On the integrability of \(q\)-oscillators based on invariants of discrete Fourier transforms (Q1372953)

When the ordinary Heisenberg relations are replaced by the \(q\)-commuting Heisenberg relations, one obtains a discrete spectrum for the \(q\)-momentum variable and for the self-adjoint extension of the \(q\)-space variable. This letter is dealing with the question : what is a harmonic oscillator in this discretized phase space quantum mechanics? The authors first postulate a number of properties that the Hamiltonian \(H\) of a harmonic oscillator (as an operator from the Hilbert space of functions over the lattice to itself) should satisfy. To find an appropriate \(H\), the relation between Fourier invariance and the oscillator algebra are considered. Then \(q\)-Fourier invariance is investigated, and applied to find representations of \(q\)-creation and annihilation operators. The analysis leads to a solution for \(H\). This Hamiltonian on the lattice is much more complicated than the classical one, but has interesting properties.