Group classification of multidimensional linear Schrödinger equations with algebraic method
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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.
IdentifiersURN: urn:nbn:se:liu:diva-125135OAI: oai:DiVA.org:liu-125135DiVA: diva2:903185