Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 046, 17 pages      arXiv:0807.4650      https://doi.org/10.3842/SIGMA.2009.046

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

Christiane Quesne
Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles, Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium

Received February 09, 2009, in final form March 19, 2009; Published online April 15, 2009

Abstract
On using the known equivalence between the presence of a position-dependent mass (PDM) in the Schrödinger 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 Schrödinger 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 Schrödinger 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.

Key words: Schrödinger equation; position-dependent mass; shape invariance; point canonical transformations.

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