Spectrum generating algebras for position-dependent mass oscillator Schrodinger equations

TL;DR

This paper develops spectrum generating algebras for position-dependent mass Schrödinger equations, enabling algebraic construction of bound states for multidimensional oscillators with specific mass functions.

Contribution

It introduces quadratic algebras for position-dependent mass oscillators and constructs spectrum generating algebras, including deformed su(1,1) generators, for these systems.

Findings

Existence of positive-discrete series representations for d ≥ 2

Two unitary irreducible representations for d=1

Algebraic generation of all bound-state wavefunctions

Abstract

The interest of quadratic algebras for position-dependent mass Schr\"odinger equations is highlighted by constructing spectrum generating algebras for a class of d-dimensional radial harmonic oscillators with $d \ge 2$ and a specific mass choice depending on some positive parameter $\alpha$. Via some minor changes, the one-dimensional oscillator on the line with the same kind of mass is included in this class. The existence of a single unitary irreducible representation belonging to the positive-discrete series type for $d \ge 2$ and of two of them for d=1 is proved. The transition to the constant-mass limit $\alpha \to 0$ is studied and deformed su(1,1) generators are constructed. These operators are finally used to generate all the bound-state wavefunctions by an algebraic procedure.