Variational perturbation theory for summing divergent tunnelling amplitudes

We present a method for evaluating divergent series with factorially growing coefficients of equal sign. The method is based on an analytic continuation of variational perturbation theory from the regime of alternating signs. We demonstrate its power first by applying it to the exactly known partition function of the anharmonic oscillator in zero space–time dimensions (the simple integral). Then we consider the quantum-mechanical case of one space–time dimension and derive the imaginary part of the ground-state energy of the anharmonic oscillator for all negative values of the coupling constant g, including the non-analytic tunnelling regime at small −g. As a highlight of the theory we extract, from the divergent perturbation expansion, the action of the critical bubble and the contribution of the higher loop fluctuations around the bubble. PACS numbers: 03.65.−w, 03.65.Xp, 03.75.Lm (Some figures in this article are in colour only in the electronic version.)