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 Using a path integral formalism we derive an approximate analytic expression for the partition function of the quartic anharmonic oscillator $V(x) = \frac{1}{2} \omega^2 x^2 + g x^4$. From the partition function we also derive explicit expressions for the ground- and first-excited state energies $E_0(g)$ and $E_1(g)$, respectively. Quite remarkably, these energies are found to be accurate for the whole range of the coupling constant $g$. In addition, both the weak- and strong-coupling expansions are qualitatively reproduced. We believe that our approximate explicit partition function reproducing the key features of the exact partition function defines an interesting model to investigate the properties of the quartic oscillator.
 
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 ### Antoine Levitt (LMO, Orsay)
 
 **Resonances: the example of the 1D chain**