Sextic and decatic anharmonic oscillator potentials: Polynomial solutions

We seek the possible polynomial solutions of the Schrödinger equation for the sextic and decatic potentials. Under certain conditions on the parameters of the potentials, we show that these potentials are exactly solvable. We evaluate the first four eigenstates for both potentials. We derive general expressions of the energy levels, for high energy levels, eigenvalues are a function of potentials' parameters and the eigenfunction's zeros.