From multi-instantons to exact results
Annales de l'Institut Fourier, Volume 53 (2003) no. 4, p. 1259-1285
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.
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.
DOI : https://doi.org/10.5802/aif.1979
Classification:  34E20,  34M37,  41A60,  81Q20
Keywords: singular perturbations, turning point theory, WKB methods, resurgence phenomena
@article{AIF_2003__53_4_1259_0,
     author = {Zinn-Justin, Jean},
     title = {From multi-instantons to exact results},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {53},
     number = {4},
     year = {2003},
     pages = {1259-1285},
     doi = {10.5802/aif.1979},
     mrnumber = {2033515},
     zbl = {1073.81043},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2003__53_4_1259_0}
}
Zinn-Justin, Jean. From multi-instantons to exact results. Annales de l'Institut Fourier, Volume 53 (2003) no. 4, pp. 1259-1285. doi : 10.5802/aif.1979. https://aif.centre-mersenne.org/item/AIF_2003__53_4_1259_0/

[1] J. Zinn-Justin Multi-instanton contributions in quantum mechanics. II., Nucl. Phys. B, Tome 218 (1983), pp. 333-348 | Article | MR 702804

[1] J. Zinn-Justin Expansion around instantons in quatum mechanics, J. Math. Phys., Tome 22 (1981), pp. 511-520 | Article | MR 611604

[2] J. Zinn-Justin Instantons in quantum mechanics: numerical evidence for conjecture, J. Math. Phys, Tome 25 (1984) no. 3, pp. 549-555 | Article | MR 737301

[3] J. Zinn-Justin Quantum Field Theory and Critical Phenomena, Oxford Univ. Press, Oxford Tome chap 43 (2002) | MR 1079938 | Zbl 1033.81006

[4] L. Boutet De Monvel Ed. Analyse algébrique des perturbations singulières, Contribution to the Proceedings of the Franco-Japanese Colloquium Marseille-Luminy, Octobre 1991, Hermann, Paris (Collection Travaux en cours) Tome 47 (1994)

[5] F. Pham Resurgence, Quantized Canonical Transformations, and Multi-Instanton Expansions, Algebraic Analysis, Tome vol. II (1988) | MR 992490 | Zbl 0686.58032

[5] F. Pham Fonctions résurgentes implicites, C. R. Acad. Sci. Paris, Tome 309 (1989), pp. 999 | MR 1054521 | Zbl 0734.32001

[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, Tome 310 (1990), pp. 141-146 | MR 1046892 | Zbl 0712.35071

[5] E. Delabaere; H. Dillinger (1991) (Thesis Université de Nice)

[6] E. Delabaere Spectre de l'opérateur de Schrödinger stationnaire unidimensionnel à potentiel polynôme trigonométrique, C.R. Acad. Sci. Paris, Tome 314 (1992), pp. 807 | MR 1166051 | Zbl 0766.34060

[7] C. Bender; T.T. Wu semi-classical calculation of order g, Phys. Rev. D, Tome 7 (1973), pp. 1620

[7] R. Damburg; R. Propin J. Chem. Phys. Tome 55 (1971), pp. 612

[8] J.C. Le; Guillou; J. Zinn-Justin Large Order Behaviour of Perturbation Theory, North-Holland, Amsterdam, Current Physics, Tome vol. 7 (1990)

[9] U.D. Jentschura; J. Zinn-Justin Higher-order corrections to instantons, J. Phys. A, Tome 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, Tome 20 (1979), pp. 1398 | MR 538715 | Zbl 0495.65002

[11] A.A. Andrianov The large N expansion as a local perturbation theory, Ann. Phys. (NY), Tome 140 (1982), pp. 82 | 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, Tome 17 (1984), pp. 3493 | MR 772336 | Zbl 0541.70025

[13] V. Buslaev; V. Grecchi Equivalence of unstable anharmonic oscillators and double wells, J. Phys. A, Tome 26 (1993), pp. 5541 | MR 1248734 | Zbl 0817.47077

[14] A. Voros The return of the quartic oscillator: the complex WKB method., Ann. IHP, A, Tome 39 (1983), pp. 211 | Numdam | MR 729194 | Zbl 0526.34046

[15] E. Brézin; G. Parisi; J. Zinn-Justin Large order calculations in gauge theories, Phys. Rev. D, Tome 16 (1977), pp. 408

[15] E.B. Bogomolny; V.A. Fateev Large order calculations in gauge theories, Phys. Lett. B, Tome 71 (1977), pp. 93 | MR 496011