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

SIGMA 3 (2007), 067, 14 pages      arXiv:0705.2577      https://doi.org/10.3842/SIGMA.2007.067

Quadratic Algebra Approach to an Exactly Solvable Position-Dependent Mass Schrödinger Equation in Two Dimensions

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 March 30, 2007, in final form May 08, 2007; Published online May 17, 2007

Abstract
An exactly solvable position-dependent mass Schrödinger equation in two dimensions, depicting a particle moving in a semi-infinite layer, is re-examined in the light of recent theories describing superintegrable two-dimensional systems with integrals of motion that are quadratic functions of the momenta. To get the energy spectrum a quadratic algebra approach is used together with a realization in terms of deformed parafermionic oscillator operators. In this process, the importance of supplementing algebraic considerations with a proper treatment of boundary conditions for selecting physical wavefunctions is stressed. Some new results for matrix elements are derived. This example emphasizes the interest of a quadratic algebra approach to position-dependent mass Schrödinger equations.

Key words: Schrödinger equation; position-dependent mass; quadratic algebra.

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