# From multi-instantons to exact results

Jean Zinn-Justin^{[1]}

- [1] CEA, Service de Physique Théorique de Saclay, CNRS URA 2306, 91191 Gif-sur-Yvette Cedex (France)

Annales de l’institut Fourier (2003)

- Volume: 53, Issue: 4, page 1259-1285
- ISSN: 0373-0956

## Access Full Article

top## Abstract

top## How to cite

topZinn-Justin, Jean. "From multi-instantons to exact results." Annales de l’institut Fourier 53.4 (2003): 1259-1285. <http://eudml.org/doc/116067>.

@article{Zinn2003,

abstract = {In these notes, conjectures about the exact semi-classical expansion of eigenvalues of
hamiltonians corresponding to potentials with degenerate minima, are recalled. They were
initially motivated by semi-classical calculations of quantum partition functions using a
path integral representation and have later been proven to a large extent, using the
theory of resurgent functions. They take the form of generalized Bohr--Sommerfeld
quantization formulae. We explain here their relation with the corresponding WKB
expansion of the Schrödinger equation. We show how these conjectures naturally emerge
from an evaluation of multi-instanton contributions in the path integral formulation of
quantum mechanics.},

affiliation = {CEA, Service de Physique Théorique de Saclay, CNRS URA 2306, 91191 Gif-sur-Yvette Cedex (France)},

author = {Zinn-Justin, Jean},

journal = {Annales de l’institut Fourier},

keywords = {singular perturbations; turning point theory; WKB methods; resurgence phenomena},

language = {eng},

number = {4},

pages = {1259-1285},

publisher = {Association des Annales de l'Institut Fourier},

title = {From multi-instantons to exact results},

url = {http://eudml.org/doc/116067},

volume = {53},

year = {2003},

}

TY - JOUR

AU - Zinn-Justin, Jean

TI - From multi-instantons to exact results

JO - Annales de l’institut Fourier

PY - 2003

PB - Association des Annales de l'Institut Fourier

VL - 53

IS - 4

SP - 1259

EP - 1285

AB - In these notes, conjectures about the exact semi-classical expansion of eigenvalues of
hamiltonians corresponding to potentials with degenerate minima, are recalled. They were
initially motivated by semi-classical calculations of quantum partition functions using a
path integral representation and have later been proven to a large extent, using the
theory of resurgent functions. They take the form of generalized Bohr--Sommerfeld
quantization formulae. We explain here their relation with the corresponding WKB
expansion of the Schrödinger equation. We show how these conjectures naturally emerge
from an evaluation of multi-instanton contributions in the path integral formulation of
quantum mechanics.

LA - eng

KW - singular perturbations; turning point theory; WKB methods; resurgence phenomena

UR - http://eudml.org/doc/116067

ER -

## References

top- J. Zinn-Justin, Multi-instanton contributions in quantum mechanics. II., Nucl. Phys. B 218 (1983), 333-348 MR702804
- J. Zinn-Justin, Instantons in quantum mechanics: numerical evidence for conjecture, J. Math. Phys 25 (1984), 549-555 MR737301
- J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, chap 43 (2002), Oxford Univ. Press, Oxford Zbl1033.81006MR1079938
- Analyse algébrique des perturbations singulières, Contribution to the Proceedings of the Franco-Japanese Colloquium Marseille-Luminy, Octobre 1991 47 (1994), Hermann, Paris
- F. Pham, Resurgence, Quantized Canonical Transformations, and Multi-Instanton Expansions, vol. II (1988) Zbl0686.58032MR992490
- C. Bender, T.T. Wu, semi-classical calculation of order g, Phys. Rev. D 7 (1973)
- J.C. Le, Guillou, J. Zinn-Justin, Large Order Behaviour of Perturbation Theory, vol. 7 (1990), North-Holland, Amsterdam
- U.D. Jentschura, J. Zinn-Justin, Higher-order corrections to instantons, J. Phys. A 34 (2001) Zbl0998.81022MR1840837
- R. Seznec, J. Zinn-Justin, Summation of divergent series by order dependent mappings: Application to the anharmonic oscillator and critical exponents in field theory, J. Math. Phys 20 (1979) Zbl0495.65002MR538715
- A.A. Andrianov, The large $N$ expansion as a local perturbation theory, Ann. Phys. (NY) 140 (1982) MR660926
- R. Damburg, R. Propin, V. Martyshchenko, Large-order perturbation theory for the $O\left(2\right)$ anharmonic oscillator with negative anharmonicity and for the double-well potential, J. Phys. A 17 (1984) Zbl0541.70025MR772336
- V. Buslaev, V. Grecchi, Equivalence of unstable anharmonic oscillators and double wells, J. Phys. A 26 (1993) Zbl0817.47077MR1248734
- A. Voros, The return of the quartic oscillator: the complex WKB method., Ann. IHP, A 39 (1983) Zbl0526.34046MR729194
- E. Brézin, G. Parisi, J. Zinn-Justin, Large order calculations in gauge theories, Phys. Rev. D 16 (1977)
- J. Zinn-Justin, Expansion around instantons in quatum mechanics, J. Math. Phys. 22 (1981), 511-520 MR611604
- E. Delabaere, Spectre de l'opérateur de Schrödinger stationnaire unidimensionnel à potentiel polynôme trigonométrique, C.R. Acad. Sci. Paris 314 (1992) Zbl0766.34060MR1166051
- F. Pham, Fonctions résurgentes implicites, C. R. Acad. Sci. Paris 309 (1989) Zbl0734.32001MR1054521
- E. Delabaere, H. Dillinger, F. Pham, Développements semi-classiques exacts des niveaux d'énergie d'un oscillateur à une dimension, C. R. Acad. Sci. Paris 310 (1990), 141-146 Zbl0712.35071MR1046892
- E. Delabaere, H. Dillinger, (1991)
- R. Damburg, R. Propin, J. Chem. Phys. 55 (1971)
- E.B. Bogomolny, V.A. Fateev, Large order calculations in gauge theories, Phys. Lett. B 71 (1977) MR496011

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.