Superintegrable and shape invariant systems with position dependent mass

TL;DR

This paper classifies 3d quantum systems with position-dependent mass that are rotation invariant and admit second order integrals of motion, revealing their shape invariance and solvability, including systems bridging Coulomb and oscillator models.

Contribution

It provides a complete classification of PDM systems with second order integrals of motion and introduces an algorithm for solving their spectra.

Findings

All classified systems are shape invariant and exactly solvable.

Some systems exhibit double shape invariance with multiple superpotentials.

An algorithm successfully solves five specific PDM systems.

Abstract

Second order integrals of motion for 3d quantum mechanical systems with position dependent masses (PDM) are classified. Namely, all PDM systems are specified which, in addition to their rotation invariance, admit at least one second order integral of motion. All such systems appear to be also shape invariant and exactly solvable. Moreover, some of them possess the property of double shape invariance and can be solved using two different superpotentials. Among them there are systems with double shape invariance which present nice bridges between the Coulomb and isotropic oscillator systems. A simple algorithm for calculation the discrete spectrum and the corresponding state vectors for the considered PDM systems is presented and applied to solve five of the found systems.