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On the spectrum of positive elliptic operators and periodic bicharacteristics

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Fourier Integral Operators and Partial Differential Equations

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References

  1. G.K. Andersson, Analytic wave front sets for solutions of linear differential equations of principal type, Trans. Am. Math. Soc. 177 (1973), 1–27.

    Article  MathSciNet  MATH  Google Scholar 

  2. V.I. Arnol’d, On a characteristic class entering in quantization conditions, Funct. Anal. Appl. 1 (1967), 1–13.

    Article  MATH  Google Scholar 

  3. M.F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes I, Ann. of Math. 86 (1967), 374–407.

    Article  MathSciNet  MATH  Google Scholar 

  4. M.F. Atiyah, R. Bott and V.K. Patodi, On the heat equation and the index theorem, Inv. Math. 19 (1973), 279–330.

    Article  MathSciNet  MATH  Google Scholar 

  5. R. Bott, On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math., 9 (1956), 176–206.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Chazarain, Formule de Poisson pour les variétés riemanniennes, Inv. Math. 24 (1974), 65–82.

    Article  MathSciNet  MATH  Google Scholar 

  7. Y. Colin de Verdière, Spectre du laplacien et longueurs des géodésiques périodiques II, Comp. Math. 27 (1973), 159–184.

    MATH  Google Scholar 

  8. M. Cotsaftis, Une propriété des orbites périodiques des systèmes hamiltoniens non-linéaires, C. R. Acad. Sc. Paris 275, Série A (1973), 911–914.

    MathSciNet  MATH  Google Scholar 

  9. J.J. Duistermaat and L. Hörmander; Fourier integral operators II, Acta Math. 128 (1972), 184–269.

    Article  MathSciNet  MATH  Google Scholar 

  10. J.J. Duistermaat, Fourier Integral Operators, Courant Institute Lecture Notes, New York 1973.

    Google Scholar 

  11. J.J. Duistermaat and V.W. Guillemin, 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. J.J. Duistermaat, On the Morse index in variational calculus, to appear in Advances in Math.

    Google Scholar 

  13. I.M. Gelfand and G.E. Shilov, Generalized Functions, I, Academic Press, New York 1964.

    Google Scholar 

  14. V. Guillemin and S. Sternberg, Geometric Asymptotics, A.M.S. Publications (in press).

    Google Scholar 

  15. G.E. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford, Clarendon Press 1960.

    MATH  Google Scholar 

  16. L. Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193–218.

    Article  MathSciNet  MATH  Google Scholar 

  17. L. Hörmander, Fourier integral operators I, Acta Math. 127 (1971), 79–183.

    Article  MathSciNet  MATH  Google Scholar 

  18. S. Minakshisundaram and Å. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds, Canadian J. Math. 1 (1949), 242–256.

    Article  MathSciNet  MATH  Google Scholar 

  19. L. Nirenberg, Lectures on Linear Partial Differential Equations, Regional Conference Series in Mathematics, No 17, Conf. Board of the Math. Sc. of the A. M. S., 1972.

    Google Scholar 

  20. M. Sato, Regularity of hyperfunction solutions of partial differential equations, Proc. Nice Congress, Vol. 2, Gauthiers-Villars, Paris 1970, pp. 785–794.

    Google Scholar 

  21. M. Sato, T. Kawai and M. Kashiwara, Microfunctions and Pseudo-Differential Equations, Lecture Notes in Math. No 287, Springer-Verlag 1973, pp. 265–529.

    Article  MathSciNet  MATH  Google Scholar 

  22. R.T. Seeley, Complex powers of an elliptic operator, A.M.S. Proc. Symp. Pure Math. 10 (1967), 288–307. Corrections in: The resolvent of an elliptic boundary problem, Am. J. Math. 91 (1969), 917–919.

    Article  MathSciNet  MATH  Google Scholar 

  23. J.-P. Serre, A Course in Arithmetic, Springer-Verlag, New York, Heidelberg, Berlin, 1973.

    Book  MATH  Google Scholar 

  24. A. Weinstein, Fourier integral operators, quantization and the spectra of Riemannian manifolds, to appear in the Proc. of the C.N.R.S. Colloque de Geometrie Symplectique et Physique Mathematique, Aix-en-Provence, June 1974.

    Google Scholar 

  25. W. Klingenberg and F. Takens, Generic properties of geodesic flows, Math. Ann. 197 (1972), 323–334.

    Article  MathSciNet  MATH  Google Scholar 

  26. L. Hörmander, Linear differential operators, Proc. Nice Congress, Vol. 1, Gauthiers-Villars, Paris 1970, pp. 121–133.

    Google Scholar 

  27. L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, Berlin, Gottingen, Heidelberg 1963.

    Book  MATH  Google Scholar 

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Jacques Chazarain

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© 1975 Springer-Verlag

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Duistermaat, J.J. (1975). On the spectrum of positive elliptic operators and periodic bicharacteristics. In: Chazarain, J. (eds) Fourier Integral Operators and Partial Differential Equations. Lecture Notes in Mathematics, vol 459. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0074190

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  • DOI: https://doi.org/10.1007/BFb0074190

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