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A construction method of infinite families of quasi-exactly solvableposition-dependent mass Schr\"odinger equations with known ground and firstexcited states is proposed in a deformed supersymmetric background. Suchfamilies correspond to extensions of known potentials endowed with a deformedshape invariance property. Two different approaches are combined. The first oneis a generating function method, which enables to construct the first twosuperpotentials of a deformed supersymmetric hierarchy, as well as the firsttwo partner potentials and the first two eigenstates of the first potential,from some generating function $W_+(x)$ [and its accompanying function$W_-(x)$]. The second approach is the conditionally deformed shape invariancemethod, wherein the deformed shape invariance property of the startingpotentials is generalized to their extensions by adding some constraints on theparameters and by imposing compatibility conditions between sets ofconstraints. Detailed results are given for some extensions of the linear andradial harmonic oscillators, as well as the Kepler-Coulomb and Morsepotentials.

Infinite families of position-dependent mass Schr\"odinger equations with known ground and first excited states