Perturbative expansions in quantum mechanics
Annales de l'Institut Fourier, Volume 59 (2009) no. 5, pp. 2061-2101.

We prove a D=1 analytic versal deformation theorem in the Heisenberg algebra. We define the spectrum of an element in the Heisenberg algebra. The quantised version of the Morse lemma already shows that the perturbation series arising in a perturbed harmonic oscillator become analytic after a formal Borel transform.

Nous démontrons un théorème de déformation verselle analytique pour l’algèbre de Heisenberg dans le cas D=1. Nous définissons le spectre d’un élément dans cette algèbre. La quantification du lemme de Morse montre que les séries perturbatives du spectre de l’oscillateur harmonique deviennent analytique après une transformation de Borel formelle.

DOI: 10.5802/aif.2483
Classification: 81Q15
Keywords: Harmonic oscillator, Borel summability, micro-local analysis, non-commutative geometry
Mot clés : oscillateur harmonique, sommabilité de Borel, analyse semi-classique, formes normales
Garay, Mauricio D. 1

1 2A, avenue Édouard Herriot 91440 Bures-sur-Yvette (France)
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Garay, Mauricio D. Perturbative expansions in  quantum mechanics. Annales de l'Institut Fourier, Volume 59 (2009) no. 5, pp. 2061-2101. doi : 10.5802/aif.2483. https://aif.centre-mersenne.org/articles/10.5802/aif.2483/

[1] Arnold, V. I.; Varchenko, A. N.; Goussein-Zade, S. Singularity of differentiable mapping, vol. I Nauka:Moscow, 1982, English transl.: Birkhauser, 382p., Basel (1986)

[2] Arnold, V. I.; Varchenko, A. N.; Goussein-Zade, S. Singularity of differentiable mapping, vol. II Nauka:Moscow, 1982, English transl.: Birkhauser, 382p., Basel(1986)

[3] Birkhoff, G. D. Dynamical systems, Colloquium Publications, IX, American Mathematical Society, Providence R.I., 1927 | MR

[4] Born, M.; Heisenberg, W.; Jordan, P. Zur Quantenmechaniks II, Z. Phys., Volume 35 (1926), pp. 557-615 | DOI

[5] Born, M.; Jordan, P. Zur Quantenmechaniks, Zeit.für Phys., Volume 34 (1925), pp. 858-888 | DOI

[6] Bourbaki, N. Espaces vectoriels topologiques, Hermann, 1966

[7] Brieskorn, E. Die Monodromie der isolierten Singularitäten von Hyperflächen, Manuscr. Math., Volume 2 (1970), pp. 103-161 | DOI | MR | Zbl

[8] Colin de Verdière, Y. Singular lagrangian manifolds and semi-classical analysis, Duke Math. Journal, Volume 116 (2003) no. 2, pp. 263-298 | DOI | MR | Zbl

[9] Colin de Verdière, Y.; Parisse, B. Equilibres instables en régime semi-classique I: concentration micro-locale, Comm. PDE, Volume 19 (1994), pp. 1535-1564 | DOI | MR | Zbl

[10] Deligne, P. Déformations de l’algèbre des fonctions d’une variété symplectique: comparaison entre Fedosov et De Wilde, Lecomte, Selecta Math. (N.S.), Volume 1 (1995) no. 4, pp. 667-697 | DOI | MR | Zbl

[11] Dieudonné, J.; Schwartz, L. La dualité dans les espaces () et (), Annales de l’Institut Fourier, Volume 1 (1949), pp. 61-101 | DOI | Numdam | Zbl

[12] Dirac, P. A. M. The fundamental equations of quantum mechanics, Proc. Roy. Soc. A, Volume 109 (1926), pp. 642-653

[13] Eisenbud, D. Commutative algebra with a view towards algebraic geometry, Springer, 1999 (797 pp.) | MR | Zbl

[14] Garay, M. D. Finiteness and constructibility in local analytic geometry (math.AG/0610409, To appear in L’Enseignement Mathématique)

[15] Garay, M. D. An isochore versal deformation theorem, Topology, Volume 43 (2004) no. 5, pp. 1081-1088 | DOI | MR | Zbl

[16] Garay, M. D. Analytic quantum mechanics, 2005 (math-ph/0502027)

[17] Garay, M. D. Analytic geometry and semi-classical analysis, Proceedings of the Steklov Insitute of Mathematics, Volume 259 (2007), pp. 35-59 | DOI | MR | Zbl

