Accurate Analytic Model for the Energy Spectrum of the Anharmonic Oscillator

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.