In this work a systematic study of finite-dimensional nonrelativistic conformal groups is carried out under two complementary points of view. First, the conformal Killing equation is solved to obtain a whole family of finite-dimensional conformal algebras corresponding to each of the Galilei and Newton–Hooke kinematical groups. Some of their algebraic and geometrical properties are studied in a second step. Among the groups included in these families one can identify, for example, the contraction of the Minkowski conformal group, the analog for a nonrelativistic de Sitter space, or the nonextended Schrödinger group.
Exact analytical solutions for the bound states of a graphene Dirac electron in various magnetic fields with translational symmetry are obtained. In order to solve the time-independent Dirac-Weyl equation the factorization method used in supersymmetric quantum mechanics is adapted to this problem. The behavior of the discrete spectrum, probability and current densities are discussed.
Polynomial deformations of the Heisenberg algebra are studied in detail. Natural realizations of them are given by the higher order susy partners of the harmonic oscillator for even order polynomials. Here, it is shown that the susy partners of the radial oscillator play a similar role when the order of the polynomial is odd. Indeed, it will be proved that the general systems ruled by such a kind of algebras, in the quadratic and cubic cases, involve Painlevé transcendents of type IV and V, respectively.
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