Remarks on the connection between the additive separation of the Hamilton-Jacobi equation and the multiplicative separation of the Schrödinger equation. II. First integrals and symmetry operatorsThe fundamental elements of the variable separation theory are revisited, including the Eisenhart and Robertson theorems, Kalnins-Miller theory, and the intrinsic characterization of the separation of the Hamilton-Jacobi equation, in a unitary and geometrical perspective. The general notion of complete integrability of first-order normal systems of PDEs leads in a natural way to completeness conditions for separated solutions of the Schrödinger equation and to the Robertson condition. Two general types of multiplicative separation for the Schrödinger equation are defined and analyzed: they are called ''free'' and ''reduced'' separation, respectively. In the free separation the coordinates are necessarily orthogonal, while the reduced separation may occur in nonorthogonal coordinates, but only in the presence of symmetries ͑Killing vectors͒.Proof: By setting ⌫ a ϭ⌫ a Ϫ2g a␣ ␣ , V ϭVϩ⌫ ␣ ␣ Ϫg ␣␤ ␣ ␤ , 5201

The commutation relations of the first-order and second-order operators associated with the first integrals in involution of a Hamiltonian separable system are examined. It is shown that these operators commute if and only if a “pre-Robertson condition” is satisfied. This condition involves the Ricci tensor of the configuration manifold and it is implied by the Robertson condition, which is necessary and sufficient for the separability of the Schrödinger equation.

Abstract. We describe a procedure to construct polynomial in the momenta first integrals of arbitrarily high degree for natural Hamiltonians H obtained as one-dimensional extensions of natural (geodesic) n-dimensional Hamiltonians L. The Liouville integrability of L implies the (minimal) superintegrability of H. We prove that, as a consequence of natural integrability conditions, it is necessary for the construction that the curvature of the metric tensor associated with L is constant. As examples, the procedure is applied to one-dimensional L, including and improving earlier results, and to two and three-dimensional L, providing new superintegrable systems.

We consider classical three-body interactions on a Euclidean line depending on the reciprocal distance of the particles and admitting four functionally independent quadratic in the momenta first integrals. These systems are multiseparable, superintegrable and equivalent (up to rescalings) to a one-particle system in the three-dimensional Euclidean space. Common features of the dynamics are discussed. We show how to determine quantum symmetry operators associated with the first integrals considered here but do not analyze the corresponding quantum dynamics. The conformal multiseparability is discussed and examples of conformal first integrals are given. The systems considered here in generality include the Calogero, Wolfes, and other threebody interactions widely studied in mathematical physics.

The additive variable separation in the Hamilton–Jacobi equation is studied for a natural Hamiltonian with scalar and vector potentials on a Riemannian manifold with positive–definite metric. The separation of this Hamiltonian is related to the separation of a suitable geodesic Hamiltonian over an extended Riemannian manifold. Thus the geometrical theory of the geodesic separation is applied and the geometrical characterization of the separation is given in terms of Killing webs, Killing tensors, and Killing vectors. The results are applicable to the case of a nondegenarate separation on a manifold with indefinite metric, where no null essential separable coordinates occur.