Multi-instantons and exact results I: conjectures, WKB expansions, and instanton interactions

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 non-analytic 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 it non-analytic factor and represents itself an infinite divergent series. We attempt to provide a unified representation or 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 Schrodinger 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. (C) 2004 Elsevier Inc. All rights reserved.