The discrete and periodic heat and harmonic oscillator equation

Abstract

In this paper we present a generalization of the famous Dirac ladder formalism for the Schrödinger harmonic oscillator equation. Whereas the classical one-dimensional harmonic oscillator acts on functions of a real continuous variable, our generalization works within the framework of Pontryagin dual groups, the discrete group hℤ and the circle group h − 1𝕊 (𝕊 unit circle). We will define so-called Hermite–Kravchuk operators that constitute a close connection between the heat and harmonic oscillator equation that is invisible in the real continuous case. This generalization is not only the result of a crude discretization or perturbation process. Since this theory preserves and expands the beautiful algebraic background, one can speak of a structural discretization or “periodization”. We hope that this theory will also serve as a model for physical phenomena.