Parity inversion property of the double ring-shaped oscillator in cylindrical coordinates

Using the functional analysis method, we present the exact solutions of the double ring-shaped oscillator (DRSO) potential with certain parity in the cylindrical coordinates. Such a quantum system is separated to two differential equations, i.e. a one-dimensional harmonic oscillator plus an inverse square term and a two-dimensional harmonic oscillator plus an inverse square term. The key point is how to find the adapted symmetrical solutions of the one-dimensional harmonic oscillator plus an inverse square term at the singular point z = 0. The obtained results are compared with those in the spherical coordinates. We also explore intimate connections ET(ip) =-ET(p) and Ez(iz) =-Ez(z) by substituting p → ip and z → iz.