Variance of number of lattice points in random narrow elliptic strip
Bleher, Pavel M. ; Lebowitz, Joel L.
Annales de l'I.H.P. Probabilités et statistiques, Tome 31 (1995), p. 27-58 / Harvested from Numdam
Publié le : 1995-01-01
@article{AIHPB_1995__31_1_27_0,
     author = {Bleher, Pavel M. and Lebowitz, Joel L.},
     title = {Variance of number of lattice points in random narrow elliptic strip},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {31},
     year = {1995},
     pages = {27-58},
     mrnumber = {1340030},
     zbl = {0915.11049},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_1995__31_1_27_0}
}
Bleher, Pavel M.; Lebowitz, Joel L. Variance of number of lattice points in random narrow elliptic strip. Annales de l'I.H.P. Probabilités et statistiques, Tome 31 (1995) pp. 27-58. http://gdmltest.u-ga.fr/item/AIHPB_1995__31_1_27_0/

[Ber] M.V. Berry, Semiclassical theory of spectral rigidity, Proc. R. Soc. Lond., Vol. A 400, 1985, pp. 229-251. | MR 805089 | Zbl 0875.35061

[BT] M.V. Berry and M. Tabor, Level clustering in the regular spectrum, Proc. R. Soc. Lond., Vol. A 356, 1977, pp. 375-394.

[BCDL] P.M. Bleher, Zh. Cheng, F.J. Dyson and J.L. Lebowitz, Distribution of the error term of number of lattice points inside a shifted circle, Commun. Math. Phys., Vol. 154, 1993, pp. 433-469. | MR 1224087 | Zbl 0781.11038

[BD1] P.M. Bleher and F.J. Dyson, The variance of the error function in the shifted circle problem is a wild function of the shift, Commun. Math. Phys. (to appear). | MR 1266060 | Zbl 0808.11058

[BD2] P.M. Bleher and F.J. Dyson, Mean square value of exponential sums related to representation of integers as sum of two squares, Acta Arithm. (to appear). | MR 1302508 | Zbl 0915.11048

[BDL] P.M. Bleher, F.J. Dyson and J.L. Lebowitz, Non-gaussian energy level statistics for some integrable systems, Phys. Rev. Lett., Vol. 71, 1993, pp. 3047-3050. | MR 1246066 | Zbl 1050.81530

[BL] P.M. Bleher and J.L. Lebowitz, Energy-level statistics of model quantum systems: universality and scaling in a lattice-point problem, J. Stat. Phys., Vol. 74, 1994, pp. 167-217. | MR 1257819 | Zbl 0946.11520

[Ble] P.M. Bleher, On the distribution of the number of lattice points inside a family of convex ovals, Duke Math. Journ., Vol. 67, 1992, pp. 461-481. | MR 1181309 | Zbl 0762.11031

[CCG] G. Casati, B.V. Chirikov and I. Guarneri, Energy-level statistics of integrable quantum systems, Phys. Rev. Lett., Vol. 54, 1985, pp. 1350-1353. | MR 780344

[CL] Zh. Cheng and J.L. Lebowitz, Statistics of energy levels in integrable quantum systems, Phys. Rev., Vol. A44, 1991, pp. R3399-3402.

[CLM] Zh. Cheng, L. Lebowitz and P. Major, On the number of lattice points between two enlarged and randomly shifted copies of an oval, Preprint, Rutgers University, 1993.

[Gu] M.C. Gutzwiller, Chaos in classical and quantum mechanics, Springer-Verlag, N. Y. e.a., 1990. | MR 1077246 | Zbl 0727.70029

[HL] G.H. Hardy and J.E. Littlewood, Tauberian theorems concerning power series and Dirichlet series whose coefficients are positive, Proc. London Math. Soc., Vol. 13, 1914, p. 174. | JFM 45.0389.02

[LS] W. Luo and P. Sarnak, Number variance for arithmetic hyperbolic surfaces, Preprint, Princeton University, 1993.

[Maj] P. Major, Poisson law for the number of lattice points in a random strip with finite area, Prob. Theory Related Fields, Vol. 92, 1992, pp. 423-464. | MR 1169014 | Zbl 0767.60010

[OdA] A.M. Ozorio De Almeida, Hamiltonian systems: chaos and quantization, Cambridge Univ. Press, New York, 1988. | MR 985103 | Zbl 0734.58002

[Sin] Ya.G. Sinai, Poisson distribution in a geometric problem, in Advances in Soviet Mathematics, Vol. 3, Ya G. SINAI Ed. (AMS, Providence, Rhode Island), 1991, pp. 199-214. | MR 1118163 | Zbl 0744.60016

[Tab] M. Tabor, Chaos and integrabhility in nonlinear dynamics. An introduction, Wiley, New York, 1989. | MR 1007309 | Zbl 0682.58003