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Group classification of linear Schrödinger equations by the algebraic method
Linköping University, Department of Mathematics. Linköping University, Faculty of Science & Engineering.
2016 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is devoted to the group classification of linear Schrödinger equations. The study of Lie symmetries of such equations was initiated more than 40 years ago using the classical Lie infinitesimal method for specific types of real-valued potentials. In first papers on this subject, most attention was paid to dynamical transformations, which necessarily change the time and space variables. This is why phase translations were missed. Later, the study of Lie symmetries was extended to nonlinear Schrödinger equations. At the same time, the group classification problem for the class of linear Schrödinger equations with complex potentials remains unsolved.

The aim of the present thesis is to carry out the group classification for the class of linear Schrödinger equations with complex potentials. These potentials are nowadays important in quantum mechanics, scattering theory, condensed matter physics, quantum field theory, optics, electromagnetics and so forth. We exhaustively solve the group classification problem for space dimensions one and two.

The thesis comprises two parts. The first part is a brief review of Lie symmetries and group classification of differential equations. Next, we outline the equivalence transformations in a class of differential equations, normalization properties of such class and the algebraic method for group classification of differential equations.

The second part consists of two research papers. In the first paper, the algebraic method is applied to solve the group classification problem for (1+1)-dimensional linear Schrödinger equations with complex potentials. With this technique, the problem of the group classification of the class under study is reduced to the classification of certain subalgebras of its equivalence algebra. As a result, we find that the inequivalent cases are exhausted by eight families of potentials and we give the corresponding maximal Lie invariance algebras.

In the second paper we carry out the preliminary symmetry analysis of the class of linear Schrödinger equations with complex potentials in the multi-dimensional case. Using the direct method, we find the equivalence groupoid and the equivalence group of this class. Due to the multi-dimensionality, the results of the computations are quite different from the ones presented in Paper I. We obtain the complete group classification of (1+2)-dimensional linear Schrödinger equations with complex potentials.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2016. , 22 p.
Series
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1743
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-125136DOI: 10.3384/lic.diva-125136ISBN: 978-91-7685-810-3 (print)OAI: oai:DiVA.org:liu-125136DiVA: diva2:903188
Presentation
2016-02-24, KY26, Hus Key, Campus Valla, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2016-02-15 Created: 2016-02-15 Last updated: 2016-02-15Bibliographically approved
List of papers
1. Algebraic method for group classification of (1+1)-dimensional linear Schrödinger equations
Open this publication in new window or tab >>Algebraic method for group classification of (1+1)-dimensional linear Schrödinger equations
(English)Manuscript (preprint) (Other academic)
Abstract [en]

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.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-125134 (URN)
Available from: 2016-02-15 Created: 2016-02-15 Last updated: 2016-02-15Bibliographically approved
2. Group classification of multidimensional linear Schrödinger equations with algebraic method
Open this publication in new window or tab >>Group classification of multidimensional linear Schrödinger equations with algebraic method
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We consider the group classification problem for multidimensional linear Schrödinger equations with complex-valued potentials. Using the algebraic approach, we compute the equivalence groupoid of the class and thus show that this class is uniformly semi-normalized. More specifically, any point transformation connecting two equations from the class is the composition of a linear superposition transformation of the corresponding initial equation and an equivalence transformation of this class. This is why the algebraic method of group classification is applied, which reduces the group classification to the classification of specific low-dimensional subalgebras of the associated equivalence algebra. Inequivalent Lie symmetry extensions are listed for dimension 1+2. Splitting into different classification cases is based on several integer parameters that are invariant with respect to the adjoint action of equivalence transformations. These parameters characterize the dimensions of parts of the corresponding Lie symmetry algebra that are related to generalized scalings, rotations and generalized Galilean boosts, respectively. As expected, the computation for dimension 1+2 is much trickier and cumbersome than for dimension 1+1 due to two reasons, appearing a new kind of transformations rotations and increasing the range for dimensions of essential subalgebras of Lie symmetry algebras, whose least upper bound is n(n+3)/2+5.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-125135 (URN)
Available from: 2016-02-15 Created: 2016-02-15 Last updated: 2016-02-15Bibliographically approved

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Kurujyibwami, Célestin

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