Spectrum of the Laplace operator and periodic geodesics: thirty years after

Colin de Verdière, Yves

Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2429-2463.

On appelle « Formule de trace semi-classique » une formule exprimant la densité d’état régularisée du laplacien d’une variété riemannienne compacte en termes de ses géodésiques périodiques. Des preuves de telles formules ont été données par plusieurs auteurs dans les années 70. Le but principal de cet article est de présenter cette formule et d’en donner une preuve complète et indépendante du difficile calcul global des opérateurs intégraux de Fourier. Cette preuve est d’un esprit assez proche de celle de la thèse de l’auteur. Elle utilise l’équation de Schrödinger dépendant du temps, des propriétés des géodésiques, la méthode de la phase stationnaire et la représentation métaplectique comme outil de calcul.

What is called the “Semi-classical trace formula” is a formula expressing the smoothed density of states of the Laplace operator on a compact Riemannian manifold in terms of the periodic geodesics. Mathematical derivation of such formulas were provided in the seventies by several authors. The main goal of this paper is to state the formula and to give a self-contained proof independent of the difficult use of the global calculus of Fourier Integral Operators. This proof is close in the spirit of the first proof given in the authors thesis. It uses the time-dependent Schrödinger equation, some facts about the geodesic flow, the stationary phase approximation and the metaplectic representation as a computational tool.

[1] Abraham, R.; Marsden, J. Foundations of Mechanics, Reading, Massachusetts, 1978 | MR | Zbl

[2] Arnold, V. Mathematical Methods of Classical Mechanics, Graduate Texts in Math., Volume 60, Springer, 1989 | MR | Zbl

[3] Arnold, V.; Varchenko, A.; Goussein-Zade, S. Singularités des applications différentiables, Mir, Moscou, 1986

[4] Balian, R.; Bloch, C. Distribution of eigenfrequencies for the wave equation in a finite domain I, Ann. of Physics, Volume 60 (1970), pp. 401 | DOI | MR | Zbl

[5] Balian, R.; Bloch, C. Distribution of eigenfrequencies for the wave equation in a finite domain II, Ann. of Physics, Volume 64 (1971), pp. 271 | DOI | MR | Zbl

[6] Balian, R.; Bloch, C. Distribution of eigenfrequencies for the wave equation in a finite domain III, Ann. of Physics, Volume 69 (1972), pp. 76 | DOI | MR | Zbl

[7] Balian, R.; Bloch, C. Solution of the Schrödinger equation in terms of classical paths, Ann. of Phys., Volume 85 (1974), pp. 514 | DOI | MR | Zbl

[8] Bates, S.; Weinstein, A. Lectures on the Geometry of Quantization, Berkeley Math. Lecture Notes, Volume 8, Amer. Math. Soc., 1997 | MR | Zbl

[9] Bellissard, J.; al. Transition to Chaos in Classical and Quantum Mechanics, Lecture Notes in Maths, Volume 1589, Springer, 1994 | MR

[10] Berger, M.; Gauduchon, P.; Mazet, E. Le spectre d’une variété riemannienne compacte, Lecture Notes in Maths, Springer, 1971 | Zbl

[11] Berger, Marcel Riemannian geometry during the second half of the twentieth century, University Lecture Series, 17, American Mathematical Society, 2000 (Reprint of the 1998 original) | MR | Zbl

[12] Berry, M. V.; Tabor, M. Closed orbits and the regular bound spectrum, Proc. Royal Soc. London Ser. A, Volume 349 (1976), pp. 101-123 | DOI | MR

[13] Bogomolny, E.; Pavloff, N.; Schmit, C. Diffractive corrections in the trace formula for polygonal billiards, Phys. Rev. E (3), Volume 61 (2000), pp. 3689-3711 | DOI | MR

[14] Bohigas, O.; Giannoni, M.-J.; Schmit, C. Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett., Volume 52 (1984), pp. 1-4 | DOI | MR | Zbl

[15] Bott, R. On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math., Volume 9 (1956), pp. 171-206 | DOI | MR | Zbl

[16] Brummelhuis, R.; Uribe, A. A trace formula for Schrödinger operators, Comm. Math. Phys., Volume 136 (1991), pp. 567-584 | DOI | MR | Zbl

