SOLVABILITY OF THE F₄ INTEGRABLE SYSTEM

Abstract

It is shown that the F 4 rational and trigonometric integrable systems are exactlysolvable for 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 spaces of polynomials and each subspace of the flag coincides with the finite-dimensional representation space of this algebra. Quasiexactly-solvable generalization of the rational F 4 model depending on two continuous and one discrete parameters is found.

Solvability of the Hamiltonians Related to Exceptional Root Spaces: Rational Case

“…It implies that the coefficient functions in front of the second derivatives in the Laplace-Beltrami operator are polynomials in invariants of the Weyl group. This result was rediscovered (and then generalized) later in [4], [5], [6] and [8] for A n , BC n , G 2 and F 4 algebras, respectively. It was shown that the algebraic structure persists for the whole Laplace-Beltrami operator: the coefficient functions in front of the first derivatives are polynomials as well.…”

“…The Hamiltonian of the rational F 4 model written in the basis of the standard F 4 roots has the form (see [8] 8 ),…”

“…From a technical point of view it can be found by eliminating certain terms in the Hamiltonian, by choosing appropriate parameters A's. In [8] the minimal flag, denoted as P (F 4 ) , is found. This flag is generated by the spaces of quasi-homogeneous polynomials 8) with the characteristic vector f…”

“…In [8] the minimal flag, denoted as P (F 4 ) , is found. This flag is generated by the spaces of quasi-homogeneous polynomials 8) with the characteristic vector f…”

“…The variables (4.11) are remarkable as they are the rational limits of certain trigonometric variables in which the trigonometric F 4 model takes an algebraic form [8].…”