Quantum spectrum of some anharmonic central potentials: Wave functions ansatz

Based on the ansatz to the wave functions, the quasi-exact solutions of the 2D Schrödinger equation with some anharmonic potentials are reviewed and analyzed if admitting restrictions on the parameters of the potential and the angular momentum m. These potentials are taken as the screened Coulomb potential V(r) = a/r + 6/(r + λ), the singular one-fraction power one V(r) = ar^-1/2 + br^-3/2 and the singular two-fraction one V(r) = ar^2/3 + br^-2/3+cr^-4/3. The latter one is found that the hidden symmetry exists if substituting r → -ir. It will reverse the signs of E and c of quantum system, leaving the remaining parameters invariant.