[17] Camus, B. Spectral estimates for degenerated critical levels, J. Fourier Anal. Appl., Volume 12 (2006), pp. 455-495 | DOI | MR | Zbl

[18] Cassanas, R. A Gutzwiller type formula for a reduced Hamiltonian within the framework of symmetry, C. R. Math. Acad. Sci. Paris, Volume 340 (2005), pp. 21-26 | MR | Zbl

[19] Charbonnel, A.-M.; Popov, G. A semi-classical trace formula for several commuting operators, Comm. Partial Differential Equations, Volume 24 (1999), pp. 283-323 | DOI | MR | Zbl

[20] Chazarain, J. Formule de Poisson pour les variétés riemanniennes, Invent. Math., Volume 24 (1974), pp. 65-82 | DOI | MR | Zbl

[21] Duistermaat, J. On the Morse index in variational calculus, Advances in Math., Volume 21 (1976), pp. 173-195 | DOI | MR | Zbl

[22] Duistermaat, J.; Guillemin, V. The spectrum of positive elliptic operators and periodic geodesics, Invent. Math., Volume 29 (1975), pp. 39-79 | DOI | MR | Zbl

[23] Faure, F. Semi-classical formula beyond the Ehrenfest time in quantum chaos. (I) Trace formula, Annales de l’Institut Fourier, Volume 57 (2007) no. 7, pp. 2525-2599 | DOI | Numdam

[24] Feynman, R.; Hibbs, A. Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965 | Zbl

[25] Folland, G. Harmonic Analysis in Phase Space, Princeton University Press, 1989 | MR | Zbl

[26] Guillemin, V. Wave-trace invariants, Duke Math. J., Volume 83 (1996), pp. 287-352 | DOI | MR | Zbl

[27] Guillemin, V.; Melrose, R. The Poisson summation formula for manifolds with boundary, Adv. in Math., Volume 32 (1979), pp. 204-232 | DOI | MR | Zbl

[28] Guillemin, Victor Clean intersection theory and Fourier integrals, Fourier integral operators and partial differential equations (Colloq. Internat., Univ. Nice, Nice, 1974), Lecture Notes in Math., Vol. 459, Springer, 1975, pp. 23-35 | MR | Zbl

[29] Gutzwiller, M. Periodic orbits and classical quantization conditions, J. Math. Phys., Volume 12 (1971), pp. 343-358 | DOI

[30] Hejhal, D. The Selberg trace formula and the Riemann $\zeta$ function, Duke Math. J., Volume 43 (1976), pp. 441-482 | DOI | MR | Zbl

[31] Hillairet, L. Contribution of periodic diffractive geodesics, J. Funct. Anal., Volume 226 (2005), pp. 48-89 | DOI | MR | Zbl

[32] Hofer, H.; Zehnder, E. Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser, 1994 | MR | Zbl

[33] Hörmander, L. The spectral function of an elliptic operator, Acta Math., Volume 121 (1968), pp. 193-218 | DOI | MR | Zbl

[34] Hörmander, L. The Analysis of Linear Partial Differential Operators I, Grundlehren, Springer, 1983 | MR | Zbl

[35] Hörmander, L. The Analysis of Linear Partial Differential Operators I, Grundlehren, Springer, 1985 | MR | Zbl

[36] Huber, H. Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen, Math. Ann., Volume 138 (1959), pp. 1-26 | DOI | MR | Zbl

[37] Kac, M. Can one hear the shape of a drum?, Amer. Math. Monthly, Volume 73 (1966), pp. 1-23 | DOI | MR | Zbl

[38] Kozlov, V.; Treshchëv, D. Billiards: a genetic introduction to the dynamics of systems with impacts, Transl. Math. Monographs, Volume 89, Amer. Math. Soc., 1991 | MR | Zbl

[39] Lax, P. D. Asymptotic solutions of oscillatory initial value problems, Duke Math. J., Volume 24 (1957), pp. 627-646 | DOI | MR | Zbl

