Quadratic Algebra Approach to an Exactly Solvable Position-Dependent Mass Schr\"odinger Equation in Two Dimensions

TL;DR

This paper applies quadratic algebra methods to solve a two-dimensional position-dependent mass Schrödinger equation, revealing new insights into its spectrum, wavefunctions, and boundary conditions, and demonstrating the approach's effectiveness.

Contribution

It introduces a quadratic algebra framework combined with deformed parafermionic oscillators for exactly solving a 2D position-dependent mass Schrödinger equation.

Findings

Derived new matrix element results

Highlighted importance of boundary conditions in algebraic solutions

Demonstrated the quadratic algebra approach's utility in PDM systems

Abstract

An exactly solvable position-dependent mass Schr\"odinger 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 Schr\"odinger equations.