  We consider specific quantum mechanical model problems for which perturbation
theory fails to explain physical properties like the eigenvalue spectrum even
qualitatively, even if the asymptotic perturbation series is augmented by
resummation prescriptions to "cure" the divergence in large orders of
perturbation theory. Generalizations of perturbation theory are necessary which
include instanton configurations, characterized by nonanalytic factors
exp(-a/g) where a is a constant and g is the coupling. In the case of
one-dimensional quantum mechanical potentials with two or more degenerate
minima, the energy levels may be represented as an infinite sum of terms each
of which involves a certain power of a nonanalytic factor and represents itself
an infinite divergent series. We attempt to provide a unified representation of
related derivations previously found scattered in the literature. For the
considered quantum mechanical problems, we discuss the derivation of the
instanton contributions from a semi-classical calculation of the corresponding
partition function in the path integral formalism. We also explain the relation
with the corresponding WKB expansion of the solutions of the Schroedinger
equation, or alternatively of the Fredholm determinant det(H-E) (and some
explicit calculations that verify this correspondence). We finally recall how
these conjectures naturally emerge from a leading-order summation of
multi-instanton contributions to the path integral representation of the
partition function. The same strategy could result in new conjectures for
problems where our present understanding is more limited.
