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Acta Applicandae Mathematicae

, Volume 157, Issue 1, pp 171–203 | Cite as

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

  • Célestin Kurujyibwami
  • Peter Basarab-Horwath
  • Roman O. Popovych
Article
  • 43 Downloads

Abstract

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.

Keywords

Group classification of differential equations Group analysis of differential equations Equivalence group Equivalence groupoid Lie symmetry Schrödinger equations 

Mathematics Subject Classification (2010)

35Q41 35B06 35A30 

Notes

Acknowledgements

The authors are pleased to thank Anatoly Nikitin, Olena Vaneeva and Vyacheslav Boyko for stimulating discussions.

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Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Célestin Kurujyibwami
    • 1
    • 2
  • Peter Basarab-Horwath
    • 2
  • Roman O. Popovych
    • 3
    • 4
  1. 1.College of Science and TechnologyUniversity of RwandaKigaliRwanda
  2. 2.Linköping UniversityLinköpingSweden
  3. 3.Wolfgang Pauli InstituteViennaAustria
  4. 4.Institute of Mathematics of NAS of UkraineKyivUkraine

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