Point Canonical Transformation versus Deformed Shape Invariance for Position-Dependent Mass Schr\"odinger Equations

TL;DR

This paper introduces a novel method combining deformed shape invariance and point canonical transformation techniques to explicitly construct wavefunctions for position-dependent mass Schr"odinger equations, enhancing solvability and understanding.

Contribution

It presents a new approach that merges deformed shape invariance with point canonical transformations to derive explicit wavefunctions for PDM Schr"odinger equations.

Findings

Wavefunctions can be constructed without solving differential-difference equations.

The method confirms the equivalence of wavefunctions from both approaches.

Enhanced solvability of PDM Schr"odinger equations is demonstrated.

Abstract

On using the known equivalence between the presence of a position-dependent mass (PDM) in the Schr\"odinger equation and a deformation of the canonical commutation relations, a method based on deformed shape invariance has recently been devised for generating pairs of potential and PDM for which the Schr\"odinger 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 Schr\"odinger 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.