Multi-Instantons and Exact Results I: Conjectures, WKB Expansions, and Instanton Interactions

摘要

We consider specific quantum mechanical model problems for which perturbationtheory fails to explain physical properties like the eigenvalue spectrum evenqualitatively, even if the asymptotic perturbation series is augmented byresummation prescriptions to "cure" the divergence in large orders ofperturbation theory. Generalizations of perturbation theory are necessary whichinclude instanton configurations, characterized by nonanalytic factorsexp(-a/g) where a is a constant and g is the coupling. In the case ofone-dimensional quantum mechanical potentials with two or more degenerateminima, the energy levels may be represented as an infinite sum of terms eachof which involves a certain power of a nonanalytic factor and represents itselfan infinite divergent series. We attempt to provide a unified representation ofrelated derivations previously found scattered in the literature. For theconsidered quantum mechanical problems, we discuss the derivation of theinstanton contributions from a semi-classical calculation of the correspondingpartition function in the path integral formalism. We also explain the relationwith the corresponding WKB expansion of the solutions of the Schroedingerequation, or alternatively of the Fredholm determinant det(H-E) (and someexplicit calculations that verify this correspondence). We finally recall howthese conjectures naturally emerge from a leading-order summation ofmulti-instanton contributions to the path integral representation of thepartition function. The same strategy could result in new conjectures forproblems where our present understanding is more limited.