Second order symmetries of the conformal laplacian and R-separation

30th International Colloquium on Group Theoretical Methods in Physics

Michel, Jean-Philippe^1 ; Radoux, Fabian^2 ; Silhan, Josef^3

Institut de Recherche en Mathematique et Physique (IRMP), Universite Catholique de Louvain (UCL), Chemin du Cyclotron 2, Louvain-la-Neuve

1348, Belgium^1

Department of Mathematics, University of Liege, Grande Traverse 12, Liege

4000, Belgium^2

Department of Algebra and Geometry, Masaryk University in Brno, Janakovo nam. 2a, Brno

662 95, Czech Republic^3

关键词: Arbitrary potentials;    Conformal symmetry;    Laplace-Beltrami operator;    Laplacians;    Riemannian manifold;    Second orders;    Zero energies;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/597/1/012058/pdf
DOI  :  10.1088/1742-6596/597/1/012058

来源: IOP

【 摘 要 】

Let (M, g) be an arbitrary pseudo-Riemannian manifold of dimension at least 3, let Δ := ∇agab∇bbe the Laplace-Beltrami operator and let ΔYbe 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 ΔY+ V (where V is an arbitrary potential) can be used to characterize the R-separation of the Schrodinger equation (ΔY+ V)Ψ = EΨ (resp. The Schrodinger equation at zero energy (ΔY+ V)Ψ = 0). Using a result exposed in our previous paper, we obtain characterizations of the R-separation of the equations ΔYΨ = 0 and ΔYΨ = EΨ uniquely in terms of (conformal) Killing tensors pertaining to (conformal) Killing-Stackel algebras.