2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/120008An 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. Finally, the two-dimensional model is extended to two integrable and exactly solvable (but not superintegrable) models in three dimensions, depicting a particle in a semi-infinite parallelepipedal or cylindrical channel, respectively.23 pages, no figureQuantum PhysicsMathematical Physicstext