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Mark D. Hamilton; Eva Miranda
Geometric quantization of integrable systems with hyperbolic singularities
(Quantification géométrique des systèmes intégrables avec singularités hyperboliques)
Annales de l'institut Fourier, 60 no. 1 (2010), p. 51-85
Article: sur abonnement
Class. Math.: 53D50
Mots clés: quantification géométrique, système intégrable, singularité non-dégénérée
[1] V. I. Arnold, S. M. Gusein-Zade & A. N. Varchenko, Singularities of Differentiable Maps, Monographs in Mathematics 1, 2, Birkhäuser, 1988 MR 966191 | Zbl 0659.58002
[2] A. V. Bolsinov & A. T. Fomenko, Integrable Hamiltonian systems: geometry, topology, classification, Chapman & Hall/CRC, 2004 MR 2036760 | Zbl 1056.37075
[3] Y. Colin de Verdière & J. Vey, “Le lemme de Morse isochore”, Topology 18 (1979), p. 283-293 MR 551010 | Zbl 0441.58003
[4] R. H. Cushman & L. M. Bates, Global aspects of classical integrable systems, Birkhäuser Verlag, Basel, 1997 MR 1438060 | Zbl 0882.58023
[5] J. P. Dufour, P. Molino & A. Toulet, “Classification des systèmes intégrables en dimension $2$ et invariants des modèles de Fomenko”, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), p. 949-952 MR 1278158 | Zbl 0808.58025
[6] L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals, thesis, Stockholm University, 1984
[7] L. H. Eliasson, “Normal forms for Hamiltonian systems with Poisson commuting integrals—elliptic case”, Comment. Math. Helv. 65 (1990), p. 4-35 MR 1036125 | Zbl 0702.58024
[8] V. Ginzburg, V. Guillemin & Y. Karshon, Moment maps, cobordisms, and Hamiltonian group actions, AMS Monographs, 2004 Zbl pre01842395
[9] V. Guillemin & S. Sternberg, “The Gel’fand-Cetlin system and quantization of the complex flag manifolds”, J. Funct. Anal. 52 (1983), p. 106-128 MR 705993 | Zbl 0522.58021
[10] M. Hamilton, “Locally toric manifolds and singular Bohr-Sommerfeld leaves”, to appear in Mem. AMS, http://arxiv.org/abs/0709.4058
arXiv
[11] B. Kostant, “On the Definition of Quantization”, Géométrie Symplectique et Physique Mathématique, Coll. CNRS, No. 237, Paris (1975), p. 187-210 MR 488137 | Zbl 0326.53047
[12] J. Marsden & T. Ratiu, Introduction to mechanics and symmetry: A basic exposition of classical mechanical systems, Second edition, Texts in Applied Mathematics 17, Springer-Verlag, New York, 1999 MR 1723696 | Zbl 0933.70003
[13] J. Milnor, Morse theory, Princeton University, 1963 MR 163331 | Zbl 0108.10401
[14] E. Miranda, On symplectic linearization of singular Lagrangian foliations, thesis, Univ. de Barcelona, 2003
[15] E. Miranda & San Vu Ngoc, “A singular Poincaré lemma”, IMRN 1 (2005), p. 27-46 MR 2130052 | Zbl 1078.58007
[16] H.-J. Petzsche, “On E. Borel’s Theorem”, Math. Ann. 282 (1988), p. 299-313 Zbl 0633.46033
[17] J. Rawnsley, “On the Cohomology Groups of a Polarization and Diagonal quantization”, Transaction of the American Mathematical Society 230 (1977), p. 235-255 MR 648775 | Zbl 0313.58016
[18] J. Śniatycki, On Cohomology Groups Appearing in Geometric Quantization, Differential Geometric Methods in Mathematical Physics, 1975 Zbl 0353.53019
[19] J. Śniatycki, Geometric quantization and quantum mechanics, Applied Mathematical Sciences 30, Springer-Verlag, New York-Berlin, 1980 MR 554085 | Zbl 0429.58007
[20] J. C. Tougeron, “Idéaux de fonctions différentiables”, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 71 (1972) MR 440598 | Zbl 0251.58001
[21] J. Williamson, “On the algebraic problem concerning the normal forms of linear dynamical systems”, Amer. J. Math. 58:1 (1936), p. 141-163 MR 1507138 | JFM 62.1795.10
[22] N. M. J. Woodhouse, Geometric quantization, Second edition. Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992 MR 1183739 | Zbl 0747.58004
Mark D. Hamilton; Eva Miranda
Geometric quantization of integrable systems with hyperbolic singularities
(Quantification géométrique des systèmes intégrables avec singularités hyperboliques)
Annales de l'institut Fourier, 60 no. 1 (2010), p. 51-85
Article: sur abonnement
Class. Math.: 53D50
Mots clés: quantification géométrique, système intégrable, singularité non-dégénérée
Résumé - Abstract
On construit la quantification géométrique d’une surface compacte en utilisant une polarisation singulière donnée par un système intégrable. Cette polarisation présente toujours des singularités qu’on suppose de type non-dégénéré. En particulier, on calcule l’effet des singularités hyperboliques qui donnent une contribution de dimension infinie à la quantification, en démontrant que cette quantification dépend fortement de la polarisation choisie.