[40] Levit, S.; Smilansky, U. A theorem on infinite products of eigenvalues of Sturm type operators, Proc. Amer. Math. Soc., Volume 65 (1977), pp. 299-303 | DOI | MR | Zbl

[41] Malgrange, B. Intégrales asymptotiques et monodromie, Ann. Sci. École Norm. Sup., Volume 7 (1974), pp. 405-430 | Numdam | MR | Zbl

[42] Marklof, J. Selberg’s trace formula: an introduction (Proceedings of the International School:“Quantum Chaos on Hyperbolic Manifolds”, Schloss Reisensburg, Gunzburg, Germany, 4–11 october 2003, to appear in Springer Lecture Notes in Physics. See also arXiv:math/ 0407288)

[43] Meinrenken, E. Semi-classical principal symbols and Gutzwiller’s trace formula, Rep. Math. Phys., Volume 31 (1992), pp. 279-295 | DOI | Zbl

[44] Meinrenken, E. Trace formulas and Conley-Zehnder index, J. Geom. Phys., Volume 13 (1994), pp. 1-15 | DOI | MR | Zbl

[45] Michel (ed.), L. Symmetry, invariants, topology, Physics reports, Volume 341 (2001), pp. 1-6 | MR | Zbl

[46] Milnor, J. Morse Theory, Princeton, 1967 | Zbl

[47] Boutet de Monvel, L.; Guillemin, V. The spectral theory of Toeplitz operators, Annals of Math. Studies, Volume 99, Princeton, 1981 | MR | Zbl

[48] Morette, C. On the definition and approximation of Feynman’s path integrals, Physical Rev. (2), Volume 81 (1951), pp. 848-852 | DOI | Zbl

[49] Ray, D. B.; Singer, I. M. $R$-torsion and the Laplacian on Riemannian manifolds, Advances in Math., Volume 7 (1971), pp. 145-210 | DOI | MR | Zbl

[50] Selberg, A. Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc., Volume 20 (1956), pp. 47-87 | MR | Zbl

[51] Serre, J.-P. Homologie singulière des espaces fibrés. Applications, Ann. of Math. (2), Volume 54 (1951), pp. 425-505 | DOI | MR | Zbl

[52] Colin de Verdière, Y. Spectre du Laplacien et longueurs des géodésiques périodiques I, Comp. Math., Volume 27 (1973), pp. 80-106 | Numdam | MR | Zbl

[53] Colin de Verdière, Y. Spectre du Laplacien et longueurs des géodésiques périodiques II, Comp. Math., Volume 27 (1973), pp. 159-184 | Numdam | MR | Zbl

[54] Colin de Verdière, Y. Sur le spectre des opérateurs elliptiques bicaractéristiques toutes périodiques, Comment. Math. Helv., Volume 54 (1979), pp. 508-522 | DOI | MR | Zbl

[55] Colin de Verdière, Y. Déterminants et intégrales de Fresnel, Ann. Inst. Fourier, Volume 49 (1999), pp. 861-881 | DOI | Numdam | MR | Zbl

[56] Colin de Verdière, Y. Bohr-Sommerfeld rules to all orders, Ann. Henri Poincaré, Volume 6 (2005), pp. 925-936 | DOI | MR | Zbl

[57] Weinstein, A. Asymptotics of eigenvalue clusters for the Laplacian plus a potential, Duke Math. J., Volume 44 (1977), pp. 883-892 | DOI | MR | Zbl

[58] Weinstein, Alan On Maslov’s quantization condition, Fourier integral operators and partial differential equations (Colloq. Internat., Univ. Nice, Nice, 1974),Lecture Notes in Math., Vol. 459, Springer, 1975, pp. 341-372 | Zbl

[59] Yorke, J. Periods of periodic solutions and the Lipschitz constant, Proc. Amer. Math. Soc., Volume 22 (1969), pp. 509-512 | DOI | MR | Zbl

[60] Zelditch, S. Wave trace invariants at elliptic closed geodesics, GAFA, Volume 7 (1997), pp. 145-213 | DOI | MR | Zbl

[61] Zelditch, S. Wave invariants for non-degenerate closed geodesics, GAFA, Volume 8 (1998), pp. 179-207 | DOI | MR | Zbl