  The Stark effect in hydrogen and the cubic anharmonic oscillator furnish
examples of quantum systems where the perturbation results in a certain
ionization probability by tunneling processes. Accordingly, the perturbed
ground-state energy is shifted and broadened, thus acquiring an imaginary part
which is considered to be a paradigm of nonperturbative behavior. Here we
demonstrate how the low order coefficients of a divergent perturbation series
can be used to obtain excellent approximations to both real and imaginary parts
of the perturbed ground state eigenenergy. The key is to use analytic
continuation functions with a built in analytic structure within the complex
plane of the coupling constant, which is tailored by means of Bender-Wu
dispersion relations. In the examples discussed the analytic continuation
functions are Gauss hypergeometric functions, which take as input fourth order
perturbation theory and return excellent approximations to the complex
perturbed eigenvalue. These functions are Borel-consistent and dramatically
outperform widely used Pad\'e and Borel-Pad\'e approaches, even for rather
large values of the coupling constant.
