Solvability of the $F_4$ Integrable System

TL;DR

This paper proves that the $F_4$ rational and trigonometric integrable systems are exactly solvable for any coupling constants, explicitly finding their spectra and eigenfunctions using algebraic methods and special variables.

Contribution

It introduces new Weyl-invariant variables that simplify the Hamiltonian to an algebraic form and demonstrates their relation to an infinite-dimensional Lie algebra, advancing the understanding of $F_4$ integrable systems.

Findings

Explicit spectra and eigenfunctions for $F_4$ systems

Introduction of Weyl-invariant algebraic variables

Connection to infinite-dimensional Lie algebra

Abstract

It is shown that the $F_4$ rational and trigonometric integrable systems are exactly-solvable for {\it arbitrary} values of the coupling constants. Their spectra are found explicitly while eigenfunctions by pure algebraic means. For both systems new variables are introduced in which the Hamiltonian has an algebraic form being also (block)-triangular. These variables are invariant with respect to the Weyl group of $F_4$ root system and can be obtained by averaging over an orbit of the Weyl group. Alternative way of finding these variables exploiting a property of duality of the $F_4$ model is presented. It is demonstrated that in these variables the Hamiltonian of each model can be expressed as a quadratic polynomial in the generators of some infinite-dimensional Lie algebra of differential operators in a finite-dimensional representation. Both Hamiltonians preserve the same flag of polynomials and each subspace of the flag coincides with the finite-dimensional representation space of this algebra. Quasi-exactly-solvable generalization of the rational $F_4$ model depending on two continuous and one discrete parameters is found.