Second order symmetries of the conformal laplacian and R-separation

Let (M, g) be an arbitrary pseudo-Riemannian manifold of dimension at least 3, let Δ := ∇_ag^ab∇_b be the Laplace-Beltrami operator and let Δ_Y be the conformal Laplacian. In some references, Kalnins and Miller provide an intrinsic characterization for R-separation of the Laplace equation ΔΨ = 0 in terms of second order conformal symmetries of Δ. The main goal of this paper is to generalize this result and to explain how the (resp. conformal) symmetries of Δ_Y + V (where V is an arbitrary potential) can be used to characterize the R-separation of the Schrodinger equation (Δ_Y + V)Ψ = EΨ (resp. the Schrödinger equation at zero energy (Δ_Y + V)Ψ = 0). Using a result exposed in our previous paper, we obtain characterizations of the R-separation of the equations Δ_YΨ = 0 and Δ_YΨ = EΨ uniquely in terms of (conformal) Killing tensors pertaining to (conformal) Killing-Stäckel algebras. (paper)