Exact solutions of the Schrödinger equation with double ring-shaped oscillator

• Solve the Schrödinger equation with double ring-shaped oscillator analytically.
• Introduce a new variable x=cosθ.
• Construct super-universal associated Legendre polynomials.
• Present results go back to those of special cases.
• Symmetry breaking caused by surrounded two ring-shaped inversed square potentials.

We present the exact solutions of the Schrödinger equation with the double ring-shaped oscillator (DRSO) potential. By introducing a new variable x=cosθ and constructing super-universal associated Legendre polynomials we express the polar angular wave functions explicitly. We observe that the present DRSO has caused the symmetry breaking from the original spherical oscillator SU(3)⊃SO(3)⊃O(2)SU(3)⊃SO(3)⊃O(2) symmetries to the present O(2)O(2) symmetry due to the surrounded two ring-shaped inversed square potentials. Some special cases are also discussed.