Self-Similar Potentials and the q-Oscillator Algebra at Roots of Unity

TL;DR

This paper explores self-similar potentials linked to the q-oscillator algebra, especially at roots of unity, providing explicit forms and their relation to special functions, enriching the understanding of q-deformed quantum systems.

Contribution

It explicitly derives the functional forms of self-similar potentials at roots of unity and connects them to special elliptic functions, advancing the analysis of q-deformed algebras.

Findings

Explicit potentials for q^3=1 and q^4=1 derived

Wave functions form bases of q-deformed Heisenberg-Weyl algebra representations

Potentials expressed via Weierstrass functions

Abstract

Properties of the simplest class of self-similar potentials are analyzed. Wave functions of the corresponding Schr\"odinger equation provide bases of representations of the $q$-deformed Heisenberg-Weyl algebra. When the parameter $q$ is a root of unity the functional form of the potentials can be found explicitly. The general $q^3=1$ and the particular $q^4=1$ potentials are given by the equianharmonic and (pseudo)lemniscatic Weierstrass functions respectively.