Inventiones mathematicae

, Volume 29, Issue 1, pp 39–79

The spectrum of positive elliptic operators and periodic bicharacteristics

  • J. J. Duistermaat
  • V. W. Guillemin


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andersson, G.K.: Analytic wave front sets for solutions of linear differential equations of principal type. Trans. Am. Math. Soc.177, 1–27 (1973)Google Scholar
  2. 2.
    Arnol'd, V.I.: On a characteristic class entering in quantization conditions. Funct. Anal. Appl.1, 1–13 (1967)Google Scholar
  3. 3.
    Atiyah, M.F., Bott, R.: A Lefschetz fixed point formula for elliptic complexes I. Ann. of Math.86, 374–407 (1967)Google Scholar
  4. 4.
    Atiyah, M.F., Bott, R., Patodi, V.K.: On the heat equation and the index theorem. Inventiones math.19, 279–330 (1973)Google Scholar
  5. 5.
    Bott, R.: On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math.9, 176–206 (1956)Google Scholar
  6. 6.
    Chazarain, J.: Formule de Poisson pour les variétés riemanniennes. Inventiones math.24, 65–82 (1974)Google Scholar
  7. 7.
    Colin de Verdière, Y.: Spectre du laplacien et longueurs des géodésiques périodiques II. Comp. Math.27, 159–184 (1973)Google Scholar
  8. 8.
    Cotsaftis, M.: Une propriété des orbites périodiques des systèmes hamiltoniens non-linéaires. C. R. Acad. Sc. Paris 275, Série A 911–914 (1973)Google Scholar
  9. 9.
    Duistermaat, J.J., Hörmander, L.: Fourier integral operators II. Acta Math.128, 184–269 (1972)Google Scholar
  10. 10.
    Duistermaat, J.J.: Fourier Integral Operators. Courant Institute Lecture Notes, New York 1973Google Scholar
  11. 11.
    Duistermaat, J.J., Guillemin, V. W.: The spectrum of positive elliptic operators and periodic geodesics. Proc. A.M.S. Summer Institute on Differential Geometry, Stanford 1973 (to appear)Google Scholar
  12. 12.
    Duistermaat, J.J.: On the Morse index in variational calculus. To appear in Advances in Math.Google Scholar
  13. 13.
    Gelfand, I.M.., Shilov, G. E.: Generalized Functions, I. New York: Academic Press 1964Google Scholar
  14. 14.
    Guillemin, V., Sternberg, S.: Geometric Asymptotics. A.M.S. Publications (in press)Google Scholar
  15. 15.
    Hardy, G.E., Wright, E. M. An Introduction to the Theory of Numbers, 4th ed. Oxford: Clarendon Press 1960Google Scholar
  16. 16.
    Hörmander, L.: The spectral function of an elliptic operator. Acta Math.121, 193–218 (1968)Google Scholar
  17. 17.
    Hörmander, L.: Fourier integral operators I. Acta Math.127, 79–183 (1971)Google Scholar
  18. 18.
    Minakshisundaram, S., Pleijel, Å.: Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds. Canadian J. Math.1, 242–256 (1949)Google Scholar
  19. 19.
    Nirenberg, L.: Lectures on Linear Partial Differential Equations, Regional Conference Series in Mathematics, No 17. Conf. Board of the Math. Sc. of the A.M.S., 1972Google Scholar
  20. 20.
    Sato, M.: Regularity of hyperfunction solutions of partial differential equations. Proc. Nice Congress, Vol. 2, pp. 785–794. Paris: Gauthiers-Villars 1970Google Scholar
  21. 21.
    Sato, M., Kawai, T., Kashiwara, M.: Microfunctions and Pseudo-Differential Equations. Lecture Notes in Mathematics287, pp. 265–529. Berlin-Heidelberg-New York: Springer 1973Google Scholar
  22. 22.
    Seeley, R. T.: Complex powers of an elliptic operator. A.M.S. Proc. Symp. Pure Math.10, 288–307 (1967). Corrections in: The resolvent of an elliptic boundary problem. Am. J. Math.91, 917–919 (1969)Google Scholar
  23. 23.
    Serre, J-P.: A Course in arithmetic. Berlin-Heidelberg-New York: Springer 1973Google Scholar
  24. 24.
    Weinstein, A.: Fourier integral operators, quantization and the spectraof Riemannian manifolds. To appear in the Proc. of the C.N.R.S. Colloque de Géométrie Symplectique et Physique Mathématique. Aix-en-Provence, June 1974Google Scholar
  25. 25.
    Klingenberg, W., Takens, F.: Generic properties of geodesic flows. Math. Ann.197, 323–334 (1972)Google Scholar
  26. 26.
    Hörmander, L.: Linear differential operators. Proc. Nice Congress, Vol. 1, pp. 121–133. Paris: Gauthiers-Villars 1970Google Scholar

Copyright information

© Springer-Verlag 1975

Authors and Affiliations

  • J. J. Duistermaat
    • 1
  • V. W. Guillemin
    • 2
  1. 1.Mathematisch InstituutUtrechtThe Netherlands
  2. 2.Mathematics DepartmentMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations