Table of contents for this issue | Previous article | Next article
Mauricio D. Garay
Perturbative expansions in quantum mechanics
(Séries perturbatives en mécanique quantique)
Annales de l'institut Fourier, 59 no. 5 (2009), p. 2061-2101, doi: 10.5802/aif.2483
Article: subscription required (your ip address: 50.16.108.167) | Reviews MR 2573197 | Zbl pre05641408
Class. Math.: 81Q15
Keywords: Harmonic oscillator, Borel summability, micro-local analysis, non-commutative geometry
[1] V. I. Arnold, A. N. Varchenko & S. Goussein-Zade, “Singularity of differentiable mapping, vol. I”, Nauka:Moscow, 1982, English transl.: Birkhauser, 382p., Basel (1986)
[2] V. I. Arnold, A. N. Varchenko & S. Goussein-Zade, “Singularity of differentiable mapping, vol. II”, Nauka:Moscow, 1982, English transl.: Birkhauser, 382p., Basel(1986)
[3] G. D. Birkhoff, Dynamical systems, Colloquium Publications IX, American Mathematical Society, 1927 MR 209095 | JFM 53.0732.01
[4] M. Born, W. Heisenberg & P. Jordan, “Zur Quantenmechaniks II”, Z. Phys. 35 (1926), p. 557-615
Article | JFM 52.0963.01
[5] M. Born & P. Jordan, “Zur Quantenmechaniks”, Zeit.für Phys. 34 (1925), p. 858-888
Article | JFM 51.0728.08
[6] N. Bourbaki, “Espaces vectoriels topologiques”, Hermann, 1966
[7] E. Brieskorn, “Die Monodromie der isolierten Singularitäten von Hyperflächen”, Manuscr. Math. 2 (1970), p. 103-161
Article | MR 267607 | Zbl 0186.26101
[8] Y. Colin de Verdière, “Singular lagrangian manifolds and semi-classical analysis”, Duke Math. Journal 116 (2003) no. 2, p. 263-298
Article | MR 1953293 | Zbl 1074.53066
[9] Y. Colin de Verdière & B. Parisse, “Equilibres instables en régime semi-classique I: concentration micro-locale”, Comm. PDE 19 (1994), p. 1535-1564
Article | MR 1294470 | Zbl 0819.35116
[10] P. Deligne, “Déformations de l’algèbre des fonctions d’une variété symplectique: comparaison entre Fedosov et De Wilde, Lecomte”, Selecta Math. (N.S.) 1 (1995) no. 4, p. 667-697
Article | MR 1383583 | Zbl 0852.58033
[11] J. Dieudonné & L. Schwartz, “La dualité dans les espaces $(\mathcal{F})$ et $(\mathcal{L}\mathcal{F})$”, Annales de l’Institut Fourier 1 (1949), p. 61-101
Cedram | Zbl 0035.35501
[12] P. A. M. Dirac, “The fundamental equations of quantum mechanics”, Proc. Roy. Soc. A 109 (1926), p. 642-653 JFM 51.0729.01
[13] D. Eisenbud, Commutative algebra with a view towards algebraic geometry, Springer, 1999, 797 pp. MR 1322960 | Zbl 0819.13001
[14] M. D. Garay, “Finiteness and constructibility in local analytic geometry”, math.AG/0610409, To appear in L’Enseignement Mathématique
arXiv
[15] M. D. Garay, “An isochore versal deformation theorem”, Topology 43 (2004) no. 5, p. 1081-1088
Article | MR 2079995 | Zbl 1100.32010
[16] M. D. Garay, “Analytic quantum mechanics”, math-ph/0502027, 2005
[17] M. D. Garay, “Analytic geometry and semi-classical analysis”, Proceedings of the Steklov Insitute of Mathematics 259 (2007), p. 35-59
Article | MR 2433676 | Zbl 1161.58013
[18] A. Grothendieck, “Topological vector spaces”, Gordon and Breach, 1973, 245 p., English Translation: Espaces vectoriels topologiques, São Paulo 1954 Zbl 0763.46002
[19] A. Grothendieck, “Résumé des résultats essentiels dans la théorie des produits tensoriels topologiques et des espaces nucléaires”, Annales de l’Institut Fourier (1952), p. 73-112
Cedram | MR 61754 | Zbl 0055.09705
[20] W. Heisenberg, “Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen”, Zeitschrift für Physik 33 (1925), p. 879-893
Article | JFM 51.0728.07
[21] B. Helffer & J. Sjöstrand, “Semiclassical analysis for Harper’s equation. III. Cantor structure of the spectrum”, Mémoire de la Société Mathématique de France 39 (1989), p. 1-124
Numdam | MR 1041490 | Zbl 0725.34099
[22] C. Houzel, “Espaces analytiques relatifs et théorème de finitude”, Math. Annalen 205 (1973), p. 13-54
Article | MR 393552 | Zbl 0264.32012
[23] R. Kiehl & J. L. Verdier, “Ein Einfacher Beweis des Kohärenzsatzes von Grauert”, Math. Annalen 195 (1971), p. 24-50
Article | MR 306555 | Zbl 0223.32010
[24] E. J. N. Looijenga, Isolated singular points on complete intersections, in Cambridge University Press, ed., Lect. Notes Series, London Math. Society, 1984, p. 200 pp. MR 747303 | Zbl 0552.14002
[25] B. Malgrange, “Intégrales asymptotiques et monodromie”, Ann. Scient. École Norm. Sup. 7 (1974) no. 4, p. 405-430
Numdam | MR 372243 | Zbl 0305.32008
[26] B. Malgrange, “Sommation des séries divergentes”, Expositiones Mathematicae 13 (1995) no. 2/3, p. 163-222 MR 1346201 | Zbl 0836.40004
[27] J. Martinet, Singularities of smooth functions and maps, Lecture Notes Series, Cambridge University Press, 1982, p. 272 pp. MR 671585 | Zbl 0522.58006
[28] J. Mather, “Stratifications and mappings”, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, 1973, pp. 195–232 Zbl 0253.58005
[29] J. E. Moyal, “Quantum mechanics as a statistical theory”, Proc. Cambridge Philos. Soc. 45 (1949), p. 99-124
Article | MR 29330 | Zbl 0031.33601
[30] F. Pham, “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 1017.34091
[31] F. Pham, “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 0686.58032
[32] P. Polesello & P. Schapira, “Stacks of quantization-deformation modules on complex symplectic manifolds”, Int. Math. Research Notices 49 (2004), p. 2637-2664
Article | MR 2077680 | Zbl 1086.53107
[33] M. Reed & B. Simon, “Methods of modern mathematical physics, vol. IV”, Academic Press, 1978 MR 493422 | Zbl 0401.47001
[34] B. Simon, “Borel summability of the ground state energy in spatially cutoff $(\varphi ^4)_2$”, Physical Review letters 25 (1970) no. 22, p. 1583-1586
Article | MR 395601
[35] B. Simon, “Determination of eigenvalues by divergent perturbation series”, Advances in Mathematics 7 (1971), p. 240-253
Article | MR 300138 | Zbl 0244.47008
[36] J. Sjöstrand, “Singularités analytiques microlocales”, Astérisque 95 (1982), p. 1-166 MR 699623 | Zbl 0524.35007
[37] J. Vey, “Sur le lemme de Morse”, Invent. Math. 40 (1977) no. 1, p. 1-9
Article | MR 453737 | Zbl 0348.58007
[38] A. Voros, “Exact quantization condition for anharmonic oscillators (in one dimension)”, J. Phys. A 27 (1994), p. 4653-4661
Article | MR 1294967 | Zbl 0842.34090
[39] van der Waerden (ed.), Sources of quantum mechanics, Dover, 1968 Zbl 1140.81002
[40] J. Zinn-Justin, “Multi-instanton contributions in quantum mechanics, 2”, Nucl.Phys. B 218 (1983), p. 333-348
Article | MR 702804
Mauricio D. Garay
Perturbative expansions in quantum mechanics
(Séries perturbatives en mécanique quantique)
Annales de l'institut Fourier, 59 no. 5 (2009), p. 2061-2101, doi: 10.5802/aif.2483
Article: subscription required (your ip address: 50.16.108.167) | Reviews MR 2573197 | Zbl pre05641408
Class. Math.: 81Q15
Keywords: Harmonic oscillator, Borel summability, micro-local analysis, non-commutative geometry
Résumé - Abstract
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.
