Differential representations of dynamical oscillator symmetries in discrete Hilbert space (Q1766390)

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.