Higher order symmetries for linear and nonlinear Schroedinger equations

TL;DR

This paper investigates high-order symmetry operators for linear Schrödinger equations across multiple spatial dimensions and provides a comprehensive classification of symmetries for nonlinear Schrödinger equations.

Contribution

It derives determining equations for symmetry operators and offers a complete group classification for nonlinear Schrödinger equations, advancing symmetry analysis in quantum mechanics.

Findings

Explicit solutions for symmetry determining equations in specific cases

Complete group classification of nonlinear Schrödinger equations

Identification of arbitrary order symmetry operators

Abstract

We study arbitrary order symmetry operators for the linear Schr\"odinger equations with arbitrary number of spatial variables. We deduce determining equations for coefficient functions of such operators and consider in detail some cases when these equations can be explicitly solved. In addition, the complete group classification of the nonlinear Schr\"odinger equation is presented.