presence of a uniformgravitational field. A further generalization is proposed by considering a$m$-dependent position-dependent mass for $0<m<2$ and deriving the associatedsemiconfined potential. This results in a family of position-dependent mass andpotential 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 energyoperator is presented and the examples of the Zhu-Kroemer andMustafa-Mazharimousavi settings are briefly discussed.

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