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Differential representations of dynamical oscillator symmetries in discrete Hilbert space. (English) Zbl 1060.81539
As a very important example for dynamical symmetries in the context of \(q\)-generalized quantum mechanics the algebra \(aa^\dagger-q^{-2}a^\dagger a=1\) is investigated. It represents the oscillator symmetry \(\text{SU}_q(1,1)\) and is regarded as commutation phenomenon of the \(q\)-Heisenberg algebra which provides a discrete spectrum of momentum and space i.e., a discrete Hilbert space structure. Generalized \(q\)-Hermite functions and systems of creation and annihilation operators are derived. The classical \(q\to 1\) is investigated. Finally the \(\text{SU}_q(1,1)\) algebra is represented by the dynamical variables of th \(q\)-Heisenberg algebra.
MSC:
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
81R15 Operator algebra methods applied to problems in quantum theory
22E70 Applications of Lie groups to the sciences; explicit representations
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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