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Register a new userLocal realizations of $q$-Oscillators in Quantum Mechanics
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Authors
A. A. Andrianov
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Representations of the quantum q-oscillator algebra are studied with particular attention to local Hamiltonian representations of the Schroedinger type. In contrast to the standard harmonic oscillators such systems exhibit a continuous spectrum. The general scheme of realization of the q-oscillator algebra on the space of wave functions for a one-dimensional Schroedinger Hamiltonian shows the existence of non-Fock irreducible representations associated to the continuous part of the spectrum and directly related to the deformation. An algorithm for the mapping of energy levels is described.
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The geometric realizations of Lusztigs symmetries of symmetrizable quantum groups are given in this paper. This construction is a generalization of that in [19].
We described the $q$-deformed phase space. The $q$-deformed Hamilton eqations of motion are derived and discussed. Some simple models are considered.
In this paper, we shall study the structure of the Grothendieck group of the category consisting of Lusztigs perverse sheaves and give a decomposition theorem of it. By using this decomposition theorem and the geometric realizations of Lusztigs symmetries on the positive part of a quantum group, we shall give geometric realizations of Lusztigs symmetries on the whole quantum group.
We compute the braiding for the `principal gradation of $U_q(hat{{it sl}_2})$ for $|q|=1$ from first principles, starting from the idea of a rigid braided tensor category. It is not necessary to assume either the crossing or the unitarity condition from S-matrix theory. We demonstrate the uniqueness of the normalisation of the braiding under certain analyticity assumptions, and show that its convergence is critically dependent on the number-theoretic properties of the number $tau$ in the deformation parameter $q=e^{2pi itau}$. We also examine the convergence using probability, assuming a uniform distribution for $q$ on the unit circle.
We study a family of modules over Kac-Moody algebras realized in multi-valued functions on a flag manifold and find integral representations for intertwining operators acting on these modules. These intertwiners are related to some expressions involving complex powers of Lie algebra generators. When applied to affine Lie algebras, these expressions give integral formulas for correlation functions with values in not necessarily highest weight modules. We write related formulas out in an explicit form in the case of $hat{gtsl_{2}}$. The latter formulas admit q-deformation producing an integral representation of q-correlation functions. We also discuss a relation of complex powers of Lie algebra (quantum group) generators and Casimir operators to ($q-$)special functions.
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