  Recently, it is realized that non-perturbative instanton effects can be
generated to all orders by perturbation theory around a degenerate minima via
Dunne-Unsal relation in several quantum mechanical systems. In this work we
verify the Dunne-Unsal relation for resonance energy levels of one-dimensional
polynomial anharmonic oscillators. We show that the relation is applicable to
cubic and quartic anharmonic oscillators which are genus one potentials.
However for higher order (higher genus) anharmonic potentials the relation is
not satisfied and is subject to a certain extension.
