[Analyse des instantons et résultats exacts]
Dans ces notes nous rappelons des conjectures sur le développement semi-classique exact du spectre des hamiltoniens quantiques avec potentiels à minima dégénérés. Ces conjectures ont été initialement motivées par une évaluation semi-classique d'intégrales de chemin. Elles prennent la forme d'une formule de quantification de Bohr-Sommerfeld modifiée. Nous expliquons ici leurs relations avec un développement de l'équation de Schrodinger. Nous montrons comment ces conjectures apparaîssent naturellement dans un calcul des contributions de type instanton à l'intégrale de chemin.
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.
Keywords: singular perturbations, turning point theory, WKB methods, resurgence phenomena
Mot clés : perturbations singulières, théorie du point tournant, méthodes WKB, phénomène de résurgence
@article{AIF_2003__53_4_1259_0,
author = {Zinn-Justin, Jean},
title = {From multi-instantons to exact results},
journal = {Annales de l'Institut Fourier},
pages = {1259--1285},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {53},
number = {4},
year = {2003},
doi = {10.5802/aif.1979},
mrnumber = {2033515},
zbl = {1073.81043},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/aif.1979/}
}
TY - JOUR AU - Zinn-Justin, Jean TI - From multi-instantons to exact results JO - Annales de l'Institut Fourier PY - 2003 SP - 1259 EP - 1285 VL - 53 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.1979/ DO - 10.5802/aif.1979 LA - en ID - AIF_2003__53_4_1259_0 ER -
%0 Journal Article %A Zinn-Justin, Jean %T From multi-instantons to exact results %J Annales de l'Institut Fourier %D 2003 %P 1259-1285 %V 53 %N 4 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.1979/ %R 10.5802/aif.1979 %G en %F AIF_2003__53_4_1259_0
Zinn-Justin, Jean. From multi-instantons to exact results. Annales de l'Institut Fourier, Colloque en l'honneur de Frédéric Pham, Tome 53 (2003) no. 4, pp. 1259-1285. doi : 10.5802/aif.1979. http://archive.numdam.org/articles/10.5802/aif.1979/
[1] Multi-instanton contributions in quantum mechanics. II., Nucl. Phys. B, Volume 218 (1983), pp. 333-348 | DOI | MR
[1] Expansion around instantons in quatum mechanics, J. Math. Phys., Volume 22 (1981), pp. 511-520 | DOI | MR
[2] Instantons in quantum mechanics: numerical evidence for conjecture, J. Math. Phys, Volume 25 (1984) no. 3, pp. 549-555 | DOI | MR
[3] Quantum Field Theory and Critical Phenomena, chap 43, Oxford Univ. Press, Oxford, 2002 | MR | Zbl
[4] Analyse algébrique des perturbations singulières, Contribution to the Proceedings of the Franco-Japanese Colloquium Marseille-Luminy, Octobre 1991 (Collection Travaux en cours), Volume 47 (1994)
[5] Resurgence, Quantized Canonical Transformations, and Multi-Instanton Expansions, Algebraic Analysis, vol. II, 1988 | MR | Zbl
[5] Fonctions résurgentes implicites, C. R. Acad. Sci. Paris, Volume 309 (1989), pp. 999 | MR | Zbl
[5] Développements semi-classiques exacts des niveaux d'énergie d'un oscillateur à une dimension, C. R. Acad. Sci. Paris, Volume 310 (1990), pp. 141-146 | MR | Zbl
[5] (1991) (Thesis Université de Nice)
[6] Spectre de l'opérateur de Schrödinger stationnaire unidimensionnel à potentiel polynôme trigonométrique, C.R. Acad. Sci. Paris, Volume 314 (1992), pp. 807 | MR | Zbl
[7] semi-classical calculation of order g, Phys. Rev. D, Volume 7 (1973), pp. 1620
[7] J. Chem. Phys., 55 (1971), pp. 612
[8] Large Order Behaviour of Perturbation Theory, Current Physics, vol. 7, North-Holland, Amsterdam, 1990
[9] Higher-order corrections to instantons, J. Phys. A, Volume 34 (2001) | MR | Zbl
[10] Summation of divergent series by order dependent mappings: Application to the anharmonic oscillator and critical exponents in field theory, J. Math. Phys, Volume 20 (1979), pp. 1398 | MR | Zbl
[11] The large expansion as a local perturbation theory, Ann. Phys. (NY), Volume 140 (1982), pp. 82 | DOI | MR
[12] Large-order perturbation theory for the anharmonic oscillator with negative anharmonicity and for the double-well potential, J. Phys. A, Volume 17 (1984), pp. 3493 | MR | Zbl
[13] Equivalence of unstable anharmonic oscillators and double wells, J. Phys. A, Volume 26 (1993), pp. 5541 | MR | Zbl
[14] The return of the quartic oscillator: the complex WKB method., Ann. IHP, A, Volume 39 (1983), pp. 211 | Numdam | MR | Zbl
[15] Large order calculations in gauge theories, Phys. Rev. D, Volume 16 (1977), pp. 408
[15] Large order calculations in gauge theories, Phys. Lett. B, Volume 71 (1977), pp. 93 | MR
Cité par Sources :
