Lie symmetry classification and qualitative analysis for the fourth-order Schr\"{o}dinger equation

TL;DR

This paper applies Lie symmetry analysis to classify potentials and simplify a fourth-order Schrödinger equation influenced by minimum length considerations, enabling qualitative analysis of its asymptotic behavior.

Contribution

It provides a detailed classification of scalar potentials with non-trivial symmetries and simplifies the equation for qualitative analysis, which is novel in the context of modified quantum mechanics.

Findings

Classification of scalar potentials with Lie symmetries

Simplification of the Schrödinger equation using symmetries

Qualitative analysis of asymptotic dynamics

Abstract

The Lie symmetry analysis for the study of a $1+n~$fourth-order Schr\"{o}dinger equation inspired by the modification of the deformation algebra in the presence of a minimum length is applied. Specifically, we perform a detailed classification for the scalar field potential function where non-trivial Lie symmetries exist and simplify the Schr\"{o}dinger equation. Then, a qualitative analysis allows for the reduced ordinary differential equation to be analyzed to understand the asymptotic dynamics.