Variational Perturbation Theory for Summing Divergent Non-Borel-Summable Tunneling Amplitudes

We present a method for evaluating divergent non-Borel-summable series by an analytic continuation of variational perturbation theory. We demonstrate the power of the method by an application to the exactly known partition function of the anharmonic oscillator in zero spacetime dimensions. In one spacetime dimension we derive the imaginary part of the ground state energy of the anharmonic oscillator for {\em all negative values of the coupling constant $g$, including the nonanalytic tunneling regime at small-$g$. As a highlight of the theory we retrieve from the divergent perturbation expansion the action of the critical bubble and the contribution of the higher loop fluctuations around the bubble.