Anharmonic Energy Spectrum for $V(x)=\pm x^{4},\pm x^{6},\pm x^{8}$

In this work we wish to revisit the energy spectrum for the anharmonic potentials% \[ H=\frac{p^{2}}{2m}\pm x^{N}, \] where $N=4,6,8$. Using the second quantized operator formalism of Dirac, we have evaluated matrix truncations of up to $100\times100$. Our results for the energy spectrum are in disagreement with the work of Bender and Boettcher (PRL 80, 5243). They studied a \emph{PT} symmetric Hamiltonian whose potential is given by $V(x)=-(ix)^{N}$ and who maintain that \textquotedblleft when $N\geq2$, the spectrum is infinite, discrete and entirely real and positive\textquotedblright. We find, for the potentials with $N=4,6,8$ that the spectrum is not completely positive and in fact has no lower bound.