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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.


[1] J. Zinn-Justin, “Multi-instanton contributions in quantum mechanics. II.”, Nucl. Phys. B 218 (1983), p. 333-348 Article |  MR 702804
[2] J. Zinn-Justin, “Instantons in quantum mechanics: numerical evidence for conjecture”, J. Math. Phys 25 (1984) no. 3, p. 549-555 Article |  MR 737301
[3] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena chap 43, Oxford Univ. Press, Oxford, 2002  MR 1079938 |  Zbl 1033.81006
[4] , Analyse algébrique des perturbations singulières, Collection Travaux en cours, Hermann, 1994
[5] F. Pham, Resurgence, Quantized Canonical Transformations, and Multi-Instanton Expansions, Algebraic Analysis vol. II, 1988  MR 992490 |  Zbl 0686.58032
[7] C. Bender & T.T. Wu, “semi-classical calculation of order g”, Phys. Rev. D 7 (1973)
[8] J.C. Le, Guillou & J. Zinn-Justin, Large Order Behaviour of Perturbation Theory, Current Physics vol. 7, North-Holland, Amsterdam, 1990
[9] U.D. Jentschura & J. Zinn-Justin, “Higher-order corrections to instantons”, J. Phys. A 34 (2001)  MR 1840837 |  Zbl 0998.81022
[10] 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)  MR 538715 |  Zbl 0495.65002
[11] A.A. Andrianov, “The large $N$ expansion as a local perturbation theory”, Ann. Phys. (NY) 140 (1982) Article |  MR 660926
[12] R. Damburg, R. Propin & V. Martyshchenko, “Large-order perturbation theory for the $O(2)$ anharmonic oscillator with negative anharmonicity and for the double-well potential”, J. Phys. A 17 (1984)  MR 772336 |  Zbl 0541.70025
[13] V. Buslaev & V. Grecchi, “Equivalence of unstable anharmonic oscillators and double wells”, J. Phys. A 26 (1993)  MR 1248734 |  Zbl 0817.47077
[14] A. Voros, “The return of the quartic oscillator: the complex WKB method.”, Ann. IHP, A 39 (1983) Numdam |  MR 729194 |  Zbl 0526.34046
[15] E. Brézin, G. Parisi & J. Zinn-Justin, “Large order calculations in gauge theories”, Phys. Rev. D 16 (1977)
[<L>1] J. Zinn-Justin, “Expansion around instantons in quatum mechanics”, J. Math. Phys. 22 (1981), p. 511-520 Article |  MR 611604
[6] E. Delabaere, “Spectre de l'opérateur de Schrödinger stationnaire unidimensionnel à potentiel polynôme trigonométrique”, C.R. Acad. Sci. Paris 314 (1992)  MR 1166051 |  Zbl 0766.34060
[<L>5] F. Pham, “Fonctions résurgentes implicites”, C. R. Acad. Sci. Paris 309 (1989)  MR 1054521 |  Zbl 0734.32001
[<L>5] 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), p. 141-146  MR 1046892 |  Zbl 0712.35071
[<L>5] E. Delabaere & H. Dillinger, “”, Thesis Université de Nice, 1991
[<L>7] R. Damburg & R. Propin, J. Chem. Phys. 55 (1971)
[<L>15] E.B. Bogomolny & V.A. Fateev, “Large order calculations in gauge theories”, Phys. Lett. B 71 (1977)  MR 496011