Completely integrable Hamiltonian systems connected with semisimple Lie algebras

Quantum systems related to root systems, and radial parts of Laplace operators

“…As usual, we shall introduce the operators of the coordinate q j and of the momentum p j = − i ∂/∂q j , and let g 2 α be positive constants on roots of equal length. As in [5], we shall consider systems with a Hamiltonian of the form…”

“…It is possible to indicate an alcove form also for the functions a 2 γ(aξ) (IV (3), see [5]). Thus we have found that for the systems I, II and V the particle moves in the Weyl chamber Λ, whereas for the systems III and IV it moves in the Weyl alcove Λ a .…”

“…Since the algebraS=Im j(D) is isomorphic to the algebra D (23) andS ⊂ S (26), the assertion of the theorem for the systems I, II and III will follow from the existence of n generators of the commutative algebra D on the symmetric spaces X 0 , X − , and X + [16]. For systems of type V the assertion follows from the complete integrability of systems of type I when we change the variables p j → p j ± i ωq j (see [19]). Corollary 1.…”

“…Then the commutators of the operators will go over into the Poisson brackets of classical variables. Let us note that in the classical case, complete integrability has been proved earler (in [5]) for classical root systems only.…”

“…In [19] we have obtained explicit formulas for certain integrals inS. With the aid of the theorem presented in Sec.…”