dorsal/arxiv
View SchemaMulti-Instantons and Exact Results I: Conjectures, WKB Expansions, and Instanton Interactions
| Authors | Jean Zinn-Justin, Ulrich D. Jentschura |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0501136 |
| URL | https://arxiv.org/abs/quant-ph/0501136 |
| DOI | 10.1016/j.aop.2004.04.004 |
| Journal | Ann.Phys.(N.Y.) 313 (2004) 197-267 |
Abstract
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.
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"abstract": "We consider specific quantum mechanical model problems for which perturbation\ntheory fails to explain physical properties like the eigenvalue spectrum even\nqualitatively, even if the asymptotic perturbation series is augmented by\nresummation prescriptions to \"cure\" the divergence in large orders of\nperturbation theory. Generalizations of perturbation theory are necessary which\ninclude instanton configurations, characterized by nonanalytic factors\nexp(-a/g) where a is a constant and g is the coupling. In the case of\none-dimensional quantum mechanical potentials with two or more degenerate\nminima, the energy levels may be represented as an infinite sum of terms each\nof which involves a certain power of a nonanalytic factor and represents itself\nan infinite divergent series. We attempt to provide a unified representation of\nrelated derivations previously found scattered in the literature. For the\nconsidered quantum mechanical problems, we discuss the derivation of the\ninstanton contributions from a semi-classical calculation of the corresponding\npartition function in the path integral formalism. We also explain the relation\nwith the corresponding WKB expansion of the solutions of the Schroedinger\nequation, or alternatively of the Fredholm determinant det(H-E) (and some\nexplicit calculations that verify this correspondence). We finally recall how\nthese conjectures naturally emerge from a leading-order summation of\nmulti-instanton contributions to the path integral representation of the\npartition function. The same strategy could result in new conjectures for\nproblems where our present understanding is more limited.",
"arxiv_id": "quant-ph/0501136",
"authors": [
"Jean Zinn-Justin",
"Ulrich D. Jentschura"
],
"categories": [
"quant-ph"
],
"doi": "10.1016/j.aop.2004.04.004",
"journal_ref": "Ann.Phys.(N.Y.) 313 (2004) 197-267",
"title": "Multi-Instantons and Exact Results I: Conjectures, WKB Expansions, and Instanton Interactions",
"url": "https://arxiv.org/abs/quant-ph/0501136"
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