Abstract
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.
•Using polynomials solution, we solve Schrödinger equation for the harmonic Oscillator.•We perform the necessary and sufficient conditions for the solution existence.•We generalize our study for the sextic and decatic anharmonic oscillators.•We derive general expressions of the energy levels.