Bibliography
[2] V. I. Arnold, A. N. Varchenko & S. Goussein-Zade, “Singularity of differentiable mapping, vol. II”, Nauka:Moscow, 1982, English transl.: Birkhauser, 382p., Basel(1986)
[3] G. D. Birkhoff, Dynamical systems, Colloquium Publications IX, American Mathematical Society, 1927 MR 209095 | JFM 53.0732.01
[4] M. Born, W. Heisenberg & P. Jordan, “Zur Quantenmechaniks II”, Z. Phys. 35 (1926), p. 557-615
Article | JFM 52.0963.01
[5] M. Born & P. Jordan, “Zur Quantenmechaniks”, Zeit.für Phys. 34 (1925), p. 858-888
Article | JFM 51.0728.08
[6] N. Bourbaki, “Espaces vectoriels topologiques”, Hermann, 1966
[7] E. Brieskorn, “Die Monodromie der isolierten Singularitäten von Hyperflächen”, Manuscr. Math. 2 (1970), p. 103-161
Article | MR 267607 | Zbl 0186.26101
[8] Y. Colin de Verdière, “Singular lagrangian manifolds and semi-classical analysis”, Duke Math. Journal 116 (2003) no. 2, p. 263-298
Article | MR 1953293 | Zbl 1074.53066
[9] Y. Colin de Verdière & B. Parisse, “Equilibres instables en régime semi-classique I: concentration micro-locale”, Comm. PDE 19 (1994), p. 1535-1564
Article | MR 1294470 | Zbl 0819.35116
[10] P. Deligne, “Déformations de l’algèbre des fonctions d’une variété symplectique: comparaison entre Fedosov et De Wilde, Lecomte”, Selecta Math. (N.S.) 1 (1995) no. 4, p. 667-697
Article | MR 1383583 | Zbl 0852.58033
[11] J. Dieudonné & L. Schwartz, “La dualité dans les espaces $(\mathcal{F})$ et $(\mathcal{L}\mathcal{F})$”, Annales de l’Institut Fourier 1 (1949), p. 61-101
Cedram | Zbl 0035.35501
[12] P. A. M. Dirac, “The fundamental equations of quantum mechanics”, Proc. Roy. Soc. A 109 (1926), p. 642-653 JFM 51.0729.01
[13] D. Eisenbud, Commutative algebra with a view towards algebraic geometry, Springer, 1999, 797 pp. MR 1322960 | Zbl 0819.13001
[14] M. D. Garay, “Finiteness and constructibility in local analytic geometry”, math.AG/0610409, To appear in L’Enseignement Mathématique
arXiv
[15] M. D. Garay, “An isochore versal deformation theorem”, Topology 43 (2004) no. 5, p. 1081-1088
Article | MR 2079995 | Zbl 1100.32010
[16] M. D. Garay, “Analytic quantum mechanics”, math-ph/0502027, 2005
[17] M. D. Garay, “Analytic geometry and semi-classical analysis”, Proceedings of the Steklov Insitute of Mathematics 259 (2007), p. 35-59
Article | MR 2433676 | Zbl 1161.58013
[18] A. Grothendieck, “Topological vector spaces”, Gordon and Breach, 1973, 245 p., English Translation: Espaces vectoriels topologiques, São Paulo 1954 Zbl 0763.46002
[19] A. Grothendieck, “Résumé des résultats essentiels dans la théorie des produits tensoriels topologiques et des espaces nucléaires”, Annales de l’Institut Fourier (1952), p. 73-112
Cedram | MR 61754 | Zbl 0055.09705
[20] W. Heisenberg, “Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen”, Zeitschrift für Physik 33 (1925), p. 879-893
Article | JFM 51.0728.07
