Group classification of linear Schrödinger equations by the algebraic method

Abstract

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.iii Populärvetenskaplig sammanfattningAvhandlingen behandlar gruppklassificeringen av linjära Schrödingerekvationer. Studiet av sådana ekvationers Lie symmetrier påbörjades för mer än 40 år sedan. Man använde då Lies klassiska infinitesimala metod och begränsade sig till speciella typer av reellvärd potential. De första arbetena ägnades huvudsakligen åt dynamiska transformationer. Eftersom dessa nödvändigtvis ändrar både tidsoch rumsvariablerna behandlades inte fastranslationer. Senare utvidgades studierna till icke-linjära Schrödingerekvationer, men gruppklassificeringsproblemet för klassen av linjä...

Group analysis of general Burgers–Korteweg–de Vries equations

“…Each normalized class is semi-normalized in the same sense, and the converse is not in general true. There are also more sophisticated notions, uniform semi-normalization and weak uniform semi-normalization, which mediate the notion of normalization and semi-normalization [19,20].…”

“…The further consideration partitions into five principal cases, depending on the value kκ and certain constraints for components of nonzeroν ∈ Nκ, I)ν 1ν3 = 0, II)ν 1 = 0,ν 3 = 0, III)ν 1 = 0, ν 2 = 0,ν 3 = 0, IV)ν 1 = 0,ν 2 = 0,ν 3 = 0 with kκ = 1 and V) kκ = 2. For each of these cases, we integrate the corresponding set of independent copies of the system (20) and thus derive the respective form ofκ parameterized by arbitrary constants. The relations (6b)-(6c) between κ andκ imply that the arbitrary-element tuple κ is of the same form asκ, but maybe with other values of parameter constants.…”