Canonical Discretization. I. Discrete faces of (an)harmonic oscillator

TL;DR

The paper introduces a canonical equivalence in quantum mechanics to relate continuous systems like the harmonic oscillator to their discrete counterparts, preserving spectral properties and polynomial eigenfunctions through discretization and $q$-deformation.

Contribution

It proposes a new notion of canonical equivalence that connects quantum systems with their discrete analogs, including harmonic and anharmonic oscillators, maintaining key spectral features.

Findings

Discrete systems canonically equivalent to harmonic oscillator identified

Discretizations preserve isospectrality and polynomial eigenfunctions

$q$-deformation leads to dilatation-covariant discretization

Abstract

A certain notion of canonical equivalence in quantum mechanics is proposed. It is used to relate quantal systems with discrete ones. Discrete systems canonically equivalent to the celebrated harmonic oscillator as well as the quartic and the quasi-exactly-solvable anharmonic oscillators are found. They can be viewed as a translation-covariant discretization of the (an)harmonic oscillator preserving isospectrality. The notion of the $q-$deformation of the canonical equivalence leading to a dilatation-covariant discretization preserving polynomiality of eigenfunctions is also presented.