[18] Grothendieck, A. Topological vector spaces (Gordon and Breach, 1973, 245 p., English Translation: Espaces vectoriels topologiques, São Paulo 1954) | Zbl

[19] Grothendieck, A. Résumé des résultats essentiels dans la théorie des produits tensoriels topologiques et des espaces nucléaires, Annales de l’Institut Fourier (1952), pp. 73-112 | DOI | Numdam | MR | Zbl

[20] Heisenberg, W. Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, Zeitschrift für Physik, Volume 33 (1925), pp. 879-893 | DOI

[21] Helffer, B.; Sjöstrand, J. Semiclassical analysis for Harper’s equation. III. Cantor structure of the spectrum, Mémoire de la Société Mathématique de France, Volume 39 (1989), pp. 1-124 | Numdam | MR | Zbl

[22] Houzel, C. Espaces analytiques relatifs et théorème de finitude, Math. Annalen, Volume 205 (1973), pp. 13-54 | DOI | MR | Zbl

[23] Kiehl, R.; Verdier, J. L. Ein Einfacher Beweis des Kohärenzsatzes von Grauert, Math. Annalen, Volume 195 (1971), pp. 24-50 | DOI | MR | Zbl

[24] Looijenga, E. J. N.; Press, Cambridge University Isolated singular points on complete intersections, Lect. Notes Series, London Math. Society, 1984 no. 77, pp. 200 pp. | MR | Zbl

[25] Malgrange, B. Intégrales asymptotiques et monodromie, Ann. Scient. École Norm. Sup., Volume 7 (1974) no. 4, pp. 405-430 | Numdam | MR | Zbl

[26] Malgrange, B. Sommation des séries divergentes, Expositiones Mathematicae, Volume 13 (1995) no. 2/3, pp. 163-222 | MR | Zbl

[27] Martinet, J. Singularities of smooth functions and maps, Lecture Notes Series, Volume 58, Cambridge University Press, 1982, pp. 272 pp. | MR | Zbl

[28] Mather, J. Stratifications and mappings Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, 1973, pp. 195–232 | Zbl

[29] Moyal, J. E. Quantum mechanics as a statistical theory, Proc. Cambridge Philos. Soc., Volume 45 (1949), pp. 99-124 | DOI | MR | Zbl

[30] Pham, F. Multiple turning points in exact WKB analysis (variations on a theme of Stokes) Towards the exact WKB analysis of differential equations, linear or non linear (C. Howls, T. Kawai, and Y. Takei, eds.), Kyoto University Press, 2000, pp. 71–85 | Zbl

[31] Pham, F. Resurgence, quantized canonical transformations, and multi-instanton expansions Algebraic analysis (M. Kashiwara and T. Kawai, eds.), vol. II, Academic Press, Boston, MA, 1988, Papers dedicated to Professor Mikio Sato on the occasion of his sixtieth birthday, pp. 699–726 | Zbl

[32] Polesello, P.; Schapira, P. Stacks of quantization-deformation modules on complex symplectic manifolds, Int. Math. Research Notices, Volume 49 (2004), pp. 2637-2664 | DOI | MR | Zbl

[33] Reed, M.; Simon, B. Methods of modern mathematical physics, vol. IV, Academic Press, 1978 | MR | Zbl

[34] Simon, B. Borel summability of the ground state energy in spatially cutoff (ϕ 4 ) 2 , Physical Review letters, Volume 25 (1970) no. 22, pp. 1583-1586 | DOI | MR

[35] Simon, B. Determination of eigenvalues by divergent perturbation series, Advances in Mathematics, Volume 7 (1971), pp. 240-253 | DOI | MR | Zbl

[36] Sjöstrand, J. Singularités analytiques microlocales, Astérisque, Volume 95 (1982), pp. 1-166 | MR | Zbl

[37] Vey, J. Sur le lemme de Morse, Invent. Math., Volume 40 (1977) no. 1, pp. 1-9 | DOI | MR | Zbl

[38] Voros, A. Exact quantization condition for anharmonic oscillators (in one dimension), J. Phys. A, Volume 27 (1994), pp. 4653-4661 | DOI | MR | Zbl

[39] der Waerden (ed.), van Sources of quantum mechanics, Dover, 1968 | Zbl

[40] Zinn-Justin, J. Multi-instanton contributions in quantum mechanics, 2, Nucl.Phys. B, Volume 218 (1983), pp. 333-348 | DOI | MR

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