The partition function for anharmonic oscillator in the strong-coupling regime

We consider a single anharmonic oscillator with frequency ω and coupling constant λ respectively, in the strong-coupling regime. We are assuming that the system is in thermal equilibrium with a reservoir at temperature β -1. Using the strong-coupling perturbative expansion, we obtain the partition function for the oscillator in the regime λ>>ω, up to the order (1)/(√(λ)). To obtain this result, we follow two steps. First, we must give meaning to the first term of the strong-coupling perturbative expansion, i.e., the static ultra-local generating functional Qβ(h). Second, we have to regularize and re normalize the kernel K(ω, σ,τ-τ') integrated over Euclidean time. In order to solve both problems, we make use of a combination of Klauder representation for the static ultra-local generating functional (Acta Phys. Austr. 41, 237 (1975), Ann. Phys. 117, 19 (1979)), and the generalized zeta-function method. The free energy and the mean energy, up to the order (1)/(√(λ)) are also presented. We are showing that the thermodynamics quantities are non analytic in the coupling constant. (author)