Discretization of the phase space for a q-deformed harmonic oscillator with q a root of unity

TL;DR

This paper investigates the phase space structure of a q-deformed harmonic oscillator with q as a root of unity, revealing discrete eigenvalues, a lattice phase space, and connections to deformed Hermite polynomials.

Contribution

It demonstrates that the position and momentum operators have discrete eigenvalues and characterizes the non-uniform lattice phase space for the q-deformed oscillator.

Findings

Eigenvalues are roots of deformed Hermite polynomials.

Phase space has a non-uniform lattice structure.

Connections to truncated and parafermionic oscillators.

Abstract

The ``position'' and ``momentum'' operators for the q-deformed oscillator with q being a root of unity are proved to have discrete eigenvalues which are roots of deformed Hermite polynomials. The Fourier transform connecting the ``position'' and ``momentum'' representations is also found The phase space of this oscillator has a lattice structure, which is a non-uniformly distributed grid. Non-equidistant lattice structures also occur in the cases of the truncated harmonic oscillator and of the q-deformed parafermionic oscillator, while the parafermionic oscillator corresponds to a uniformly distributed grid.