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We explore the quantization of classical models with position-dependent mass(PDM) terms constrained to a bounded interval in the canonical position. Thisis achieved through the Weyl-Heisenberg covariant integral quantization byproperly choosing a regularizing function $\Pi(q,p)$ on the phase space thatsmooths the discontinuities present in the classical model. We thus obtainwell-defined operators without requiring the construction of self-adjointextensions. Simultaneously, the quantization mechanism leads naturally to asemi-classical system, that is, a classical-like model with a well-definedHamiltonian structure in which the effects of the Planck's constant are notnegligible. Interestingly, for a non-separable function $\Pi(q,p)$, a purelyquantum minimal-coupling term arises in the form of a vector potential for boththe quantum and semi-classical models.

Quantum and semi-classical aspects of confined systems with variable mass