Differential representations of dynamical oscillator symmetries in discrete Hilbert space

Abstract

As a very important example for dynamical symmetries in the context ofq-generalized quantum mechanics the algebraaa†−q−2a†a=1is investigated. It represents the oscillator symmetrySUq(1,1)and is regarded as a commutation phenomenon of theq-Heisenberg algebra which provides a discrete spectrum of momentum and space,i.e., a discrete Hilbert space structure. Generalizedq-Hermite functions and systems of creation and annihilation operators are derived. The classical limitq→1is investigated. Finally theSUq(1,1)algebra is represented by the dynamical variables of theq-Heisenberg algebra.