[21] B. Helffer & J. Sjöstrand, “Semiclassical analysis for Harper’s equation. III. Cantor structure of the spectrum”, Mémoire de la Société Mathématique de France 39 (1989), p. 1-124
Numdam | MR 1041490 | Zbl 0725.34099
[22] C. Houzel, “Espaces analytiques relatifs et théorème de finitude”, Math. Annalen 205 (1973), p. 13-54
Article | MR 393552 | Zbl 0264.32012
[23] R. Kiehl & J. L. Verdier, “Ein Einfacher Beweis des Kohärenzsatzes von Grauert”, Math. Annalen 195 (1971), p. 24-50
Article | MR 306555 | Zbl 0223.32010
[24] E. J. N. Looijenga, Isolated singular points on complete intersections, in Cambridge University Press, ed., Lect. Notes Series, London Math. Society, 1984, p. 200 pp. MR 747303 | Zbl 0552.14002
[25] B. Malgrange, “Intégrales asymptotiques et monodromie”, Ann. Scient. École Norm. Sup. 7 (1974) no. 4, p. 405-430
Numdam | MR 372243 | Zbl 0305.32008
[26] B. Malgrange, “Sommation des séries divergentes”, Expositiones Mathematicae 13 (1995) no. 2/3, p. 163-222 MR 1346201 | Zbl 0836.40004
[27] J. Martinet, Singularities of smooth functions and maps, Lecture Notes Series, Cambridge University Press, 1982, p. 272 pp. MR 671585 | Zbl 0522.58006
[28] J. Mather, “Stratifications and mappings”, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, 1973, pp. 195–232 Zbl 0253.58005
[29] J. E. Moyal, “Quantum mechanics as a statistical theory”, Proc. Cambridge Philos. Soc. 45 (1949), p. 99-124
Article | MR 29330 | Zbl 0031.33601
[30] F. Pham, “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 1017.34091
[31] F. Pham, “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 0686.58032
[32] P. Polesello & P. Schapira, “Stacks of quantization-deformation modules on complex symplectic manifolds”, Int. Math. Research Notices 49 (2004), p. 2637-2664
Article | MR 2077680 | Zbl 1086.53107
[33] M. Reed & B. Simon, “Methods of modern mathematical physics, vol. IV”, Academic Press, 1978 MR 493422 | Zbl 0401.47001
[34] B. Simon, “Borel summability of the ground state energy in spatially cutoff $(\varphi ^4)_2$”, Physical Review letters 25 (1970) no. 22, p. 1583-1586
Article | MR 395601
[35] B. Simon, “Determination of eigenvalues by divergent perturbation series”, Advances in Mathematics 7 (1971), p. 240-253
Article | MR 300138 | Zbl 0244.47008
[36] J. Sjöstrand, “Singularités analytiques microlocales”, Astérisque 95 (1982), p. 1-166 MR 699623 | Zbl 0524.35007
[37] J. Vey, “Sur le lemme de Morse”, Invent. Math. 40 (1977) no. 1, p. 1-9
Article | MR 453737 | Zbl 0348.58007
[38] A. Voros, “Exact quantization condition for anharmonic oscillators (in one dimension)”, J. Phys. A 27 (1994), p. 4653-4661
Article | MR 1294967 | Zbl 0842.34090
[39] van der Waerden (ed.), Sources of quantum mechanics, Dover, 1968 Zbl 1140.81002
[40] J. Zinn-Justin, “Multi-instanton contributions in quantum mechanics, 2”, Nucl.Phys. B 218 (1983), p. 333-348
Article | MR 702804
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310