Point Canonical Transformation versus Deformed Shape Invariance for Position-Dependent Mass Schrödinger Equations

Abstract

On using the known equivalence between the presence of a position-dependent mass (PDM) in the Schrodinger equation and a deformation of the canonical commutation relations, a method based on deformed shape invariance has recently been devised for genera- ting pairs of potential and PDM for which the Schrodinger equation is exactly solvable. This approach has provided the bound-state energy spectrum, as well as the ground-state and the first few excited-state wavefunctions. The general wavefunctions have however remained unknown in explicit form because for their determination one would need the solutions of a rather tricky differential-difference equation. Here we show that solving this equation may be avoided by combining the deformed shape invariance technique with the point canonical transformation method in a novel way. It consists in employing our previous knowledge of the PDM problem energy spectrum to construct a constant-mass Schrodinger equation with similar characteristics and in deducing the PDM wavefunctions from the known constant- mass ones. Finally, the equivalence of the wavefunctions coming from both approaches is checked.