Laplace-Beltrami operator on symmetric space X ≡ SOo(2, 3)/SO(2) × SO(3) of rank 2 and the related 2-body one dimensional quantum systems

The explicit form of the Laplace-Beltrami operator ΔLB on symmetric space X  SO0(2,3)/SO(2) x SO(3) of rank 2 is given. Using the representation of SO0(2,3) the eigenfunctions of δLB, i.e. plane waves on X are constructed. The integral representation of the zonal spherical functions of the group SOo(2, 3) is presented and the related 2-body one-dimensional quantum systems are considered.