Bibliographie
[2] A. V. Bolsinov & A. T. Fomenko, Integrable Hamiltonian systems: geometry, topology, classification, Chapman & Hall/CRC, 2004 MR 2036760 | Zbl 1056.37075
[3] Y. Colin de Verdière & J. Vey, “Le lemme de Morse isochore”, Topology 18 (1979), p. 283-293 MR 551010 | Zbl 0441.58003
[4] R. H. Cushman & L. M. Bates, Global aspects of classical integrable systems, Birkhäuser Verlag, Basel, 1997 MR 1438060 | Zbl 0882.58023
[5] J. P. Dufour, P. Molino & A. Toulet, “Classification des systèmes intégrables en dimension $2$ et invariants des modèles de Fomenko”, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), p. 949-952 MR 1278158 | Zbl 0808.58025
[6] L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals, thesis, Stockholm University, 1984
[7] L. H. Eliasson, “Normal forms for Hamiltonian systems with Poisson commuting integrals—elliptic case”, Comment. Math. Helv. 65 (1990), p. 4-35 MR 1036125 | Zbl 0702.58024
[8] V. Ginzburg, V. Guillemin & Y. Karshon, Moment maps, cobordisms, and Hamiltonian group actions, AMS Monographs, 2004 Zbl pre01842395
[9] V. Guillemin & S. Sternberg, “The Gel’fand-Cetlin system and quantization of the complex flag manifolds”, J. Funct. Anal. 52 (1983), p. 106-128 MR 705993 | Zbl 0522.58021
[10] M. Hamilton, “Locally toric manifolds and singular Bohr-Sommerfeld leaves”, to appear in Mem. AMS, http://arxiv.org/abs/0709.4058
arXiv
[11] B. Kostant, “On the Definition of Quantization”, Géométrie Symplectique et Physique Mathématique, Coll. CNRS, No. 237, Paris (1975), p. 187-210 MR 488137 | Zbl 0326.53047
[12] J. Marsden & T. Ratiu, Introduction to mechanics and symmetry: A basic exposition of classical mechanical systems, Second edition, Texts in Applied Mathematics 17, Springer-Verlag, New York, 1999 MR 1723696 | Zbl 0933.70003
[13] J. Milnor, Morse theory, Princeton University, 1963 MR 163331 | Zbl 0108.10401
[14] E. Miranda, On symplectic linearization of singular Lagrangian foliations, thesis, Univ. de Barcelona, 2003
[15] E. Miranda & San Vu Ngoc, “A singular Poincaré lemma”, IMRN 1 (2005), p. 27-46 MR 2130052 | Zbl 1078.58007
[16] H.-J. Petzsche, “On E. Borel’s Theorem”, Math. Ann. 282 (1988), p. 299-313 Zbl 0633.46033
[17] J. Rawnsley, “On the Cohomology Groups of a Polarization and Diagonal quantization”, Transaction of the American Mathematical Society 230 (1977), p. 235-255 MR 648775 | Zbl 0313.58016
[18] J. Śniatycki, On Cohomology Groups Appearing in Geometric Quantization, Differential Geometric Methods in Mathematical Physics, 1975 Zbl 0353.53019
[19] J. Śniatycki, Geometric quantization and quantum mechanics, Applied Mathematical Sciences 30, Springer-Verlag, New York-Berlin, 1980 MR 554085 | Zbl 0429.58007
[20] J. C. Tougeron, “Idéaux de fonctions différentiables”, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 71 (1972) MR 440598 | Zbl 0251.58001
[21] J. Williamson, “On the algebraic problem concerning the normal forms of linear dynamical systems”, Amer. J. Math. 58:1 (1936), p. 141-163 MR 1507138 | JFM 62.1795.10
[22] N. M. J. Woodhouse, Geometric quantization, Second edition. Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992 MR 1183739 | Zbl 0747.58004
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