  Certain quantum mechanical potentials give rise to a vanishing perturbation
series for at least one energy level (which as we here assume is the ground
state), but the true ground-state energy is positive. We show here that in a
typical case, the eigenvalue may be expressed in terms of a generalized
perturbative expansion (resurgent expansion). Modified Bohr-Sommerfeld
quantization conditions lead to generalized perturbative expansions which may
be expressed in terms of nonanalytic factors of the form exp(-a/g), where a > 0
is the instanton action, and power series in the coupling g, as well as
logarithmic factors. The ground-state energy, for the specific Hamiltonians, is
shown to be dominated by instanton effects, and we provide numerical evidence
for the validity of the related conjectures.
