Polynomial Heisenberg algebras and higher order supersymmetry

TL;DR

This paper demonstrates that higher order supersymmetric partners of the harmonic oscillator realize polynomial Heisenberg algebras, and introduces a linearized operator version that preserves the Fock space structure.

Contribution

It shows that higher order supersymmetric quantum mechanics provides simple realizations of polynomial Heisenberg algebras with a consistent Fock representation.

Findings

Higher order SUSY partners realize polynomial Heisenberg algebras

Linearized operators maintain harmonic oscillator Fock space

Fock representation remains unchanged for SUSY partners

Abstract

It is shown that the higher order supersymmetric partners of the harmonic oscillator Hamiltonian provide the simplest non-trivial realizations of the polynomial Heisenberg algebras. A linearized version of the corresponding annihilation and creation operator leads to a Fock representation which is the same as for the harmonic oscillator Hamiltonian