Symplectic Path Intersections and the Leray Index

  • Maurice de Gosson
  • Serge de Gosson
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 52)


Historically, the embryo of what is called today the “Maslov quantization condition” was already proposed by the physicist Albert Einstein in his 1917 paper [1 2]. Einstein had remarked that the Bohr quantization rule


Symplectic Form Homotopy Class Trace Formula Maslov Index Lagrangian Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Arnold, V.I., A characteristic class entering in quantization conditions, Funkt. Anal. i. Priloz. 1(1), 1–14 (in Russian) (1967); Funct. Anal Appl. 1, 1–14 (English translation) (1967).Google Scholar
  2. 2.
    Brack, M. and Bhaduri R.K., Semiclassical Physics, Addison-Wesley, 1997.Google Scholar
  3. 3.
    Booss-Bavnbek B. and Furutani K., The Maslov Index: A Functional Analytical Definition and the Spectral Flow Formula, Tokyo J. Math. 21(1) (1998).Google Scholar
  4. 4.
    Creagh, S.C., Robbins, J.M. and Littlejohn, R.G., Geometrical properties of Maslov indices in the semiclassical trace formula for the density of states, Phys. Rev. A, 42(4), 1990.Google Scholar
  5. 5.
    de Gosson, M., La définition de l’indice de Maslov sans hypothèse de transversalité, C.R. Acad. Sci., Paris, 309, Série I, (1990), 279–281.Symplectic Path Intersections and the Leray Index 95Google Scholar
  6. 6.
    de Gosson, M., La relation entre Spoo, revêtement universel du groupe symplectique Sp et Sp x Z, C.R. Acad. Sci., Paris, 310, Série I, (1990), 245–248.Google Scholar
  7. 7.
    de Gosson, M., The structure of q-symplectic geometry, J. Math. Pures et Appl. 71 (1992), 429–453.zbMATHGoogle Scholar
  8. 8.
    de Gosson, M., Lagrangian path intersections and the Leray index: Aarhus Geometry and Topology Conference, Contemp. Math. 258, Amer. Math. Soc., Providence, RI, 2000, pp. 177–184.Google Scholar
  9. 9.
    de Gosson, M., de Gosson, S., The Cohomological Meaning of Maslov’s Lagrangian Path Intersection Index. In Proceedings of the Conference in the Honor of Jean Leray, Karlskrona 1999, (M. de Gosson, ed.), Kluwer Acad. Publ., 2001.Google Scholar
  10. 10.
    de Gosson, S., Lagrangian Path Intersection Indices, MSc Thesis, Kalmar, 1998.Google Scholar
  11. 11.
    de Gosson, S., Maslov Indices for Symplectic Paths and Applications to Trace Formulae, Ph.D. Thesis, Växjö, 2002.Google Scholar
  12. 12.
    Einstein, A., Zum Quantensatz von Sommerfeld und Epstein, Verhandlungen der Deutschen Phys. Ges., nr. 9–10 (1917).Google Scholar
  13. 13.
    Keller, J.B., Corrected Bohr—Sommerfeld Quantum Conditions for Nonseparable Systems, Ann. of Physics 4 (1958), 180–188.zbMATHCrossRefGoogle Scholar
  14. 14.
    Leray, J., Lagrangian Analysis and Quantum Mechanics, a mathematical structure related to asymptotic expansions and the Maslov index, MIT Press, Cambridge, Mass., 1981; translated from Analyse Lagrangienne RCP 25, Strasbourg Collège de France, 1976–1977.Google Scholar
  15. 15.
    Long, Y., Precise Iteration Formulae of the Maslov-type Index Theory and Ellipticity of Closed Characteristics, Advances in Mathematics 154 (2000), 76–131.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    McDuff, D and Salamon, D., Introduction to Symplectic Topology, Oxford Science Publications, 1998.Google Scholar
  17. 17.
    Maslov, V.P., Théorie des Perturbations et Méthodes Asymptotiques, Dunod, Paris, 1972; translated from Russian [original Russian edition 1965].Google Scholar
  18. 18.
    Maslov, V.P. and Fedoriuk, M.V., Semi-Classical Approximations in Quantum Mechanics, Reidel, Boston, 1981.CrossRefGoogle Scholar
  19. 19.
    Robbin, J.W. and Salamon, D.A, The Maslov index for paths, Topology 32 (1993), 827–44.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Souriau, J.-M., Construction explicite de l’indice de Maslov, Group Theoretical Methods in Physics, Lecture Notes in Physics 50, Springer-Verlag, 1975, pp. 17–148.Google Scholar
  21. 21.
    Souriau, J.-M., Indice de Maslov des variétés lagrangiennes orientables, C. R. Acad. Sci., Paris, Série A, 276 (1973), 1025–1026.MathSciNetzbMATHGoogle Scholar
  22. 22.
    Sugita, A., Geometrical properties of Maslov indices in periodic-orbit theory, Phys. Lett. A 266 (2000), 321–330.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Maurice de Gosson
    • 1
  • Serge de Gosson
    • 2
  1. 1.Blekinge Institute of Technology 371 79 KarlsronaSweden
  2. 2.University MSI, Mathematics and System Engineering 351 95 VäxjöSweden

Personalised recommendations