Algebraic method for group classification of (1+1)-dimensional linear Schr\"odinger equations

TL;DR

This paper performs a complete algebraic classification of (1+1)-dimensional linear Schrödinger equations with complex potentials, using a new method based on uniform semi-normalization to simplify the process.

Contribution

It introduces the concept of uniform semi-normalization for differential equations and applies it to classify Schrödinger equations, simplifying the algebraic classification process.

Findings

The equivalence groupoid of the class is uniformly semi-normalized.

Admissible transformations are compositions of superposition and equivalence transformations.

The classification reduces to analyzing low-dimensional subalgebras of the equivalence algebra.

Abstract

We carry out the complete group classification of the class of (1+1)-dimensional linear Schr\"odinger equations with complex-valued potentials. After introducing the notion of uniformly semi-normalized classes of differential equations, we compute the equivalence groupoid of the class under study and show that it is uniformly semi-normalized. More specifically, each admissible transformation in the class is the composition of a linear superposition transformation of the corresponding initial equation and an equivalence transformation of this class. This allows us to apply the new version of the algebraic method based on uniform semi-normalization and reduce the group classification of the class under study to the classification of low-dimensional appropriate subalgebras of the associated equivalence algebra. The partition into classification cases involves two integers that characterize Lie symmetry extensions and are invariant with respect to equivalence transformations.