Cramér's formula for Heisenberg manifolds
Annales de l'Institut Fourier, Volume 55 (2005) no. 7, pp. 2489-2520.

Let R(λ) be the error term in Weyl’s law for a 3-dimensional Riemannian Heisenberg manifold. We prove that 1 T |R(t)| 2 dt=cT 5 2 +O δ (T 9 4+δ ), where c is a specific nonzero constant and δ is an arbitrary small positive number. This is consistent with the conjecture of Petridis and Toth stating that R(t)=O δ (t 3 4+δ ).The idea of the proof is to use the Poisson summation formula to write the error term in a form which can be estimated by the method of the stationary phase. The similar result will be also proven in the 2n+1-dimensional case.

Soit R(λ) le terme d’erreur de la loi de Weyl pour une variété riemannienne de Heisenberg de dimension 3. Nous prouvons que 1 T |R(t)| 2 dt=cT 5 2 +O δ (T 9 4+δ ), où c est une constante spécifique non nulle et δ est un nombre positif arbitrairement petit. Ce résultat est une avancée vers la conjecture de Petridis et Toth, qui énonce que R(t)=O δ (t 3 4+δ ). L’idée de la preuve est d’utiliser la formule de sommation poisson pour réécrire le terme d’erreur sous une forme qui est majorable au moyen de la méthode des phases stationnaires. Le même résultat sera prouvé pour la dimension 2n+1.

DOI: 10.5802/aif.2168
Classification: 35P20, 58J50
Keywords: Heisenberg manifolds, Weyl's law, Cramér's formula, poisson summation formula, Heisenberg manifolds, Weyl's law, Cramér's formula, poisson summation formula
Mot clés : variété d'Heisenberg, loi de Weyl, formule de Cramér, formule de sommation de Poisson
Khosravi, Mahta 1; Toth, John A. 1

1 McGill University, Department of Mathematics and Statistics, 805 Sherbrooke St. West, Montreal, Quebec H3A 2K6 (Canada)
@article{AIF_2005__55_7_2489_0,
     author = {Khosravi, Mahta and Toth, John A.},
     title = {Cram\'er's formula for {Heisenberg} manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {2489--2520},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {55},
     number = {7},
     year = {2005},
     doi = {10.5802/aif.2168},
     mrnumber = {2207391},
     zbl = {1090.58018},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2168/}
}
TY  - JOUR
AU  - Khosravi, Mahta
AU  - Toth, John A.
TI  - Cramér's formula for Heisenberg manifolds
JO  - Annales de l'Institut Fourier
PY  - 2005
SP  - 2489
EP  - 2520
VL  - 55
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2168/
DO  - 10.5802/aif.2168
LA  - en
ID  - AIF_2005__55_7_2489_0
ER  - 
%0 Journal Article
%A Khosravi, Mahta
%A Toth, John A.
%T Cramér's formula for Heisenberg manifolds
%J Annales de l'Institut Fourier
%D 2005
%P 2489-2520
%V 55
%N 7
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2168/
%R 10.5802/aif.2168
%G en
%F AIF_2005__55_7_2489_0
Khosravi, Mahta; Toth, John A. Cramér's formula for Heisenberg manifolds. Annales de l'Institut Fourier, Volume 55 (2005) no. 7, pp. 2489-2520. doi : 10.5802/aif.2168. http://archive.numdam.org/articles/10.5802/aif.2168/

[Be] Bérard, P.H. On the wave equation on a compact Riemannian manifold without conjugate points, Math. Z., Volume 155 (1977) no. 3, pp. 249-276 | DOI | MR | Zbl

[BG] Bentkus, V.; Götze, F. Lattice point problems and distribution of values of quadratic forms, Ann. of Math., Volume 50:3 (1999) no. 2, pp. 977-1027 | DOI | MR | Zbl

[Bl] Bleher, P. On the distribution of the number of lattice points inside a family of convex ovals, Duke Math. J., Volume 67 (1992) no. 3, pp. 461-481 | DOI | MR | Zbl

[Bu] Butler, L. Integrable geodesic flows on n-step nilmanifolds, J. Geom. Phys., Volume 36 (2000) no. 3-4, pp. 315-323 | DOI | MR | Zbl

[Co] Verdière, Y. Colin de Spectre conjoint d'opérateurs pseudo-diifférentiels qui commtent. II. Le cas intégrable, Math. Z., Volume 171 (1980) no. 1, pp. 51-73 | DOI | MR | Zbl

[Cop] Copson, E.T. Asymptotic Expansions, Cambridge University Press (1965), pp. 29-33 | MR | Zbl

[CPT] Chung, D.; Petridis, Y.N.; Toth, J.A. The remained in Weyl's law for Heisenberg manifolds II, Bonner Mathematische Schriften, Bonn, Volume 360 (2003), pp. 16 | MR | Zbl

[Cr] Cramér, H. Über zwei Sätze von Herrn G.H. Hardy, Math. Z., Volume 15 (1922), pp. 201-210 | DOI | JFM | MR

[DG] Duistermaat, J.J.; Guillemin, V. The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math., Volume 29 (1975) no. 1, pp. 39-79 | DOI | MR | Zbl

[Fo] Folland, G.B. Harmonic Analysis in Phase Space, Princeton University Press (1989), pp. 9-73 | MR | Zbl

[Fr] Fricker, F. Einführung in die Gitterpunketlehre, [Introduction to lattice point theory], Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften(LMW) (Mathematische Reihe [Textbooks and Monographs in the Exact Sciences]), Volume 73, Birkhäuser Verlag, Basel-Boston, Mas., 1982 | MR | Zbl

[Go] Götze, F. Lattice point problems and values of quadratic forms, Inventiones Mathematicae, Volume 157 (2004) no. 1, pp. 195-226 | DOI | MR | Zbl

[GW] Gordon, C.; Wilson, E. The spectrum of the Laplacian on Reimannian Heisenberg manifolds, Michigan Math. J., Volume 33 (1986) no. 2, pp. 253-271 | DOI | MR | Zbl

[Ha] Hardy, G.H. On the expression of a number as the sum of two squares, Quart. J. Math., Volume 46 (1915), pp. 263-283 | JFM

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

[Hu] Huxley, M.N. Exponential sums and lattice points, II, Proc. London Math. Soc., Volume 66 (1993) no. 2, pp. 279-301 | DOI | MR | Zbl

[Iv] Ivrii, V.YA. Precise Spectral Asymptotics for elliptic Operators Acting in Fibrings over Manifolds with Boundary, Springer Lecture Notes in Mathematics, Volume 1100 (1984) | MR | Zbl

[KP] Khosravi, M.; Petridis, Y.N. The remainder in Weyl's law for n-Dimensional Heisenberg manifolds Proc. of AMS (to appear) | Zbl

[PT] Petridis, Y.N.; Toth, J.A. The remainder in Weyl's law for Heisenberg manifolds, J. Diff. Geom., Volume 60 (2002), pp. 455-483 | MR | Zbl

[St] Stein, E.M. Harmonic Analysis, Princeton University Press (1993), pp. 527-574 | MR | Zbl

[Vo] Volovoy, A.V. Improved two-term asymptotics for the eigenvalue distribution function of an ellitic operator on a compact manifold, Comm. Partial Differential Equations, Volume 15 (1990) no. 11, pp. 1509-1563 | DOI | MR | Zbl

Cited by Sources: