Algebraic Method for Group Classification of (1+1)-Dimensional Linear Schrödinger Equations

• Published in: Acta applicandae mathematicae 2018-10, Vol.157 (1), p.171-203
• Main Authors: Kurujyibwami, Célestin, Basarab-Horwath, Peter, Popovych, Roman O.
• Format: Article
• Language: English
• Subjects: Algebra; Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Classification; Computational Mathematics and Numerical Analysis; Differential equations; Equivalence; Group theory; Integers; Mathematical analysis; Mathematics; Mathematics and Statistics; Partial Differential Equations; Probability Theory and Stochastic Processes; Schrodinger equation; Superposition (mathematics); Transformations (mathematics)
• ISSN: 0167-8019; 1572-9036; 1572-9036
• DOI: 10.1007/s10440-018-0169-y

We carry out the complete group classification of the class of (1+1)-dimensional linear Schrödinger 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.