Exactly solvable schrödinger equations with a position-dependent mass: Null potential

In this work, an alternative approach for finding exact solutions of Schrödinger equations with a position-dependent mass is presented. Essentially, the method consists in transforming the position-dependent mass Schrödinger equation into a standard constant mass Schrödinger equation. The constant mass equation is formulated in terms of a new variable and includes, as effective potential, a term related to the position-dependent mass and another term connected with the original potential. For exactly solvable potentials, our proposal leads to a Riccati equation for an equivalent Witten superpotential, which means that this kind of problems can be studied within the quantum supersymmetric framework with the aim to find the explicit solutions and the energy spectra. To show the usefulness of the proposed approach, we consider the particular case of exactly solvable Schrödinger equations for different position-dependent mass situations in a null potential. However, the method can directly be applied to other effective potentials as well as to find isospectral potentials for specific position-dependent mass distributions.