Quadratic algebras and position-dependent mass Schr\"odinger equations

TL;DR

This paper explores the use of quadratic algebras to solve position-dependent mass Schrödinger equations, offering an alternative to traditional methods like Lie algebraic techniques and supersymmetry, and constructs spectrum generating algebras for certain oscillators.

Contribution

It introduces quadratic algebra methods for position-dependent mass Schrödinger equations and constructs spectrum generating algebras for specific radial oscillators, expanding the algebraic toolkit.

Findings

Constructed spectrum generating algebras for d-dimensional radial oscillators.

Demonstrated quadratic algebra approach as an alternative to Lie algebra methods.

Extended the su(1,1) algebraic approach to position-dependent mass oscillators.

Abstract

During recent years, exact solutions of position-dependent mass Schr\"odinger equations have inspired intense research activities, based on the use of point canonical transformations, Lie algebraic methods or supersymmetric quantum mechanical techniques. Here we highlight the interest of another approach to such problems, relying on quadratic algebras. We illustrate this point by constructing spectrum generating algebras for a class of $d$-dimensional radial harmonic oscillators with $d\ge2$ (including the one-dimensional oscillator on the line via some minor changes) and a specific mass choice. This provides us with a counterpart of the well-known su(1,1) Lie algebraic approach to the constant-mass oscillators.