[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
Zinn-Justin, Jean 1
@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},
zbl = {1073.81043},
mrnumber = {2033515},
language = {en},
url = {https://aif.centre-mersenne.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 - https://aif.centre-mersenne.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 https://aif.centre-mersenne.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, Tome 53 (2003) no. 4, pp. 1259-1285. doi : 10.5802/aif.1979. https://aif.centre-mersenne.org/articles/10.5802/aif.1979/
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