Journal of Differential Geometry

The inverse spectral problem for surfaces of revolution

Steve Zelditch

Full-text: Open access

Article information

Source
J. Differential Geom., Volume 49, Number 2 (1998), 207-264.

Dates
First available in Project Euclid: 26 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1214461019

Digital Object Identifier
doi:10.4310/jdg/1214461019

Mathematical Reviews number (MathSciNet)
MR1664907

Zentralblatt MATH identifier
0938.58027

Subjects
Primary: 58G25
Secondary: 53C22: Geodesics [See also 58E10] 58D27: Moduli problems for differential geometric structures

Citation

Zelditch, Steve. The inverse spectral problem for surfaces of revolution. J. Differential Geom. 49 (1998), no. 2, 207--264. doi:10.4310/jdg/1214461019. https://projecteuclid.org/euclid.jdg/1214461019


Export citation

References

  • [1] V. I. Arnold, Mathematical methods of classical mechanics, Springer, New York, 1977.
  • [2] S. Bates and A. Weinstein, Lectures on the geometry of quantization, Berkeley lecture notes, 1996.
  • [3] P. Berard, Quelques remarques sur les surfaces de revolution dans R3, C. R. Acad. Sci. Paris 282 (1976) 159-161.
  • [4] A. Besse, On manifolds all of whose geodesics are closed, Ergeb. Math. Grenoble (3) 93, Springer, New York, 1978.
  • [5] P. Bleher, Distribution of energy levels of a quantum free particle on a surface of revolution, Duke Math. J. 74 (1994) 45-93.
  • [6] L. Boutet de Monvel and V. Guillemin, The Spectral Theory of Toeplitz Operators, Ann. of Math. Stud. 99 Princeton U. Press (1981).
  • [7] J. Bruning and E. Heintze, Spektrale starrheit gewisser Dreh achen, Math. Ann. 269 (1984) 95-101.
  • [8] G. Darboux, Theorie des surfaces. III, Gauthier-Villars, Paris 1894.
  • [9] Y. Colin de Verdiere, Spectre conjoint d'operateurs pseudo-differentiels qui commutent I: Le cas non-integrable, Duke Math. J. 46 (1979) 169-182.
  • [10] Y. Colin de Verdiere, Sur les longuers des trajectoires periodiques d'un billard, (P. Dazord and N. Desolneux-Moulis eds.), Geometrie Symplectique et de Contact: Autour du Theoreme de Poincare-Birkho. Travaux en Cours, Sem. Sud-Rhodanien de Geometrie III Pairs: Herman, 1984, 122-139.
  • [11] Y. Colin de Verdiere, Quasi-modes sur les varietes Riemanniennes, Invent. Math. 43 (1977) 15-52.
  • [12] C. Croke and B. Kleiner, Conjugacy and rigidity for manifolds with a parallel vector eld, J. Differential Geom. 39 (1994) 659-680.
  • [13] J. J. Duistermaat, On the existence of global action-angle variables, Comm. Pure. Appl. Math. 33 (1980) 687-706.
  • [14] J. J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975) 29-79.
  • [15] J. P. Francoise and V. Guillemin, On the period spectrum of a symplectic mapping, J. Funct. Anal. 100 (1991) 317-358.
  • [16] I. M. Gel'fand and G. E. Shilov, Generalized Functions, Vol. 1, Academic Press, New York, 1964.
  • [17] V. Guillemin, Wave trace invariants, Duke Math. J. 83 (1996) 287-52.
  • [18] V. Guillemin and S.Sternberg, Symplectic techniques in Physics. Cambridge Univ. Press, New York, 1985.
  • [19] V. Guillemin and S.Sternberg, Homogeneous quantization and multiplicities of group representations, J. Funct. Anal. 47 (1982) 344-380.
  • [20] D.Gurarie, Semiclassical eigenvalues and shape problems on surfaces of revolution, J. Math. Phys. 36 (1995) 1934-1944.
  • [21] M.Kac, On applying mathematics: re ections and examples, Mark Kack: Probability, Number Theory, and Statistical Physics, (K.Baclawski and M.D.Donskder eds.), MIT Press, Cambridge, 1979.
  • [22] W. Klingenberg, Lectures on Closed Geodesics, Grundlehren Math. Wiss. Vol. 230, Springer, 1978.
  • [23] V. P. Maslov, Theorie des Perturbations et Methods Asymptotiques, Dunod, Paris, 1972.
  • [24] G. Popov, Length spectrum invariants of Riemannian manifolds, Math. Z. 213 (1993) 311-351.
  • [25] S.Zelditch, Wave invariants at elliptic closed geodesics, Geom. Funct. Anal. 7 (1997) 145-213.
  • [26] S.Zelditch, Wave invariants at non-degenerate closed geodesics, Geom. Funct. Anal. 8 (1998) 179-217.
  • [27] S.Zelditch, Isospectrality in the FIO category, J. Differential Geom. 35 (1992) 689-710.
  • [28] S.Zelditch, Kuznecov sum formulae and Szego limit formulae on manifolds, Comm. Partial Differential Equations 17 (1992) 221-260.