Algebraic method of group classification for semi-normalized classes of differential equations

TL;DR

This paper extends algebraic group classification methods to semi-normalized classes of differential equations, demonstrating their effectiveness through applications to Schrödinger equations with complex potentials.

Contribution

It introduces a generalized algebraic method for classifying semi-normalized differential equations and applies it to Schrödinger equations, including new results on their symmetry structures.

Findings

The class of linear Schrödinger equations with complex potentials is uniformly semi-normalized.

Group classification reduces to classifying low-dimensional subalgebras of the equivalence algebra.

Results are relevant for Schrödinger equations with real-valued potentials.

Abstract

We generalize the notion of semi-normalized classes of systems of differential equations, study properties of such classes and extend the algebraic method of group classification to them. In particular, we prove the important theorems on factoring out symmetry groups and invariance algebras of systems from semi-normalized classes and on splitting such groups and algebras within disjointedly semi-normalized classes. Nontrivial particular examples of classes that arise in real-world applications and showcase the relevance of the developed theory are provided. To convincingly illustrate the efficiency of the proposed method, we apply it to the group classification problem for the class of linear Schr\"odinger equations with complex-valued potentials and the general value of the space dimension. We compute the equivalence groupoid of the class by the direct method and thus show that this class is uniformly semi-normalized with respect to the linear superposition of solutions. This is why the group classification problem reduces to the classification of specific low-dimensional subalgebras of the associated equivalence algebra, which is completely realized for the case of space dimension two. Splitting into different classification cases is based on three integer parameters that are invariant with respect to equivalence transformations. We also single out those of the obtained results that are relevant to linear Schr\"odinger equations with real-valued potentials.