A note on a class of exact solutions for a doubly anharmonic oscillator

We examine a class of exact solutions for the eigenvalues and eigenfunctions of a doubly anharmonic oscillator defined by the potential $V(x)=\omega^2/2 x^2+\lambda x^4/4+\eta x^6/6$, $\eta>0$. These solutions hold provided certain constraints on the coupling parameters $\omega^2$, $\lambda$ and $\eta$ are satisfied.