Solvability of $F_4$ quantum integrable systems

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.