The quantum $H_4$ integrable system

TL;DR

This paper analyzes the quantum $H_4$ integrable system, revealing its algebraic form, polynomial eigenfunctions, and spectral properties related to an anisotropic harmonic oscillator with a rich polynomial invariant structure.

Contribution

It demonstrates that the gauge-rotated $H_4$ Hamiltonian and integrals are algebraic, with polynomial eigenfunctions and a detailed invariant subspace structure.

Findings

Hamiltonian has polynomial coefficients in derivatives

Eigenfunctions are polynomial times ground state

Spectra match anisotropic harmonic oscillator

Abstract

The quantum $H_4$ integrable system is a 4D system with rational potential related to the non-crystallographic root system $H_4$ with 600-cell symmetry. It is shown that the gauge-rotated $H_4$ Hamiltonian as well as one of the integrals, when written in terms of the invariants of the Coxeter group $H_4$, is in algebraic form: it has polynomial coefficients in front of derivatives. Any eigenfunctions is a polynomial multiplied by ground-state function (factorization property). Spectra corresponds to one of the anisotropic harmonic oscillator. The Hamiltonian has infinitely-many finite-dimensional invariant subspaces in polynomials, they form the infinite flag with the characteristic vector $\vec \al\ =\ (1,5,8,12)$.