Physics – Quantum Physics
Scientific paper
2005-01-24
Annals Phys.313:197-267,2004
Physics
Quantum Physics
66 pages, LaTeX; refs. to part II preprint updated
Scientific paper
10.1016/j.aop.2004.04.004
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.
Jentschura Ulrich D.
Zinn-Justin Jean
No associations
LandOfFree
Multi-Instantons and Exact Results I: Conjectures, WKB Expansions, and Instanton Interactions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Multi-Instantons and Exact Results I: Conjectures, WKB Expansions, and Instanton Interactions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Multi-Instantons and Exact Results I: Conjectures, WKB Expansions, and Instanton Interactions will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-513777