Differential representations of dynamical oscillator symmetries in discrete Hilbert space

As a very important example for dynamical symmetries in the context of q -generalized quantum mechanics the algebra a a † − q − 2 a † a = 1 is investigated. It represents the oscillator symmetry S U q ( 1 , 1 ) and is regarded as a 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 limit q → 1 is investigated. Finally the S U q ( 1 , 1 ) algebra is represented by the dynamical variables of the q -Heisenberg algebra.