Superintegrable systems with position dependent mass

TL;DR

This paper classifies Schrödinger equations with position-dependent mass based on their symmetries, identifying integrable and superintegrable systems, including some with exact solutions and rich algebraic structures.

Contribution

It provides a comprehensive classification of first order integrals of motion for these equations, revealing new superintegrable systems with specific symmetry properties.

Findings

Seventeen classes of equations with non-equivalent symmetries identified

Three systems solved exactly

Discovery of systems with Lorentz invariance and so(4) algebra

Abstract

First order integrals of motion for Schr\"odinger equations with position dependent masses are classified. Seventeen classes of such equations with non-equivalent symmetries are specified. They include integrable, superintegrable and maximally superintegrable systems. Among them is a system invariant with respect to the Lie algebra of Lorentz group and a system whose integrals of motion form algebra so(4). Three of the obtained systems are solved exactly.