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 Δ_LB on symmetric space X  SO₀(2,3)/SO(2) xSO(3) of rank 2 is given. Using the representation of SO₀(2,3) the eigenfunctions of δ_LB, i.e. plane waves on X are constructed. The integral representation of the zonal spherical functions of the group SO_o(2, 3) is presented and the related 2-body one-dimensional quantum systems are considered.