Group classification of Schr\"odinger equations with position dependent mass

TL;DR

This paper classifies the symmetry groups of 2D Schrödinger equations with position-dependent mass, identifying seven classes with distinct continuous symmetries, including a previously missing class for constant mass cases.

Contribution

It provides a complete classification of maximal kinematical invariance groups for 2D Schrödinger equations with position-dependent mass, including new classes and a test for equivalence to constant mass systems.

Findings

Seven classes of equations with non-equivalent symmetry groups identified

A new class missing in previous classifications for constant mass case

A constructive test for system equivalence to constant mass systems proposed

Abstract

Maximal kinematical invariance groups of $2d$ Schr\"odinger equation with a position dependent mass and arbitrary potential are classified. It is demonstrated that there exist seven classes of such equations possessing non-equivalent continuous symmetry group. Three of these classes include arbitrary functions while the remaining ones are defined up to arbitrary parameters. In particular, for the case of a constant mass the class missing in the Boyer classification (Boyer C P 1974 Helv. Phys. Acta{\bf 47}, 450) is indicated. A constructive test of (non)equivalence of a PDM system to a constant mass system is proposed.