  In a recent work we have derived an analytic expression for the partition
function of the quartic oscillator using a path integral formalism. Quite
remarkably, the free energy was found to be accurate to a few percent over the
entire range of temperatures and quartic coupling constant. In addition, the
key features of the exact partition function were successfully reproduced.
Accurate analytic expressions for the ground- and first-excited state energies
as function of $g$ were derived. In this work, we extend our results to the
calculation of the full energy spectrum. We also generalize our study of the
quartic oscillator to the case of the anharmonic oscillator with sextic and
octic couplings. The energy levels found are accurate for all couplings and
principal quantum numbers considered here (up to $n=8$), confirming this model
partition function as a good and faithful approximation of the exact one.
