Symmetry of the Schr\"odinger equation with variable potential

TL;DR

This paper investigates the symmetry properties of the Schrödinger equation with variable potential, including systems with conditions on the potential and equations with convection terms, using contact transformations.

Contribution

It introduces a comprehensive analysis of symmetry transformations for the Schrödinger equation with potential as a dependent variable, extending to systems and convection terms.

Findings

Identifies symmetry transformations that preserve the form of the Schrödinger equation with variable potential.

Derives contact transformations for the Schrödinger equation with potential.

Analyzes symmetry properties of Schrödinger systems with specific potential conditions.

Abstract

We study symmetry properties of the Schr\"odinger equation with the potential as a new dependent variable, i.e., the transformations which do not change the form of the class of equations. We also consider systems of the Schr\"odinger equations with certain conditions on the potential. In addition we investigate symmetry properties of the equation with convection term. The contact transformations of the Schr\"odinger equation with potential are obtained.