The theta series is a classical example of a modular form. In this article we argue that the trace , where is a self-adjoint elliptic pseudo-differential operator of order 1 with periodic bicharacteristic flow, may be viewed as a natural generalization. In particular, we establish approximate functional relations under the action of the modular group. This allows a detailed analysis of the asymptotics of near the real axis, and the proof of logarithm laws and limit theorems for its value distribution. These asymptotics are in fact distinctly different from those for the ‘wave trace’ whose singularities are well known to be located at the lengths of the periodic orbits of the bicharacteristic flow.
La série thêta est un exemple classique de forme modulaire. Dans cet article, nous montrons que la trace , où est un opérateur pseudo-différentiel elliptique auto-adjoint d’ordre 1 à flot bicaractéristique périodique, en est une généralisation naturelle. En particulier, nous établissons des égalités fonctionnelles approchées sous l’action du groupe modulaire. Ceci permet une analyse détaillée de l’asymptotique de au voisinage de l’axe réel, et prouve des lois du logarithme et des théorèmes limites pour la distribution de ses valeurs. Ces asymptotiques diffèrent de celles relatives à la trace de l’opérateur des ondes , dont les singularités sont portées par les longueurs des bicaractéristiques périodiques.
Keywords: Spectral theta series, Zoll manifolds, periodic geodesic flow, Shale-Weil representation, horocycle flow, logarithm laws
Mot clés : série théta spectrale, variété de Zoll, flot géodésique périodique, représentation de Shale-Weil, flot horocyclique, lois du logarithme
Marklof, Jens 1
@article{AIF_2007__57_7_2401_0, author = {Marklof, Jens}, title = {Spectral theta series of operators with periodic bicharacteristic flow}, journal = {Annales de l'Institut Fourier}, pages = {2401--2427}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {57}, number = {7}, year = {2007}, doi = {10.5802/aif.2338}, mrnumber = {2394547}, zbl = {1133.35075}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2338/} }
TY - JOUR AU - Marklof, Jens TI - Spectral theta series of operators with periodic bicharacteristic flow JO - Annales de l'Institut Fourier PY - 2007 SP - 2401 EP - 2427 VL - 57 IS - 7 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2338/ DO - 10.5802/aif.2338 LA - en ID - AIF_2007__57_7_2401_0 ER -
%0 Journal Article %A Marklof, Jens %T Spectral theta series of operators with periodic bicharacteristic flow %J Annales de l'Institut Fourier %D 2007 %P 2401-2427 %V 57 %N 7 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2338/ %R 10.5802/aif.2338 %G en %F AIF_2007__57_7_2401_0
Marklof, Jens. Spectral theta series of operators with periodic bicharacteristic flow. Annales de l'Institut Fourier, Volume 57 (2007) no. 7, pp. 2401-2427. doi : 10.5802/aif.2338. https://aif.centre-mersenne.org/articles/10.5802/aif.2338/
[1] Manifolds all of whose geodesics are closed, Springer-Verlag, Berlin-New York, 1978 | MR | Zbl
[2] Formule de Poisson pour les variétés riemanniennes, Invent. Math., Volume 24 (1974), pp. 65-82 | DOI | MR | Zbl
[3] Spectre du laplacien et longueurs des géodésiques périodiques. I., Compositio Math., Volume 27 (1973), pp. 83-106 | Numdam | MR | Zbl
[4] Spectre du laplacien et longueurs des géodésiques périodiques. II., Compositio Math., Volume 27 (1973), pp. 159-184 | Numdam | MR | Zbl
[5] Sur le spectre des opérateurs elliptiques à bicaractéristiques toutes périodiques, Comment. Math. Helv., Volume 54 (1979) no. 3, pp. 508-522 | DOI | MR | Zbl
[6] The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math., Volume 29 (1975) no. 1, pp. 39-79 | DOI | MR | Zbl
[7] Mixing, counting, and equidistribution in Lie groups, Duke Math. J., Volume 71 (1993) no. 1, pp. 181-209 | DOI | MR | Zbl
[8] Asymptotic expansions of finite theta series, Acta Arithm., Volume 32 (1977), pp. 129-146 | MR | Zbl
[9] Table of integrals, series, and products, Academic Press, New York-London, 1965 | Zbl
[10] Some spectral results for the Laplace operator with potential on the -sphere, Advances in Math., Volume 27 (1978) no. 3, pp. 273-286 | DOI | MR | Zbl
[11] Some spectral results on rank one symmetric spaces, Advances in Math., Volume 28 (1978) no. 2, pp. 129-137 | DOI | MR | Zbl
[12] Chaos in classical and quantum mechanics. Interdisciplinary Applied Mathematics, 1., Springer-Verlag, New York, 1990 | MR | Zbl
[13] The proof of the central limit theorem for theta sums, Duke Math. J. (1981), pp. 873-885 | DOI | MR | Zbl
[14] On the central limit theorem for theta series, Michigan Math. J., Volume 29 (1982), pp. 65-77 | DOI | MR | Zbl
[15] The uniform central limit theorem for theta sums, Duke Math. J., Volume 50 (1983), pp. 649-666 | DOI | MR | Zbl
[16] Logarithm laws for flows on homogeneous spaces, Invent. Math., Volume 138 (1999) no. 3, pp. 451-494 | DOI | MR | Zbl
[17] The Weil representation, Maslov index and theta series, Progress in Mathematics, 6. Birkhäuser, Boston, Mass., 1980 | MR | Zbl
[18] Spectral form factors of rectangle billiards, Comm. Math. Phys., Volume 199 (1998) no. 1, pp. 169-202 | DOI | MR | Zbl
[19] Limit theorems for theta sums, Duke Math. J., Volume 97 (1999) no. 1, pp. 127-153 | DOI | MR | Zbl
[20] Theta sums, Eisenstein series, and the semiclassical dynamics of a precessing spin, Emerging applications of number theory (Minneapolis, MN, 1996), IMA Vol. Math. Appl., Springer, New York, Volume 109 (1999), pp. 405-450 | MR | Zbl
[21] Pair correlation densities of inhomogeneous quadratic forms, Ann. of Math., Volume 158 (2003) no. 2, pp. 419-471 | DOI | MR | Zbl
[22] Ratner’s theorems on unipotent flows, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2005 | Zbl
[23] Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series, Comm. Pure Appl. Math., Volume 34 (1981) no. 6, pp. 719-739 | DOI | MR | Zbl
[24] Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.), Volume 20 (1956), pp. 47-87 | MR | Zbl
[25] Limit distributions of expanding translates of certain orbits on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci., Volume 106 (1996) no. 2, pp. 105-125 | DOI | MR | Zbl
[26] On the uniform equidistribution of long closed horocycles, Duke Math. J., Volume 123 (2004) no. 3, pp. 507-547 | DOI | MR | Zbl
[27] Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math., Volume 149 (1982) no. 1, pp. 215-237 | DOI | MR | Zbl
[28] Spectral statistics on Zoll surfaces, Comm. Math. Phys., Volume 154 (1993) no. 2, pp. 313-346 | DOI | MR | Zbl
[29] Asymptotics of eigenvalue clusters for the Laplacian plus a potential, Duke Math. J., Volume 44 (1977) no. 4, pp. 883-892 | DOI | MR | Zbl
[30] Fine structure of Zoll spectra, J. Funct. Anal., Volume 143 (1997) no. 2, pp. 415-460 | DOI | MR | Zbl
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