The quantum $H_3$ integrable system

TL;DR

This paper analyzes the quantum $H_3$ integrable system, revealing its algebraic structure, invariant subspaces, and hidden algebra, and introduces a quasi-exactly-solvable generalization and a related discrete model.

Contribution

It provides the algebraic form of the quantum $H_3$ Hamiltonian, identifies its hidden algebra, and constructs a quasi-exactly-solvable extension and a related discrete integrable model.

Findings

Hamiltonian expressed in algebraic form with polynomial coefficients

Identified infinite-dimensional invariant subspaces with characteristic vector (1,2,3)

Constructed a quasi-exactly-solvable generalization and a discrete isospectral model

Abstract

The quantum $H_3$ integrable system is a 3D system with rational potential related to the non-crystallographic root system $H_3$. It is shown that the gauge-rotated $H_3$ Hamiltonian as well as one of the integrals, when written in terms of the invariants of the Coxeter group $H_3$, is in algebraic form: it has polynomial coefficients in front of derivatives. The Hamiltonian has infinitely-many finite-dimensional invariant subspaces in polynomials, they form the infinite flag with the characteristic vector $\vec \al\ =\ (1,2,3)$. One among possible integrals is found (of the second order) as well as its algebraic form. A hidden algebra of the $H_3$ Hamiltonian is determined. It is an infinite-dimensional, finitely-generated algebra of differential operators possessing finite-dimensional representations characterized by a generalized Gauss decomposition property. A quasi-exactly-solvable integrable generalization of the model is obtained. A discrete integrable model on the uniform lattice in a space of $H_3$-invariants "polynomially"-isospectral to the quantum $H_3$ model is defined.