From multi-instantons to exact results

Zinn-Justin, Jean

Annales de l'Institut Fourier, Tome 53 (2003) no. 4, pp. 1259-1285.

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.

Classification : 34E20, 34M37, 41A60, 81Q20
Keywords: singular perturbations, turning point theory, WKB methods, resurgence phenomena
Mot clés : perturbations singulières, théorie du point tournant, méthodes WKB, phénomène de résurgence

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