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Register a new userRealizations of Galilei algebras
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All inequivalent realizations of the Galilei algebras of dimensions not greater than five are constructed using the algebraic approach proposed by I. Shirokov. The varieties of the deformed Galilei algebras are discussed and families of one-parametric deformations are presented in explicit form. It is also shown that a number of well-known and physically interesting equations and systems are invariant with respect to the considered Galilei algebras or their deformations.
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Essentially generalizing Lies results, we prove that the contact equivalence groupoid of a class of (1+1)-dimensional generalized nonlinear Klein-Gordon equations is the first-order prolongation of its point equivalence groupoid, and then we carry out the complete group classification of this class. Since it is normalized, the algebraic method of group classification is naturally applied here. Using the specific structure of the equivalence group of the class, we essentially employ the classical Lie theorem on realizations of Lie algebras by vector fields on the line. This approach allows us to enhance previous results on Lie symmetries of equations from the class and substantially simplify the proof. After finding a number of integer characteristics of cases of Lie-symmetry extensions that are invariant under action of the equivalence group of the class under study, we exhaustively describe successive Lie-symmetry extensions within this class.
A complete set of inequivalent realizations of three- and four-dimensional real unsolvable Lie algebras in vector fields on a space of an arbitrary (finite) number of variables is obtained.
We study realizations of Lie algebras by vector fields. A correspondence between classification of transitive local realizations and classification of subalgebras is generalized to the case of regular local realizations. A reasonable classification problem for general realizations is rigorously formulated and an algorithm for construction of such classification is presented.
Nonrelativistic conformal groups, indexed by l=N/2, are analyzed. Under the assumption that the mass parametrizing the central extension is nonvanishing the coadjoint orbits are classified and described in terms of convenient variables. It is shown that the corresponding dynamical system describes, within Ostrogradski framework, the nonrelativistic particle obeying (N+1)-th order equation of motion. As a special case, the Schroedinger group and the standard Newton equations are obtained for N=1 (l=1/2).
We study sufficient conditions for the absence of positive eigenvalues of magnetic Schrodinger operators in $mathbb{R}^d,, dgeq 2$. In our main result we prove the absence of eigenvalues above certain threshold energy which depends explicitly on the magnetic and electric field. A comparison with the examples of Miller--Simon shows that our result is sharp as far as the decay of the magnetic field is concerned. As applications, we describe several consequences of the main result for two-dimensional Pauli and Dirac operators, and two and three dimensional Aharonov--Bohm operators.
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