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Jean Zinn-Justin
From multi-instantons to exact results
(Analyse des instantons et résultats exacts)
Annales de l'institut Fourier, 53 no. 4 (2003), p. 1259-1285, doi: 10.5802/aif.1979
Article PDF | Reviews MR 2033515 | Zbl 1073.81043
Class. Math.: 34E20, 34M37, 41A60, 81Q20
Keywords: singular perturbations, turning point theory, WKB methods, resurgence phenomena

Résumé - 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.

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