Solvability of $F_4$ quantum integrable systems

TL;DR

This paper demonstrates that the $F_4$ rational and trigonometric quantum integrable systems are exactly solvable for any coupling constants, providing explicit spectra and algebraic eigenfunctions using invariants of the $F_4$ Weyl group.

Contribution

It introduces new algebraic variables based on $F_4$ invariants that simplify the Hamiltonian and establish exact solvability for these systems.

Findings

Explicit spectra for $F_4$ systems are obtained.

Eigenfunctions are constructed algebraically.

Hamiltonians preserve a specific polynomial flag.

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 are obtained 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 a certain invariants of the $F_4$ Weyl group. Both Hamiltonians preserve the same (minimal) flag of spaces of polynomials, which is found explicitly.