# Group classification of (1+3)-dimensional Schr\"odinger equations with position dependent mass

## TL;DR

This paper classifies the symmetry groups of 3D Schr"odinger equations with position-dependent mass, revealing 94 classes, and discusses their invariance properties, extended symmetries, and an exact solution for a deformed harmonic oscillator.

## Contribution

It provides a comprehensive classification of kinematical invariance groups for PDM Schr"odinger equations, including new symmetry structures and an explicit exact solution.

## Key findings

- 94 classes of equations up to the generic equivalence group
- Extended invariance algebras of up to eight dimensions
- Exact solution describing a deformed 3D isotropic harmonic oscillator

## Abstract

Kinematical invariance groups of the 3d Schr\"odinger equations with position dependent masses (PDM) and arbitrary potentials are classified. It is shown that there exist 94 classes of such equations defined up to the generic equivalence group, and 70 classes defined up to the equivalence groupoid. The maximally extended kinematical invariance algebras of such equations appears to be eight dimensional. The specific symmetries connected with the presence of the ambiguity parameters are discussed and an extended class of systems which keep their forms for arbitrary or particular changes of these parameters is specified. The exact solution of the selected PDM Schr\"odinger equation is presented. This equation describes a deformed 3d isotropic harmonic oscillator and possesses extended continuous symmetries and hidden supersymmetries with two different superpotentials as well.