Schr\"{o}dinger Equation with the Potential $V(r)=a r^2+b r^{-4}+c   r^{-6}$
  By making use of an ${\it ansatz}$ for the eigenfunction, we obtain the exact solutions to the Schr\"{o}dinger equation with the anharmonic potential, $V(r)=a r^2+b r^{-4}+c r^{-6}$, both in three dimensions and in two dimensions, where the parameters $a$, $b$, and $c$ in the potential satisfy some constraints. 
Shi-Hai Dong, Xi-wen Hou and Zhong-Qi Ma